WEBVTT 00:00:14.710 --> 00:00:16.460 PROFESSOR: So I'm going to begin by trying 00:00:16.460 --> 00:00:20.540 to build some intuition for how one might be able to do staging 00:00:20.540 --> 00:00:24.400 from cross-sectional data, and we'll 00:00:24.400 --> 00:00:27.910 return to this question of combined staging subtyping only 00:00:27.910 --> 00:00:28.480 much later. 00:00:31.880 --> 00:00:34.760 So imagine that we had data that lived in one dimension. 00:00:34.760 --> 00:00:37.840 Here, each data point is an individual, 00:00:37.840 --> 00:00:40.610 we observe their data at just one point in time, 00:00:40.610 --> 00:00:44.250 and suppose we knew exactly which biomarker to look at. 00:00:44.250 --> 00:00:44.750 Right? 00:00:44.750 --> 00:00:46.333 So I gave you an example of that here, 00:00:46.333 --> 00:00:50.280 when you might look at some antibody expression level, 00:00:50.280 --> 00:00:54.878 and that might be what I call biomarker A, 00:00:54.878 --> 00:00:56.920 is if you knew exactly what biomarker to look at, 00:00:56.920 --> 00:01:00.730 you might just put each person along this line 00:01:00.730 --> 00:01:04.629 and you might conjecture that maybe on one side of the line, 00:01:04.629 --> 00:01:07.120 this is the early disease, and that the other sort of line, 00:01:07.120 --> 00:01:09.480 maybe that's the late disease. 00:01:09.480 --> 00:01:11.371 Why might that be a reasonable conjecture? 00:01:14.020 --> 00:01:16.088 What would be an alternative conjecture? 00:01:24.500 --> 00:01:26.000 Why don't you talk to your neighbors 00:01:26.000 --> 00:01:27.500 and see if you guys can come up with 00:01:27.500 --> 00:01:29.840 some alternative conjectures. 00:01:29.840 --> 00:01:30.800 Let's go. 00:01:37.050 --> 00:01:38.150 All right, that's enough. 00:01:38.150 --> 00:01:39.830 So hopefully simple questions, so I 00:01:39.830 --> 00:01:41.250 won't give you too much time. 00:01:41.250 --> 00:01:43.280 All right, so what would be another conjecture? 00:01:43.280 --> 00:01:47.197 So again, our goal is we have one observation per individual, 00:01:47.197 --> 00:01:49.530 each individual is in some unknown stage of the disease, 00:01:49.530 --> 00:01:51.863 we would like to be able to sort individuals and turn it 00:01:51.863 --> 00:01:55.872 into early and late stages of the disease. 00:01:55.872 --> 00:01:58.080 I give you one conjecture of how to do that, sorting, 00:01:58.080 --> 00:02:00.203 what would be another reasonable conjecture? 00:02:00.203 --> 00:02:00.870 Raise your hand. 00:02:03.420 --> 00:02:04.030 Yep? 00:02:04.030 --> 00:02:04.790 AUDIENCE: That there's the different-- 00:02:04.790 --> 00:02:07.270 that they have different types of the same diseases. 00:02:07.270 --> 00:02:09.919 They all have the same disease and it could-- 00:02:09.919 --> 00:02:13.103 just one of the subtypes might be sort of the-- 00:02:13.103 --> 00:02:13.770 PROFESSOR: Yeah. 00:02:13.770 --> 00:02:18.230 So you're going back to the example I gave here where you 00:02:18.230 --> 00:02:20.300 could conflate these things. 00:02:20.300 --> 00:02:22.240 I want to stick with a simpler story, 00:02:22.240 --> 00:02:25.903 let's suppose there's only one subtype of the disease. 00:02:25.903 --> 00:02:28.070 What would be another way to sort the patients given 00:02:28.070 --> 00:02:32.740 this data where the data is these points that you see here? 00:02:32.740 --> 00:02:34.261 Yeah? 00:02:34.261 --> 00:02:37.140 AUDIENCE: For any disease in the middle range, and then 00:02:37.140 --> 00:02:40.418 as you [INAUDIBLE] 00:02:40.418 --> 00:02:41.960 PROFESSOR: OK, so maybe early disease 00:02:41.960 --> 00:02:49.010 is right over here, and when things get bad, the patient-- 00:02:49.010 --> 00:02:51.170 this biomarker starts to become abnormal, 00:02:51.170 --> 00:02:53.060 and abnormality, for whatever reason, 00:02:53.060 --> 00:02:55.100 might be sort of to the right or to the left. 00:02:59.190 --> 00:03:02.990 Now I think that is a conjecture one could have. 00:03:02.990 --> 00:03:04.730 I would argue that that's perhaps 00:03:04.730 --> 00:03:06.230 not a very natural conjecture given 00:03:06.230 --> 00:03:08.960 what we know about common biomarkers that are measured 00:03:08.960 --> 00:03:11.270 from the human body and the way that they 00:03:11.270 --> 00:03:13.965 respond to disease progression. 00:03:13.965 --> 00:03:16.340 Unless you're in the situation of having multiple disease 00:03:16.340 --> 00:03:18.675 subtypes where, for example, going to the right marker 00:03:18.675 --> 00:03:20.300 might correspond to one disease subtype 00:03:20.300 --> 00:03:22.175 and going to the left marker might correspond 00:03:22.175 --> 00:03:24.640 to another disease subtype. 00:03:24.640 --> 00:03:26.273 What would be another conjecture? 00:03:31.696 --> 00:03:33.390 You guys are missing the easy one. 00:03:33.390 --> 00:03:34.140 Yeah, in the back. 00:03:34.140 --> 00:03:35.420 AUDIENCE: Well, it might just be one 00:03:35.420 --> 00:03:37.350 where the high values are [INAUDIBLE] stage 00:03:37.350 --> 00:03:38.780 and low values are later ones? 00:03:38.780 --> 00:03:39.710 PROFESSOR: Exactly. 00:03:39.710 --> 00:03:44.350 So this might be early disease and that might be late disease. 00:03:44.350 --> 00:03:46.580 AUDIENCE: It says vice versa on this slide. 00:03:46.580 --> 00:03:48.030 PROFESSOR: Oh, does it really? 00:03:48.030 --> 00:03:49.044 Oh shoot. 00:03:49.044 --> 00:03:50.200 [LAUGHTER] 00:03:50.200 --> 00:03:54.840 AUDIENCE: [INAUDIBLE] 00:03:54.840 --> 00:03:56.300 PROFESSOR: Right, right, OK, OK. 00:03:56.300 --> 00:03:56.630 Thank you. 00:03:56.630 --> 00:03:58.547 Next time I'll take out that lower vice versa. 00:03:58.547 --> 00:03:59.085 [LAUGHTER] 00:03:59.085 --> 00:04:00.710 That's why you guys aren't saying that. 00:04:00.710 --> 00:04:02.420 OK. 00:04:02.420 --> 00:04:04.670 OK, so this is good. 00:04:04.670 --> 00:04:06.590 Now I think we're all on the same page, 00:04:06.590 --> 00:04:09.170 and we had some idea of what are some of the assumptions 00:04:09.170 --> 00:04:10.670 that one might need to make in order 00:04:10.670 --> 00:04:11.880 to actually do anything here. 00:04:11.880 --> 00:04:16.019 Like for example, we are making some-- 00:04:16.019 --> 00:04:20.089 we'll probably have to make some assumption about continuity, 00:04:20.089 --> 00:04:23.300 that there might be some gradual progression of the biomarker 00:04:23.300 --> 00:04:26.660 relevance from early to late, and it might be getting larger, 00:04:26.660 --> 00:04:29.000 it might be getting smaller. 00:04:29.000 --> 00:04:32.390 If it's indeed the scenario that we talked about earlier where 00:04:32.390 --> 00:04:34.460 we said like early disease might be here 00:04:34.460 --> 00:04:38.450 and late disease might be going to either side, in that case, 00:04:38.450 --> 00:04:41.840 I think one could easily argue that with just information 00:04:41.840 --> 00:04:45.110 we have here, disease progression-- disease stage 00:04:45.110 --> 00:04:46.770 is unidentifiable, right? 00:04:46.770 --> 00:04:50.240 Because you wouldn't know where would it-- 00:04:50.240 --> 00:04:53.840 where should you-- where should that transition point be? 00:04:53.840 --> 00:04:55.787 So here, here, here, here, here, here. 00:04:55.787 --> 00:04:57.370 In fact, the same problem arises here. 00:04:57.370 --> 00:04:58.960 Like you don't know, is it early disease-- 00:04:58.960 --> 00:05:00.620 is it going this way or is it going that way? 00:05:00.620 --> 00:05:02.990 What would be one way to try to disentangle this just 00:05:02.990 --> 00:05:05.540 to try to get us all on the same page, right? 00:05:05.540 --> 00:05:08.828 So suppose it was only going this direction 00:05:08.828 --> 00:05:10.370 or going that direction, how could we 00:05:10.370 --> 00:05:11.453 figure out which is which? 00:05:20.470 --> 00:05:21.370 Yeah? 00:05:21.370 --> 00:05:28.260 AUDIENCE: Maybe we had data on low key and other data 00:05:28.260 --> 00:05:32.023 about how much time we had taken 00:05:32.023 --> 00:05:32.690 PROFESSOR: Yeah. 00:05:32.690 --> 00:05:33.398 No, that's great. 00:05:33.398 --> 00:05:38.230 So maybe we have data on let's say death information, 00:05:38.230 --> 00:05:40.480 or even just age. 00:05:40.480 --> 00:05:44.140 And if we started from a very, very rough assumption 00:05:44.140 --> 00:05:53.050 that disease stage let's say grows monotonically with age, 00:05:53.050 --> 00:05:55.060 then-- 00:05:55.060 --> 00:05:57.310 and if you had made an additional assumption 00:05:57.310 --> 00:06:01.215 that the disease stages are-- 00:06:01.215 --> 00:06:02.590 that the people who are coming in 00:06:02.590 --> 00:06:07.120 are uniformly drawn from across disease stages, with those two 00:06:07.120 --> 00:06:09.460 assumptions alone, then you could, for example, look 00:06:09.460 --> 00:06:12.970 at the average age of individuals over here 00:06:12.970 --> 00:06:15.580 and the average age of individuals over here, 00:06:15.580 --> 00:06:17.980 and you'd say, the one with the larger average age 00:06:17.980 --> 00:06:21.810 is the late disease one. 00:06:21.810 --> 00:06:26.090 Or you could look at time to death if you had for each-- 00:06:26.090 --> 00:06:27.900 for each data point you also knew 00:06:27.900 --> 00:06:30.210 how long until that individual died, 00:06:30.210 --> 00:06:31.920 you could look at average time to death 00:06:31.920 --> 00:06:33.330 for these individuals versus those individuals 00:06:33.330 --> 00:06:34.913 and try to tease it apart in that way. 00:06:37.270 --> 00:06:39.436 That's what you meant. 00:06:39.436 --> 00:06:41.930 OK, so I'm just trying to give you some intuition for how 00:06:41.930 --> 00:06:43.550 this might be possible. 00:06:43.550 --> 00:06:45.635 What about if your data looked like this? 00:06:51.700 --> 00:06:53.530 So now you have two biomarkers. 00:06:53.530 --> 00:06:58.040 So we've only gone up by one dimension only, 00:06:58.040 --> 00:07:00.500 and we want to figure out where's early, where's late? 00:07:03.118 --> 00:07:05.160 Already starts to become more challenging, right? 00:07:07.780 --> 00:07:10.702 So the intuition that I want you to have 00:07:10.702 --> 00:07:12.160 is that we're going to have to make 00:07:12.160 --> 00:07:16.090 some assumptions about disease progression, 00:07:16.090 --> 00:07:17.970 such as the ones we were just discussing, 00:07:17.970 --> 00:07:19.750 and we also have to get lucky in some way. 00:07:19.750 --> 00:07:22.750 So for example, one way of getting lucky 00:07:22.750 --> 00:07:25.490 would be to have a real lot of data. 00:07:25.490 --> 00:07:27.310 So if you had a ton, ton of data, 00:07:27.310 --> 00:07:30.820 and you made an additional assumption that your data lives 00:07:30.820 --> 00:07:33.832 in some low dimensional manifold where on one side of manifold 00:07:33.832 --> 00:07:35.790 is early disease and the other side of manifold 00:07:35.790 --> 00:07:38.170 is late disease, then you might be 00:07:38.170 --> 00:07:43.858 able to discover that manifold from this data, 00:07:43.858 --> 00:07:46.150 and you might conjecture that the manifold is something 00:07:46.150 --> 00:07:49.120 like that, that trajectory that I'm outlining there 00:07:49.120 --> 00:07:50.800 with my hand. 00:07:50.800 --> 00:07:52.870 But for you to be able to do that, of course, 00:07:52.870 --> 00:07:55.240 you need to have enough data, all right? 00:07:55.240 --> 00:07:58.570 So it's going to be now a trade-off between just 00:07:58.570 --> 00:08:01.360 having cross-sectional data, it might be OK so long 00:08:01.360 --> 00:08:03.010 as you have a real lot of that data 00:08:03.010 --> 00:08:04.630 so you can sort of fill in the spaces 00:08:04.630 --> 00:08:08.300 and really identify that manifold. 00:08:08.300 --> 00:08:11.210 A different approach might be, well maybe you 00:08:11.210 --> 00:08:13.460 don't have just pure cross-sectional data, 00:08:13.460 --> 00:08:17.000 maybe you have two or maybe three samples 00:08:17.000 --> 00:08:18.290 from each patient. 00:08:18.290 --> 00:08:19.915 And then you can color code this. 00:08:19.915 --> 00:08:24.875 So you might say, OK, green is patient 1-- 00:08:24.875 --> 00:08:28.220 or patient A, we'll call it, and this is the first time 00:08:28.220 --> 00:08:31.040 point from patient A, second time point from patient A, 00:08:31.040 --> 00:08:34.009 third and fourth time points from patient A. Red 00:08:34.009 --> 00:08:37.039 is patient B, and you have two time points for patient B, 00:08:37.039 --> 00:08:38.539 and blue here is patient C, and you 00:08:38.539 --> 00:08:44.120 have 1, 2, 3 time points from patient C. OK? 00:08:44.120 --> 00:08:46.580 Now again, it's not very dense data, 00:08:46.580 --> 00:08:49.337 we can't really draw curves out, But now 00:08:49.337 --> 00:08:51.170 we can start to get a sense of the ordering. 00:08:51.170 --> 00:08:55.280 And again, now we can-- even though we don't-- we're not 00:08:55.280 --> 00:08:57.240 in a dense setting like we were here, here, 00:08:57.240 --> 00:08:59.657 we'd still nonetheless be able to figure out that probably 00:08:59.657 --> 00:09:02.558 the manifold looks a little bit like this, right? 00:09:05.480 --> 00:09:09.250 And so again, I'm just trying to build intuition 00:09:09.250 --> 00:09:11.260 around when disease progression modeling 00:09:11.260 --> 00:09:14.030 for cross-sectional data might be possible, 00:09:14.030 --> 00:09:17.080 but this is a wide open field. 00:09:17.080 --> 00:09:18.880 And so today, I'll be telling you 00:09:18.880 --> 00:09:20.585 about a few algorithms that try to build 00:09:20.585 --> 00:09:22.960 on some of these intuitions for doing disease progression 00:09:22.960 --> 00:09:27.090 modeling, but they will break down very, very easily. 00:09:27.090 --> 00:09:29.730 They'll break down when these assumptions 00:09:29.730 --> 00:09:32.550 I gave you don't hold, they'll break down 00:09:32.550 --> 00:09:34.530 when your data is high dimensional, 00:09:34.530 --> 00:09:37.073 they'll break down when your data looks 00:09:37.073 --> 00:09:39.240 like this where you don't just have a single subtype 00:09:39.240 --> 00:09:41.440 of perhaps a multiple subtypes. 00:09:41.440 --> 00:09:44.530 and so this is a really very active area of research, 00:09:44.530 --> 00:09:47.430 and it's an area that I think we can make a lot of progress 00:09:47.430 --> 00:09:51.530 on in the field in the next several years. 00:09:51.530 --> 00:09:57.060 So I'll begin with one case study coming from my own work 00:09:57.060 --> 00:09:59.100 where we developed an algorithm for learning 00:09:59.100 --> 00:10:03.590 from cross-sectional data, and we valued it 00:10:03.590 --> 00:10:08.460 in the context of chronic obstructive pulmonary disorder 00:10:08.460 --> 00:10:10.030 or COPD. 00:10:10.030 --> 00:10:13.890 COPD is a condition of the lungs typically caused 00:10:13.890 --> 00:10:20.260 by air pollution or smoking, and it has a reasonably good 00:10:20.260 --> 00:10:22.210 staging mechanism. 00:10:22.210 --> 00:10:25.960 One uses what's called a spirometry device in order 00:10:25.960 --> 00:10:29.080 to measure the lung function of individual at any one point 00:10:29.080 --> 00:10:29.780 in time. 00:10:29.780 --> 00:10:32.230 So for example, you take this spirometry device, 00:10:32.230 --> 00:10:38.230 you stick it in your mouth, and you breathe in, 00:10:38.230 --> 00:10:43.550 and then you exhale, and one measure 00:10:43.550 --> 00:10:47.770 is how long it takes in order to exhale all your air, 00:10:47.770 --> 00:10:49.970 and that is going to be a measure of how 00:10:49.970 --> 00:10:52.310 good your lungs are. 00:10:52.310 --> 00:10:57.500 And so then one can take that measure of your function 00:10:57.500 --> 00:11:02.990 and one can stage how severe the person's COPD is, 00:11:02.990 --> 00:11:06.020 and that goes by what's called the gold criteria. 00:11:06.020 --> 00:11:09.140 So for example, in stage 1 of the COPD, 00:11:09.140 --> 00:11:13.460 common treatments involve just vaccinations 00:11:13.460 --> 00:11:16.250 using a short-acting bronchodilator only when 00:11:16.250 --> 00:11:18.140 needed. 00:11:18.140 --> 00:11:22.360 When the disease stage gets much more severe, like stage 4, 00:11:22.360 --> 00:11:25.460 than often treatment is recommended 00:11:25.460 --> 00:11:28.040 to be inhaling glucocorticosteroids 00:11:28.040 --> 00:11:31.260 if there are repeated aspirations of the disease, 00:11:31.260 --> 00:11:36.920 long-term oxygen If respiratory failure occurs, and so on. 00:11:36.920 --> 00:11:39.710 And so this is a disease that's reasonably well-understood 00:11:39.710 --> 00:11:44.160 because there exists a good staging mechanism. 00:11:44.160 --> 00:11:46.180 And I would argue that when we want 00:11:46.180 --> 00:11:48.340 to understand how to do disease staging 00:11:48.340 --> 00:11:51.550 in a data-driven fashion, we should first 00:11:51.550 --> 00:11:54.933 start by working with either synthetic data, 00:11:54.933 --> 00:11:57.100 or we should start with working with a disease where 00:11:57.100 --> 00:12:01.103 we have some idea of what the actual true disease staging is. 00:12:01.103 --> 00:12:03.520 And that way, we can look to see what our algorithms would 00:12:03.520 --> 00:12:05.110 recover in those scenarios, and does 00:12:05.110 --> 00:12:08.110 it align with what we would expect 00:12:08.110 --> 00:12:10.030 either from the way the data was generated 00:12:10.030 --> 00:12:12.810 or from the existing medical literature. 00:12:12.810 --> 00:12:14.170 And that's why we chose COPD. 00:12:14.170 --> 00:12:17.200 Because it is well-understood, and there's 00:12:17.200 --> 00:12:20.440 a wealth of literature on it, and because we 00:12:20.440 --> 00:12:23.650 have data on it which is much messier than the type of data 00:12:23.650 --> 00:12:26.038 that went into the original studies, and we could ask, 00:12:26.038 --> 00:12:27.580 could we come to the same conclusions 00:12:27.580 --> 00:12:28.720 as those original studies? 00:12:35.180 --> 00:12:38.540 So in this work, we're going to use data 00:12:38.540 --> 00:12:41.860 from the electronic medical record. 00:12:41.860 --> 00:12:44.500 We're only going to look at a subset of the EMR, 00:12:44.500 --> 00:12:47.200 in particular, diagnosis codes that 00:12:47.200 --> 00:12:50.223 are recorded for a patient at any point in time, 00:12:50.223 --> 00:12:52.390 and we're going to assume that we do not have access 00:12:52.390 --> 00:12:53.800 to spirometry data at all. 00:12:53.800 --> 00:12:57.470 So we don't have any obvious way of staging the patient's 00:12:57.470 --> 00:12:57.970 disease. 00:13:00.412 --> 00:13:01.870 The general approach is going to be 00:13:01.870 --> 00:13:09.840 to build a generative model for disease progression. 00:13:09.840 --> 00:13:13.290 At a very high level, this is a Markov model. 00:13:13.290 --> 00:13:19.380 It's a model that specifies the distribution of the patient's 00:13:19.380 --> 00:13:21.600 data, which is shown here in the bottom, 00:13:21.600 --> 00:13:25.750 as it evolves over time. 00:13:25.750 --> 00:13:28.180 According to a number of hidden variables 00:13:28.180 --> 00:13:30.550 that are shown in the top, these S variables that 00:13:30.550 --> 00:13:33.640 denote disease stages, and these X variables that denote 00:13:33.640 --> 00:13:35.140 comorbidities that the patient might 00:13:35.140 --> 00:13:39.780 have at that point in time, these X and S variables 00:13:39.780 --> 00:13:41.550 are always assumed to the unobserved. 00:13:41.550 --> 00:13:44.040 So if you were to clump them together into one variable, 00:13:44.040 --> 00:13:47.860 this would look exactly like a hidden Markov model. 00:13:47.860 --> 00:13:50.800 And moreover, we're not going to assume 00:13:50.800 --> 00:13:53.320 that we have a lot of longitudinal data 00:13:53.320 --> 00:13:54.520 for a patient. 00:13:54.520 --> 00:13:58.720 In particular, COPD evolves over a 10 to 20 years, and the data 00:13:58.720 --> 00:14:02.140 that we'll be learning from here has data only over one 00:14:02.140 --> 00:14:04.360 to three-year time range. 00:14:04.360 --> 00:14:08.200 The challenge will be, can we take data in this one 00:14:08.200 --> 00:14:11.975 to three-year time range and somehow stitch it together 00:14:11.975 --> 00:14:13.600 across large numbers of patients to get 00:14:13.600 --> 00:14:15.933 a picture of what the 20-year progression of the disease 00:14:15.933 --> 00:14:17.332 might look like? 00:14:17.332 --> 00:14:18.790 The way that we're going to do that 00:14:18.790 --> 00:14:22.372 is by learning the parameters of this probabilistic model. 00:14:22.372 --> 00:14:23.830 And then from the parameters, we're 00:14:23.830 --> 00:14:27.310 going to either infer the patient's actual disease stage 00:14:27.310 --> 00:14:30.760 and thus sort them, or actually simulate data 00:14:30.760 --> 00:14:33.370 from this model to see what a 20-year trajectory might 00:14:33.370 --> 00:14:35.180 look like. 00:14:35.180 --> 00:14:35.930 Is the goal clear? 00:14:38.660 --> 00:14:39.160 All right. 00:14:39.160 --> 00:14:40.577 So now what I'm going to do is I'm 00:14:40.577 --> 00:14:43.090 going to step into this model piece 00:14:43.090 --> 00:14:45.460 by piece to tell you what each of these components 00:14:45.460 --> 00:14:49.390 are, and I'll start out with a very topmost piece shown here 00:14:49.390 --> 00:14:51.790 by the red box. 00:14:51.790 --> 00:14:56.280 So this is the model of the patient's disease progression 00:14:56.280 --> 00:14:58.060 at any one point in time. 00:14:58.060 --> 00:15:00.180 So this variable, S1, for example, 00:15:00.180 --> 00:15:06.810 might denote the patient's disease stage on March 2011; 00:15:06.810 --> 00:15:10.470 S2 might denote the patient's disease stage April 2011; 00:15:10.470 --> 00:15:12.870 S capital T might denote the patient's disease stage 00:15:12.870 --> 00:15:14.850 June 2012. 00:15:14.850 --> 00:15:18.420 So we're going to have one random variable 00:15:18.420 --> 00:15:22.620 for each observation of the patient's data that we have. 00:15:22.620 --> 00:15:24.870 And notice that the observations of the patient's data 00:15:24.870 --> 00:15:27.610 might be at very irregular time intervals, 00:15:27.610 --> 00:15:30.240 and that's going to be OK with this approach, OK? 00:15:30.240 --> 00:15:33.750 So notice that there is a one-month gap between S1 00:15:33.750 --> 00:15:42.220 and S2, but a four-month gap between St minus 1 and St, OK? 00:15:42.220 --> 00:15:44.350 So we're going to model the patient's disease stage 00:15:44.350 --> 00:15:45.725 at the point in time when we have 00:15:45.725 --> 00:15:49.510 an observation for the patient. 00:15:49.510 --> 00:15:54.820 S denotes a discrete disease stage in this model. 00:15:54.820 --> 00:16:00.910 So S might be a value from 1 up to 4, maybe 1 up to 10 00:16:00.910 --> 00:16:07.150 where 1 is denoting a early disease stage and 4 or 10 00:16:07.150 --> 00:16:08.920 might denote a much later disease stage. 00:16:11.430 --> 00:16:13.980 If we have a sequence of observations per patient-- 00:16:13.980 --> 00:16:16.650 for example, we might have an observation on March 00:16:16.650 --> 00:16:19.620 and then in April, we're going to denote the disease 00:16:19.620 --> 00:16:23.010 stage by S1 and S2, what this model is going to talk about 00:16:23.010 --> 00:16:26.340 is the probability distribution of transitioning from whatever 00:16:26.340 --> 00:16:28.560 the disease stage at S1 is to whatever 00:16:28.560 --> 00:16:31.200 the disease stage at S2. 00:16:31.200 --> 00:16:34.140 Now because the time interval is between stages 00:16:34.140 --> 00:16:39.260 are not homogeneous, we have to have a transition distribution 00:16:39.260 --> 00:16:43.290 that takes into consideration that time gap. 00:16:43.290 --> 00:16:46.180 And to do that, we use what's known as a continuous time 00:16:46.180 --> 00:16:48.700 Markov process. 00:16:48.700 --> 00:16:54.790 Formally, we say that the transition distribution-- 00:16:54.790 --> 00:16:59.560 so the probability of transitioning from stage I 00:16:59.560 --> 00:17:06.710 at time t minus 1 to state j at time t, 00:17:06.710 --> 00:17:11.960 given as input the difference in time intervals-- 00:17:11.960 --> 00:17:14.060 the difference in time points delta-- 00:17:14.060 --> 00:17:18.390 so delta is the number of months between the two observations. 00:17:18.390 --> 00:17:20.569 So this conditional distribution is 00:17:20.569 --> 00:17:26.599 going to be given by the matrix exponential of this time 00:17:26.599 --> 00:17:35.560 interval times a matrix Q. 00:17:35.560 --> 00:17:38.890 And then here, the matrix Q gives us the parameters 00:17:38.890 --> 00:17:40.360 that we want to learn. 00:17:40.360 --> 00:17:42.070 So let me contrast this to things 00:17:42.070 --> 00:17:45.700 that you might already be used to. 00:17:45.700 --> 00:17:51.720 In a typical hidden Markov model, 00:17:51.720 --> 00:17:57.125 you might have asked t goes to St-- 00:17:57.125 --> 00:18:00.730 or St minus 1 goes to St, and you might imagine just 00:18:00.730 --> 00:18:07.570 parametrizing St given St minus 1 just by a lookup table. 00:18:07.570 --> 00:18:11.600 So for example, if the number of states-- 00:18:11.600 --> 00:18:17.130 for each running variable is 3, then you would have a 3 00:18:17.130 --> 00:18:23.076 by 3 table where for each state St minus 1, 00:18:23.076 --> 00:18:25.110 you have some probability of transition 00:18:25.110 --> 00:18:30.730 to the corresponding state St, so this might be something 00:18:30.730 --> 00:18:40.510 like 0.9, 0.9, 0.9, where notice, 00:18:40.510 --> 00:18:43.060 I'm having a very large value along the diagonal, 00:18:43.060 --> 00:18:46.260 because if, let's say, a very small period-- 00:18:46.260 --> 00:18:48.940 so a priori, we might believe that patients 00:18:48.940 --> 00:18:51.070 stay in the same disease, and then 00:18:51.070 --> 00:18:53.230 one might imagine that the probably 00:18:53.230 --> 00:18:59.080 transitioning from state 1 at time t minus 1 00:18:59.080 --> 00:19:08.075 to state 2 at time t might be something like 0.09, 00:19:08.075 --> 00:19:10.900 and the probability of skipping state 2, going directly 00:19:10.900 --> 00:19:12.880 to state 3 from state 1 might be something 00:19:12.880 --> 00:19:15.880 much smaller like 0.01, OK? 00:19:15.880 --> 00:19:18.130 And we might say something like that the probability-- 00:19:18.130 --> 00:19:19.672 we might imagine that the probability 00:19:19.672 --> 00:19:21.310 of going in a backwards direction, 00:19:21.310 --> 00:19:25.660 going from stage 2 at time t minus 1 00:19:25.660 --> 00:19:30.160 to let's say stage 1 at time t, that might be 0 all right? 00:19:30.160 --> 00:19:35.810 So you might imagine that actually this is the model, 00:19:35.810 --> 00:19:39.260 and what that's saying is something like you never 00:19:39.260 --> 00:19:42.560 go in the backwards direction, and you're 00:19:42.560 --> 00:19:44.690 more likely to transition to the state 00:19:44.690 --> 00:19:47.270 immediately adjacent to the current stage 00:19:47.270 --> 00:19:49.783 and very unlikely to skip a stage. 00:19:49.783 --> 00:19:51.200 So this would be an example of how 00:19:51.200 --> 00:19:54.230 you would parametrize the transition distribution 00:19:54.230 --> 00:19:58.450 in a typical discrete time Markov model, 00:19:58.450 --> 00:20:00.970 and the story here is going to be different specifically 00:20:00.970 --> 00:20:03.530 because we don't know the time intervals. 00:20:03.530 --> 00:20:07.840 So intuitively, if a lot of time has passed between the two 00:20:07.840 --> 00:20:11.120 observations, then we want to allow for an accelerated 00:20:11.120 --> 00:20:11.620 process. 00:20:11.620 --> 00:20:13.203 We want to allow for the fact that you 00:20:13.203 --> 00:20:15.280 might want to skip many different stages to go 00:20:15.280 --> 00:20:17.738 to your next time step, to go to the stage of the next time 00:20:17.738 --> 00:20:19.800 step, because so much time has passed. 00:20:19.800 --> 00:20:22.960 And that intuitively is what this scaling of this matrix 00:20:22.960 --> 00:20:25.270 Q by delta corresponds to. 00:20:25.270 --> 00:20:27.730 So the number of parameters in this parameterization 00:20:27.730 --> 00:20:29.950 is actually identical to the number of parameters 00:20:29.950 --> 00:20:31.300 in this parametrization, right? 00:20:31.300 --> 00:20:36.790 So you have a matrix Q which is given to you in essence 00:20:36.790 --> 00:20:39.160 by the number of states squared-- 00:20:39.160 --> 00:20:40.667 really, the number of states-- 00:20:40.667 --> 00:20:42.250 there's an additional redundancy there 00:20:42.250 --> 00:20:45.780 because it has to sum up to 1, but that's irrelevant. 00:20:45.780 --> 00:20:47.980 And so the same story here, but we're 00:20:47.980 --> 00:20:50.500 going to now parametrize the process 00:20:50.500 --> 00:20:56.552 by in some sense the infinitesimally small time 00:20:56.552 --> 00:20:57.760 probability of transitioning. 00:20:57.760 --> 00:21:00.250 So if you were to take the derivative of this transition 00:21:00.250 --> 00:21:03.430 distribution as the time interval shrinks, 00:21:03.430 --> 00:21:10.153 and then you were to integrate over the time interval that 00:21:10.153 --> 00:21:12.820 was observed and the probability of transitioning from any state 00:21:12.820 --> 00:21:16.420 to any other state with that infinitesimally small 00:21:16.420 --> 00:21:18.340 probability transitioning, what you get out 00:21:18.340 --> 00:21:20.790 is exactly this form. 00:21:20.790 --> 00:21:25.240 And I'll leave-- this paper is in the optional readings 00:21:25.240 --> 00:21:27.240 for today's lecture, and you can read through it 00:21:27.240 --> 00:21:32.080 to get more intuition about the continuous time Markov process. 00:21:32.080 --> 00:21:33.804 Any questions so far? 00:21:33.804 --> 00:21:34.760 Yep? 00:21:34.760 --> 00:21:37.260 AUDIENCE: Those Q are the same for both versions or-- 00:21:37.260 --> 00:21:37.950 PROFESSOR: Yes. 00:21:37.950 --> 00:21:41.970 And this model Q is essentially the same for all patients. 00:21:41.970 --> 00:21:45.120 And you might imagine, if there were disease subtypes, which 00:21:45.120 --> 00:21:47.190 there aren't in this approach, that Q might 00:21:47.190 --> 00:21:48.450 be different for each subtype. 00:21:48.450 --> 00:21:51.000 For example, you might transition between stages 00:21:51.000 --> 00:21:54.820 much more quickly for some subtypes than for others. 00:21:54.820 --> 00:21:55.968 Other questions? 00:21:59.880 --> 00:22:00.730 Yep? 00:22:00.730 --> 00:22:03.625 AUDIENCE: So-- OK, so Q you said had like-- 00:22:03.625 --> 00:22:06.058 it's just like a screen number used beforehand you 00:22:06.058 --> 00:22:08.100 kind of like specified these stages that you pick 00:22:08.100 --> 00:22:08.600 [INAUDIBLE] 00:22:08.600 --> 00:22:09.755 PROFESSOR: Correct. 00:22:09.755 --> 00:22:10.255 Yes. 00:22:10.255 --> 00:22:15.220 So you pre-specify the number of stages that you want to model, 00:22:15.220 --> 00:22:19.925 and there are many ways to try to choose that parameter. 00:22:19.925 --> 00:22:22.050 For example, you could look at how about likelihood 00:22:22.050 --> 00:22:23.730 under this model, which is learned 00:22:23.730 --> 00:22:26.100 for the different of stages. 00:22:26.100 --> 00:22:28.830 You could use typical model selection techniques 00:22:28.830 --> 00:22:31.410 from machine learning as another approach 00:22:31.410 --> 00:22:35.010 where you try to penalize complexity in some way. 00:22:35.010 --> 00:22:37.582 Or, what we found here, because of some of the other things 00:22:37.582 --> 00:22:39.540 that I'm about to tell you, it doesn't actually 00:22:39.540 --> 00:22:41.220 matter that much. 00:22:41.220 --> 00:22:44.520 So similarly to when one does hard [INAUDIBLE] clustering 00:22:44.520 --> 00:22:47.580 or even K-means clustering or even learning 00:22:47.580 --> 00:22:49.680 a problematic topic model, if you 00:22:49.680 --> 00:22:52.590 use a very small number of topics or number of clusters, 00:22:52.590 --> 00:22:54.960 you tend to learn very coarse-grained topics 00:22:54.960 --> 00:22:55.920 or clusters. 00:22:55.920 --> 00:22:57.530 If you use very many more-- 00:22:57.530 --> 00:22:59.280 if you use a much larger number of topics, 00:22:59.280 --> 00:23:01.830 you tend to learn much more fine-grained topics. 00:23:01.830 --> 00:23:03.720 Same story is going to happen here. 00:23:03.720 --> 00:23:05.638 If you use a small number of disease stages, 00:23:05.638 --> 00:23:07.680 you're going to learn very coarse-grained notions 00:23:07.680 --> 00:23:09.990 of disease stages; if you use more disease stages, 00:23:09.990 --> 00:23:12.085 you're going to learn a fine-grained notion; 00:23:12.085 --> 00:23:13.710 but the overall sorting of the patients 00:23:13.710 --> 00:23:16.680 is going to end up being very similar. 00:23:16.680 --> 00:23:18.180 But to make that statement, we're 00:23:18.180 --> 00:23:20.513 going to need to make some additional assumptions, which 00:23:20.513 --> 00:23:23.297 I'm going to show you in a few minutes. 00:23:23.297 --> 00:23:24.130 Any other questions? 00:23:24.130 --> 00:23:27.200 These are great questions. 00:23:27.200 --> 00:23:28.100 Yep? 00:23:28.100 --> 00:23:30.800 AUDIENCE: So do we know the staging of the disease 00:23:30.800 --> 00:23:31.440 because I 00:23:31.440 --> 00:23:33.320 PROFESSOR: No, and that's critical here. 00:23:33.320 --> 00:23:37.060 So I'm assuming that these variables-- these S's are all 00:23:37.060 --> 00:23:39.290 hidden variables here. 00:23:39.290 --> 00:23:41.980 And the way that we're going to learn this model 00:23:41.980 --> 00:23:43.730 is by maximum likelihood estimation 00:23:43.730 --> 00:23:46.090 where we marginalize over the hidden variables, 00:23:46.090 --> 00:23:50.870 just like you would do in any EM type algorithm. 00:23:50.870 --> 00:23:52.064 Any other questions? 00:23:55.382 --> 00:23:56.810 All right, so what I've just shown 00:23:56.810 --> 00:24:00.180 you is the topmost part of the model, 00:24:00.180 --> 00:24:02.530 now I'm going to talk about a horizontal slice. 00:24:02.530 --> 00:24:06.110 So I'm going to talk about one of these time points. 00:24:09.988 --> 00:24:15.840 So if you were to look at the translation-- 00:24:15.840 --> 00:24:18.720 the rotation of one of those time points, what you 00:24:18.720 --> 00:24:20.730 would get out is this model. 00:24:20.730 --> 00:24:24.360 These X's are also hidden variables, 00:24:24.360 --> 00:24:28.133 and we have pre-specified them to characterize different axes 00:24:28.133 --> 00:24:30.300 by which we want to understand the patient's disease 00:24:30.300 --> 00:24:32.655 progression. 00:24:32.655 --> 00:24:34.405 So in Thursday's lecture, we characterized 00:24:34.405 --> 00:24:43.070 the patient's disease as subtype by just a single number, 00:24:43.070 --> 00:24:46.325 and similarly in this example is just by a single number, 00:24:46.325 --> 00:24:48.200 but we might want to understand what's really 00:24:48.200 --> 00:24:50.150 unique about each subtype. 00:24:50.150 --> 00:24:53.730 So for example-- sorry, what's really 00:24:53.730 --> 00:24:56.080 unique about each disease stage. 00:24:56.080 --> 00:24:59.760 So for example, how is the patient's endocrine function 00:24:59.760 --> 00:25:01.440 in that disease stage? 00:25:01.440 --> 00:25:09.645 How is the patient's psychiatric status in that disease stage? 00:25:12.350 --> 00:25:15.170 Has the patient developed lung cancer yet and that disease 00:25:15.170 --> 00:25:16.790 stage? 00:25:16.790 --> 00:25:18.270 And so on. 00:25:18.270 --> 00:25:20.720 And so we're going to ask that we 00:25:20.720 --> 00:25:22.890 want to be able to read out from this model 00:25:22.890 --> 00:25:24.940 according to these axes, and this 00:25:24.940 --> 00:25:27.470 will become very clear at the end of this section 00:25:27.470 --> 00:25:29.150 where I show you a simulation of what 00:25:29.150 --> 00:25:33.500 20 years looks like for COPD according to these quantities. 00:25:33.500 --> 00:25:35.930 When does the patient typically develop diabetes, 00:25:35.930 --> 00:25:38.450 when does the patient typically become depressed, 00:25:38.450 --> 00:25:41.730 when does the patient typically develop cancer, and so on. 00:25:41.730 --> 00:25:43.310 So these are the quantities in which 00:25:43.310 --> 00:25:45.290 we want to be able to really talk 00:25:45.290 --> 00:25:48.650 about what happens to a patient at any one disease stage, 00:25:48.650 --> 00:25:50.780 but the challenge is, we never actually observe 00:25:50.780 --> 00:25:53.150 these quantities in the data that we have. 00:25:53.150 --> 00:25:56.450 Rather, all we observe are things like laboratory test 00:25:56.450 --> 00:25:58.820 results or diagnosis codes or procedures 00:25:58.820 --> 00:26:00.770 that have been formed and so on, which 00:26:00.770 --> 00:26:03.190 I'm going to call the clinical findings in the bottom. 00:26:03.190 --> 00:26:05.930 And as we've been discussing throughout this course, 00:26:05.930 --> 00:26:08.300 one could think about things as diagnosis codes 00:26:08.300 --> 00:26:10.760 as giving you information about the disease 00:26:10.760 --> 00:26:12.560 status of the patient, but they're not 00:26:12.560 --> 00:26:14.660 one and the same as the diagnosis, 00:26:14.660 --> 00:26:16.760 because there's so much noise and bias that 00:26:16.760 --> 00:26:20.015 goes into the assigning of diagnosis codes for patients. 00:26:22.600 --> 00:26:26.820 And so the way that we're going to model the raw data 00:26:26.820 --> 00:26:28.470 as a function of these hidden variables 00:26:28.470 --> 00:26:31.050 that we want to characterize is using what's 00:26:31.050 --> 00:26:33.165 known as a noisy-OR network. 00:26:35.710 --> 00:26:37.620 So we're going to suppose that there 00:26:37.620 --> 00:26:41.880 is some generative distribution where the observations you 00:26:41.880 --> 00:26:44.730 see-- for example, diagnosis codes 00:26:44.730 --> 00:26:48.510 are likely to be observed as a function of whether the patient 00:26:48.510 --> 00:26:51.450 has these phenotypes or comorbidities 00:26:51.450 --> 00:26:53.970 with some probability, and that probability can be specified 00:26:53.970 --> 00:26:56.550 by these edge weights. 00:26:56.550 --> 00:27:00.930 So for example, a diagnosis code for diabetes 00:27:00.930 --> 00:27:03.540 is very likely to be observed in the patient data 00:27:03.540 --> 00:27:06.270 if the patient truly has diabetes, 00:27:06.270 --> 00:27:07.860 but of course, it may not be recorded 00:27:07.860 --> 00:27:10.170 in the data for every single visit the patient has 00:27:10.170 --> 00:27:12.060 to a clinician, there might be some visits 00:27:12.060 --> 00:27:14.790 to clinicians that have nothing to do with their patients 00:27:14.790 --> 00:27:17.730 endocrine function and diabetes-- the diagnosis 00:27:17.730 --> 00:27:19.920 code might not be recorded for that visit. 00:27:19.920 --> 00:27:21.600 So it's going to be a noisy process, 00:27:21.600 --> 00:27:24.240 and that noise rate is going to be captured by that edge. 00:27:31.100 --> 00:27:33.080 So part of the learning algorithm 00:27:33.080 --> 00:27:35.940 is going to be to learn that transition distributions-- 00:27:35.940 --> 00:27:38.570 for example, that Q matrix I showed you in the earlier 00:27:38.570 --> 00:27:41.570 slide, but the other role-- learning algorithm 00:27:41.570 --> 00:27:43.700 is to learn all of the parameters 00:27:43.700 --> 00:27:46.490 of this noisy-OR distribution, namely these edge weights. 00:27:46.490 --> 00:27:49.820 So that's going to be discovered as part of the learning 00:27:49.820 --> 00:27:51.800 algorithm. 00:27:51.800 --> 00:27:53.930 And a key question that you have to ask 00:27:53.930 --> 00:27:58.040 me is, if I know I want to read out from the model 00:27:58.040 --> 00:28:01.390 according to these axes, but these axes are never-- 00:28:01.390 --> 00:28:03.140 I'm never assuming that they're explicitly 00:28:03.140 --> 00:28:06.380 observed in the data, how do I ground the learning 00:28:06.380 --> 00:28:09.050 algorithm to give meaning to these hidden variables? 00:28:09.050 --> 00:28:12.350 Because otherwise if we left them otherwise unconstrained 00:28:12.350 --> 00:28:14.960 and you did maximum likelihood estimation just 00:28:14.960 --> 00:28:17.360 like in any factor analysis-type model, 00:28:17.360 --> 00:28:18.920 you might discover some factors here, 00:28:18.920 --> 00:28:21.040 but they might not be the factors you care about, 00:28:21.040 --> 00:28:24.260 and if the learning problem was not identifiable, 00:28:24.260 --> 00:28:26.300 as is often the case in unsupervised learning, 00:28:26.300 --> 00:28:29.660 then you might not discover what you're interested in. 00:28:29.660 --> 00:28:32.690 So to ground the hidden variables, 00:28:32.690 --> 00:28:36.740 we introduced-- we used a technique that you already 00:28:36.740 --> 00:28:41.480 saw in an earlier lecture from lecture 8 called anchors. 00:28:41.480 --> 00:28:43.700 So a domain expert is going to specify 00:28:43.700 --> 00:28:48.980 for each one of the comorbidites one or more anchors, which 00:28:48.980 --> 00:28:52.010 are observations, which we are going to conjecture 00:28:52.010 --> 00:28:57.590 could only have arisen from the corresponding hidden variable. 00:28:57.590 --> 00:28:59.180 So notice here that this diagnosis 00:28:59.180 --> 00:29:01.430 code, which is for type 2 diabetes, 00:29:01.430 --> 00:29:05.270 has only an edge from X1. 00:29:05.270 --> 00:29:07.097 That is an assumption that we're making 00:29:07.097 --> 00:29:08.180 in the learning algorithm. 00:29:08.180 --> 00:29:10.160 We are actually explicitly zeroing 00:29:10.160 --> 00:29:14.210 out all of the other edges from all of the other comorbidities 00:29:14.210 --> 00:29:16.230 to a 1. 00:29:16.230 --> 00:29:18.680 We're not going to pre-specify what this edge rate is, 00:29:18.680 --> 00:29:21.140 we're going to allow for the fact that this might be noisy, 00:29:21.140 --> 00:29:23.907 it's not always observed even if the patient has diabetes, 00:29:23.907 --> 00:29:25.490 but we're going to say, this could not 00:29:25.490 --> 00:29:29.830 be explained by any of the other comorbidities. 00:29:29.830 --> 00:29:32.830 And so for each one of the comorbidites or phenotypes 00:29:32.830 --> 00:29:34.510 that we want to model, we're going 00:29:34.510 --> 00:29:37.270 to specify some small number of anchors 00:29:37.270 --> 00:29:40.060 which correspond to a type of sparsity assumption 00:29:40.060 --> 00:29:42.470 on that graph. 00:29:42.470 --> 00:29:46.210 And these are the anchors that we chose for asthma, we 00:29:46.210 --> 00:29:48.670 chose a diagnosis code corresponding to asthma; 00:29:48.670 --> 00:29:50.890 for lung cancer, we chose several diagnosis 00:29:50.890 --> 00:29:53.320 codes correspond to lung cancer; for obesity, we 00:29:53.320 --> 00:29:54.820 chose a diagnosis code corresponding 00:29:54.820 --> 00:29:57.340 to morbid obesity; and so on. 00:29:57.340 --> 00:29:58.960 And so these are ways that we're going 00:29:58.960 --> 00:30:01.180 to give meaning to the hidden variables, 00:30:01.180 --> 00:30:03.580 but as you'll see in just a few minutes, 00:30:03.580 --> 00:30:05.980 it is not going to pre-specify too much of the model. 00:30:05.980 --> 00:30:07.897 The model's still going to learn a whole bunch 00:30:07.897 --> 00:30:10.510 of other interesting things. 00:30:10.510 --> 00:30:14.340 By the way, the way that we actually came up with this set 00:30:14.340 --> 00:30:17.290 was by an iterative process. 00:30:17.290 --> 00:30:21.840 We specified some of the hidden variables to have anchors, 00:30:21.840 --> 00:30:24.780 but we also left some of them to be unanchored, 00:30:24.780 --> 00:30:27.090 meaning free variables. 00:30:27.090 --> 00:30:29.282 We did our learning algorithm, and just 00:30:29.282 --> 00:30:30.740 like you would do in a topic model, 00:30:30.740 --> 00:30:33.257 we discovered that there were some phenotypes that 00:30:33.257 --> 00:30:35.340 really seemed to be characterized by the patient's 00:30:35.340 --> 00:30:36.060 disease-- 00:30:36.060 --> 00:30:38.940 that seemed to characterize a patient's disease progression. 00:30:38.940 --> 00:30:43.380 Then in order to really dig deeper, working collaboratively 00:30:43.380 --> 00:30:46.150 with a domain expert, we specified anchors for those 00:30:46.150 --> 00:30:47.580 and we iterated, and in this way, 00:30:47.580 --> 00:30:50.310 we discovered the full set of interesting variables 00:30:50.310 --> 00:30:51.380 that we wanted to model. 00:30:51.380 --> 00:30:51.880 Yep? 00:30:51.880 --> 00:30:55.240 AUDIENCE: Did you measure how good an anchor these were? 00:30:55.240 --> 00:30:58.042 Like are some comorbidities better anchors than others? 00:30:58.042 --> 00:30:58.750 PROFESSOR: Great. 00:30:58.750 --> 00:31:01.042 You'll see-- I think we'll answer that question in just 00:31:01.042 --> 00:31:03.360 a few minutes when I show you what the graph looks 00:31:03.360 --> 00:31:04.752 like that's learned. 00:31:04.752 --> 00:31:05.400 Yep? 00:31:05.400 --> 00:31:06.900 AUDIENCE: Were all the other weights 00:31:06.900 --> 00:31:09.490 in that X to O network 0? 00:31:09.490 --> 00:31:12.103 They weren't part of it here. 00:31:12.103 --> 00:31:13.520 So it looks like a pretty sparse-- 00:31:13.520 --> 00:31:14.590 PROFESSOR: They're explicitly nonzero, actually, 00:31:14.590 --> 00:31:15.920 it's opposite. 00:31:15.920 --> 00:31:22.350 So for an anchor, we say that it only has a single parent. 00:31:22.350 --> 00:31:24.180 Everything that's not an anchor can 00:31:24.180 --> 00:31:27.060 have arbitrarily many parents. 00:31:27.060 --> 00:31:28.166 Is that clear? 00:31:28.166 --> 00:31:29.080 OK. 00:31:29.080 --> 00:31:30.120 Yeah? 00:31:30.120 --> 00:31:32.900 AUDIENCE: Do the anchors that you have in that linear table, 00:31:32.900 --> 00:31:35.360 you itereated yourself on that or did the doctors say 00:31:35.360 --> 00:31:37.757 that these are the [INAUDIBLE]? 00:31:37.757 --> 00:31:39.590 PROFESSOR: We started out with just a subset 00:31:39.590 --> 00:31:40.676 of these conditions. 00:31:40.676 --> 00:31:43.232 As things that we wanted to model-- 00:31:43.232 --> 00:31:44.690 things that we wanted to understand 00:31:44.690 --> 00:31:46.327 what happens along disease progression 00:31:46.327 --> 00:31:48.410 according to these axes, but just a subset of them 00:31:48.410 --> 00:31:49.670 originally. 00:31:49.670 --> 00:31:53.240 And then we included a few additional hidden variables 00:31:53.240 --> 00:31:55.460 that didn't have any anchors associated to them, 00:31:55.460 --> 00:31:57.510 and after doing unsupervised learning and just 00:31:57.510 --> 00:32:01.197 a preliminary development stage, they discovered some topics 00:32:01.197 --> 00:32:03.530 and we realized, oh shoot, we should have included those 00:32:03.530 --> 00:32:06.920 in there, and then we added them in with corresponding anchors. 00:32:09.407 --> 00:32:11.740 And so you could think about this as an exploratory data 00:32:11.740 --> 00:32:14.200 analysis pipeline. 00:32:14.200 --> 00:32:14.700 Yep? 00:32:14.700 --> 00:32:18.120 AUDIENCE: Is there a chance that these aren't anchors? 00:32:18.120 --> 00:32:20.375 PROFESSOR: Yes. 00:32:20.375 --> 00:32:22.500 So there's definitely the chance that these may not 00:32:22.500 --> 00:32:23.917 be anchors related to the question 00:32:23.917 --> 00:32:25.150 was asked a second ago. 00:32:25.150 --> 00:32:31.040 So for example, there might be some chance 00:32:31.040 --> 00:32:33.800 that the morbid obesity diagnosis 00:32:33.800 --> 00:32:37.610 code might be coded for a patient 00:32:37.610 --> 00:32:45.258 more often for a patient who has, let's say, asthma-- 00:32:45.258 --> 00:32:46.175 this is a bad example. 00:32:50.070 --> 00:32:53.270 And in which case, that would correspond to there truly 00:32:53.270 --> 00:32:56.450 existing an edge from asthma to this anchor, which 00:32:56.450 --> 00:32:58.980 would be a violation of anchor assumption. 00:32:58.980 --> 00:33:03.680 All right, so we chose these to make that unlikely, 00:33:03.680 --> 00:33:05.880 but it could happen. 00:33:05.880 --> 00:33:07.890 And it's not easily testable. 00:33:07.890 --> 00:33:10.280 So this is another example of an untestable assumption, 00:33:10.280 --> 00:33:12.230 just like we saw lots of other examples 00:33:12.230 --> 00:33:15.735 already in today's lecture and the causal inference lectures. 00:33:15.735 --> 00:33:17.360 If we had some ground truth data-- like 00:33:17.360 --> 00:33:19.652 if we had done chart review for some number of patients 00:33:19.652 --> 00:33:21.540 and we actually label these conditions, 00:33:21.540 --> 00:33:23.570 then we could test that anchor assumption. 00:33:23.570 --> 00:33:26.195 But here, we're assuming that we don't actually know the ground 00:33:26.195 --> 00:33:27.686 truth of these conditions. 00:33:27.686 --> 00:33:29.680 AUDIENCE: Is there a reason why you choose 00:33:29.680 --> 00:33:31.390 such high-level comorbidites? 00:33:31.390 --> 00:33:33.528 Like I imagine you could go more specific. 00:33:33.528 --> 00:33:34.945 Even, say, the diabetes, you could 00:33:34.945 --> 00:33:38.000 try to subtype the diabetes based on this other model, sort 00:33:38.000 --> 00:33:41.442 of use that as a single layer, but it seems to-- 00:33:41.442 --> 00:33:43.900 at least this model seems to choose [INAUDIBLE] high level. 00:33:43.900 --> 00:33:45.000 I was just curious of the reason. 00:33:45.000 --> 00:33:45.625 PROFESSOR: Yes. 00:33:45.625 --> 00:33:47.890 So that was a design choice that we made. 00:33:47.890 --> 00:33:50.330 There are many, many directions for follow-up work, 00:33:50.330 --> 00:33:53.720 one of which would be to use a hierarchical model here. 00:33:53.720 --> 00:33:55.290 But we hadn't gone that direction. 00:33:55.290 --> 00:33:57.230 Another obvious direction for follow-up work 00:33:57.230 --> 00:34:01.070 would be to do something within the subtyping with this staging 00:34:01.070 --> 00:34:03.050 by introducing another random variable, which 00:34:03.050 --> 00:34:04.672 is, let's say, the disease subtype, 00:34:04.672 --> 00:34:06.380 and making everything a function of that. 00:34:11.080 --> 00:34:14.273 OK, so I've talked about the vertical slice 00:34:14.273 --> 00:34:15.940 and I've talked about the topmost slice, 00:34:15.940 --> 00:34:17.565 but what I still need to tell you about 00:34:17.565 --> 00:34:23.874 is how these phenotypes relate to the observed disease stage. 00:34:27.139 --> 00:34:28.325 So for this, we use-- 00:34:33.090 --> 00:34:36.150 I don't remember the exact technical terminology-- 00:34:36.150 --> 00:34:38.639 a factored Markov model? 00:34:38.639 --> 00:34:41.130 Is that right, Pete? 00:34:41.130 --> 00:34:42.540 Factorized Markov model-- I mean, 00:34:42.540 --> 00:34:44.707 this is a term that existed in the graphical model's 00:34:44.707 --> 00:34:47.290 literature, but I don't remember right now. 00:34:47.290 --> 00:34:51.580 So what we're saying is that each of these Markov chains-- 00:34:51.580 --> 00:35:06.880 so each of these X1 up to Xt-- 00:35:12.500 --> 00:35:16.770 so this, will say, is the first one I call diabetes. 00:35:19.730 --> 00:35:23.180 This is the second one which I'll say is depression. 00:35:30.573 --> 00:35:32.990 We're going to assume that each one of these Markov chains 00:35:32.990 --> 00:35:35.630 is conditionally independent of each other given the disease 00:35:35.630 --> 00:35:36.710 stage. 00:35:36.710 --> 00:35:45.590 So it's the disease stage which ties everything together. 00:35:45.590 --> 00:35:57.160 So the graphical model looks like this, and so on, OK? 00:35:57.160 --> 00:36:00.130 So in particular, there are no edges between, let's say, 00:36:00.130 --> 00:36:03.010 the diabetes variable and the depression variable. 00:36:03.010 --> 00:36:05.470 All correlations between these conditions 00:36:05.470 --> 00:36:09.760 is assumed to be mediated by the disease stage variable. 00:36:09.760 --> 00:36:12.970 And that's a critical assumption that we had to make. 00:36:12.970 --> 00:36:14.110 Does anyone know why? 00:36:17.540 --> 00:36:21.242 What would go wrong if we didn't make that assumption? 00:36:21.242 --> 00:36:22.700 So for example, what would go wrong 00:36:22.700 --> 00:36:28.560 if we had something look like this, x1-- 00:36:28.560 --> 00:36:31.580 what was my notation? 00:36:31.580 --> 00:36:41.060 X1,1, X1,2, X1,3, and suppose we had edges between them, 00:36:41.060 --> 00:36:42.740 a complete graph, and we had, let's say, 00:36:42.740 --> 00:36:49.130 also the S variable with edges to everything? 00:36:49.130 --> 00:36:51.808 What would happen in that case where 00:36:51.808 --> 00:36:54.350 we're not assuming that the X's are conditionally independent 00:36:54.350 --> 00:36:55.160 given S? 00:37:02.180 --> 00:37:07.200 So I want you to think about this in terms of distributions. 00:37:07.200 --> 00:37:11.520 So remember, we're going to learn 00:37:11.520 --> 00:37:13.500 how to do disease progression through learning 00:37:13.500 --> 00:37:14.760 the parameters of this model. 00:37:14.760 --> 00:37:17.093 And so if we set this up and-- if we set up the learning 00:37:17.093 --> 00:37:19.230 problem in a way which is unidentifiable, 00:37:19.230 --> 00:37:20.220 then we're going to be screwed, we're 00:37:20.220 --> 00:37:21.678 not going to able to learn anything 00:37:21.678 --> 00:37:23.420 about disease progression. 00:37:23.420 --> 00:37:25.570 So what would happen in this situation? 00:37:30.120 --> 00:37:31.830 Someone who hasn't spoken today ideally. 00:37:38.760 --> 00:37:40.872 So any of you remember from, let's say, 00:37:40.872 --> 00:37:42.330 perhaps an earlier machine learning 00:37:42.330 --> 00:37:45.420 class what types of distribution's 00:37:45.420 --> 00:37:47.160 a complete graph-- 00:37:47.160 --> 00:37:49.658 a complete Bayesian network could represent? 00:37:56.990 --> 00:38:01.140 So the answer is all distributions, 00:38:01.140 --> 00:38:03.300 because it corresponds to any factorization 00:38:03.300 --> 00:38:05.220 of the joint distribution. 00:38:05.220 --> 00:38:08.670 And so if you allowed these x variables 00:38:08.670 --> 00:38:11.020 to be fully connected to each other-- so for example, 00:38:11.020 --> 00:38:13.620 saying that depression depends on diabetes in addition 00:38:13.620 --> 00:38:16.050 to the stage, then in fact, you don't even 00:38:16.050 --> 00:38:17.580 need this stage variable in here. 00:38:17.580 --> 00:38:20.340 The marginal-- you can fit any distribution 00:38:20.340 --> 00:38:23.460 on these X variables even without the S variable at all. 00:38:23.460 --> 00:38:26.160 And so the model could learn to simply ignore 00:38:26.160 --> 00:38:30.240 the S variable, which would be exactly not our goal, 00:38:30.240 --> 00:38:34.050 because our goal is to learn something about the disease 00:38:34.050 --> 00:38:37.690 stage, and in fact, we're going to be wanting 00:38:37.690 --> 00:38:41.430 to make assumptions on the progression of disease 00:38:41.430 --> 00:38:44.990 stage, which is going to help us learn. 00:38:44.990 --> 00:38:48.650 So by assuming conditional independence between these X 00:38:48.650 --> 00:38:51.770 variables, it's going to force all of the correlations 00:38:51.770 --> 00:38:54.230 to have to be mediated by that S variable, 00:38:54.230 --> 00:38:56.805 and it's going to remove some of that unidentifiability that 00:38:56.805 --> 00:38:59.840 would otherwise exist. 00:38:59.840 --> 00:39:01.760 It's this subtle but very important point. 00:39:05.520 --> 00:39:07.580 So the way that we're going to parametrize 00:39:07.580 --> 00:39:09.570 the conditional distribution-- so first of all, 00:39:09.570 --> 00:39:12.470 I'm going to assume these X's are all binary. 00:39:12.470 --> 00:39:14.420 So either the patient has diabetes 00:39:14.420 --> 00:39:16.900 or they don't have diabetes. 00:39:16.900 --> 00:39:18.490 I'm going to suppose that-- 00:39:18.490 --> 00:39:21.520 and this is, again, another assumption we're making, 00:39:21.520 --> 00:39:27.430 I'm going to suppose that once you already have a comorbidity, 00:39:27.430 --> 00:39:28.430 then you always have it. 00:39:28.430 --> 00:39:31.632 So for example, once this is 1, then all subsequent ones 00:39:31.632 --> 00:39:32.590 are also going to be 1. 00:39:35.190 --> 00:39:37.710 Hold the questions for just a second. 00:39:37.710 --> 00:39:39.390 I'm also going to make an assumption 00:39:39.390 --> 00:39:42.420 that later stages of the disease are more likely to develop 00:39:42.420 --> 00:39:43.680 the comorbidity. 00:39:43.680 --> 00:39:46.860 So in particular, one can formalize that mathematically 00:39:46.860 --> 00:39:53.775 as probability of X-- 00:39:56.360 --> 00:40:04.155 I'll just say Xi being 1 given S-- 00:40:04.155 --> 00:40:16.190 I'll say St equals little s, comma, Xt minus 1 equals 00:40:16.190 --> 00:40:22.010 0, and suppose that this is larger 00:40:22.010 --> 00:40:29.420 than or equal to probability of Xt equals 1 given St equals 00:40:29.420 --> 00:40:45.200 S prime and Xt minus 1 equals 0 for all S prime less than S, 00:40:45.200 --> 00:40:46.190 OK? 00:40:46.190 --> 00:40:48.580 So I'm saying, as you get further along in the disease 00:40:48.580 --> 00:40:52.450 stage, you're more likely to observe 00:40:52.450 --> 00:40:54.280 one of these complications. 00:40:56.828 --> 00:40:58.620 And again, this is an assumption that we're 00:40:58.620 --> 00:41:00.780 putting into the learning algorithm, 00:41:00.780 --> 00:41:04.525 but what we found that these types of assumptions 00:41:04.525 --> 00:41:06.900 are really critical in order to learn disease progression 00:41:06.900 --> 00:41:11.270 models when you don't have a large amount of data. 00:41:11.270 --> 00:41:16.350 And note that this is just a linear inequality 00:41:16.350 --> 00:41:19.300 on the parameters of the model. 00:41:19.300 --> 00:41:24.690 And so one can use a convex optimization algorithm 00:41:24.690 --> 00:41:27.525 during learning-- 00:41:27.525 --> 00:41:29.400 during the maximum likelihood estimation step 00:41:29.400 --> 00:41:32.610 with this algorithm, we just put a linear inequality 00:41:32.610 --> 00:41:34.500 into the convex optimization problem 00:41:34.500 --> 00:41:37.170 to enforce this constraint. 00:41:37.170 --> 00:41:38.640 There are a couple of questions. 00:41:38.640 --> 00:41:40.890 AUDIENCE: Is there generally like a quick way to check 00:41:40.890 --> 00:41:43.965 whether a model is unidentifiable or-- 00:41:48.510 --> 00:41:51.480 PROFESSOR: So there are ways to try 00:41:51.480 --> 00:41:54.960 to detect to see if a model is unidentifiable. 00:41:54.960 --> 00:41:56.440 It's beyond the scope of the class, 00:41:56.440 --> 00:41:58.980 but I'll just briefly mention one of the techniques. 00:42:04.630 --> 00:42:08.530 So one could-- so you can ask the identifiability question 00:42:08.530 --> 00:42:10.570 by looking at moments of the distribution. 00:42:10.570 --> 00:42:12.430 For example, you could talk about it 00:42:12.430 --> 00:42:14.320 as a function of all of the observed 00:42:14.320 --> 00:42:17.227 moments of distribution that you get from the data. 00:42:17.227 --> 00:42:19.810 Now the observed data here are not the S's and X's, but rather 00:42:19.810 --> 00:42:20.728 the O's. 00:42:20.728 --> 00:42:22.270 So you look at the joint distribution 00:42:22.270 --> 00:42:26.860 on the O's, and then you can ask questions about-- 00:42:26.860 --> 00:42:27.820 if I was to-- 00:42:27.820 --> 00:42:30.280 so suppose I was to choose a random set of parameters 00:42:30.280 --> 00:42:33.040 in the model, is there any way to do 00:42:33.040 --> 00:42:35.440 a perturbation of the parameters in the model which 00:42:35.440 --> 00:42:39.220 leave the observed marginal distribution on the O's 00:42:39.220 --> 00:42:41.290 identical? 00:42:41.290 --> 00:42:44.830 And often when you're in the setting of non-identifiability, 00:42:44.830 --> 00:42:50.712 you can take the gradient of a function and see-- 00:42:50.712 --> 00:42:53.170 and you could sort of find that there is some wiggle space, 00:42:53.170 --> 00:42:56.590 and then you show that OK, there are actually-- 00:42:56.590 --> 00:42:59.530 this objective function is actually unidentifiable. 00:42:59.530 --> 00:43:01.690 Now that type of technique is widely 00:43:01.690 --> 00:43:03.940 used when studying what are known as method of moments 00:43:03.940 --> 00:43:05.730 algorithms or estimation algorithms 00:43:05.730 --> 00:43:08.650 in learning verbal models, but they would be much, much 00:43:08.650 --> 00:43:13.892 harder to apply in this type of setting because first of all, 00:43:13.892 --> 00:43:15.350 these are much more complex models, 00:43:15.350 --> 00:43:17.330 and estimating the corresponding moments 00:43:17.330 --> 00:43:21.350 is going to be very hard because they're very high dimensional. 00:43:21.350 --> 00:43:25.088 And second, because they're-- 00:43:25.088 --> 00:43:27.130 I'm actually conflating two different things when 00:43:27.130 --> 00:43:28.760 I talk about identifiability. 00:43:28.760 --> 00:43:33.110 One statement is the infinite data identifiability, 00:43:33.110 --> 00:43:37.670 and the second question is your ability 00:43:37.670 --> 00:43:42.230 to actually learn a good model from a small amount of data, 00:43:42.230 --> 00:43:44.660 which is a sample complexity. 00:43:44.660 --> 00:43:48.440 And these constraints that I'm putting in, 00:43:48.440 --> 00:43:51.470 even if they don't affect the actual identifiability 00:43:51.470 --> 00:43:53.415 of the model, they could be extremely 00:43:53.415 --> 00:43:55.790 important for improving the sample complexity of learning 00:43:55.790 --> 00:43:57.637 algorithm. 00:43:57.637 --> 00:43:58.720 Is there another question? 00:44:03.715 --> 00:44:05.090 So we valued to this using a data 00:44:05.090 --> 00:44:10.050 set of almost 4,000 patients where, 00:44:10.050 --> 00:44:16.310 again, each patient we observed for only a couple of years-- 00:44:16.310 --> 00:44:18.310 one to three years. 00:44:18.310 --> 00:44:20.370 And the observations that we observed 00:44:20.370 --> 00:44:23.220 were 264 diagnosis codes, the presence 00:44:23.220 --> 00:44:25.950 or absence of each of those diagnosis codes 00:44:25.950 --> 00:44:28.080 during any three-month interval. 00:44:28.080 --> 00:44:32.760 Overall, there were almost 200,000 observations 00:44:32.760 --> 00:44:35.172 of diagnosis codes in this data set. 00:44:35.172 --> 00:44:36.630 The learning algorithm that we used 00:44:36.630 --> 00:44:38.650 was expectation maximization. 00:44:38.650 --> 00:44:40.900 Remember, there are a number of hidden variables here, 00:44:40.900 --> 00:44:42.330 and so if you want to maximize the likely-- 00:44:42.330 --> 00:44:44.372 if you want to learn the parameters that maximize 00:44:44.372 --> 00:44:46.230 the likelihood of those observations O, 00:44:46.230 --> 00:44:48.522 then you have to marginize over those hidden variables, 00:44:48.522 --> 00:44:53.010 and EM is one way to try to find a local optima of that 00:44:53.010 --> 00:44:59.500 likelihood function, with the key caveat that one has to do 00:44:59.500 --> 00:45:02.620 approximate inference during the E step here, 00:45:02.620 --> 00:45:05.500 because this model is not tractable, 00:45:05.500 --> 00:45:07.330 there's no closed form-- 00:45:07.330 --> 00:45:10.090 for example, dynamic programming algorithm for doing posterior 00:45:10.090 --> 00:45:13.420 inference in this model given its complexity. 00:45:13.420 --> 00:45:15.520 And so what we used was a Gibbs sampler 00:45:15.520 --> 00:45:17.690 to do approximate inference within that E step, 00:45:17.690 --> 00:45:19.090 and we used-- 00:45:19.090 --> 00:45:21.460 we did block sampling of the Markov chains 00:45:21.460 --> 00:45:23.590 where we combined a Gibbs sampler 00:45:23.590 --> 00:45:25.420 with a dynamic programming algorithm, which 00:45:25.420 --> 00:45:27.400 improved the mixing rate of the Markov chain 00:45:27.400 --> 00:45:30.460 for those of you who are familiar with those concepts. 00:45:30.460 --> 00:45:34.520 And in the end step of the learning algorithm when 00:45:34.520 --> 00:45:38.150 one has to learn the parameters of the distribution, 00:45:38.150 --> 00:45:40.070 the only complex part of this model 00:45:40.070 --> 00:45:42.440 is the continuous time Markov process, 00:45:42.440 --> 00:45:44.330 and there's actually been previous literature 00:45:44.330 --> 00:45:47.540 from the physics community which shows how you can really-- 00:45:47.540 --> 00:45:49.610 which gives you analytic closed-form solutions 00:45:49.610 --> 00:45:53.480 for that M step of that continuous time Markov process. 00:45:53.480 --> 00:45:55.790 Now if I were to do this again today, 00:45:55.790 --> 00:45:58.530 I would have done it a little bit differently. 00:45:58.530 --> 00:46:00.680 I would still think about modeling this problem 00:46:00.680 --> 00:46:03.230 in a very similar way, but I would 00:46:03.230 --> 00:46:06.530 do learning using a variational lower bound 00:46:06.530 --> 00:46:09.350 of the likelihood with a recognition network 00:46:09.350 --> 00:46:12.590 in order to very quickly get you a lower bound 00:46:12.590 --> 00:46:14.330 in the likelihood. 00:46:14.330 --> 00:46:16.580 And for those of you who are familiar 00:46:16.580 --> 00:46:18.200 with variational autoencoders, that's 00:46:18.200 --> 00:46:20.167 precisely the idea that is used there 00:46:20.167 --> 00:46:21.750 for learning variational autoencoders. 00:46:21.750 --> 00:46:22.880 So that's the way I would approach this 00:46:22.880 --> 00:46:23.880 if I was to do it again. 00:46:26.230 --> 00:46:29.730 There's just one or two other extensions I want to mention. 00:46:29.730 --> 00:46:32.500 The first one is something which we-- 00:46:32.500 --> 00:46:35.570 one, more customization we made for COPD, 00:46:35.570 --> 00:46:38.900 which is that we enforced monotonic stage progression. 00:46:38.900 --> 00:46:41.930 So we said that-- 00:46:41.930 --> 00:46:45.195 so here I talked about a type of monotonically 00:46:45.195 --> 00:46:47.070 in terms of the conditional distribution of X 00:46:47.070 --> 00:46:54.540 given S, but one could also put an assumption in the-- 00:46:54.540 --> 00:46:58.020 I already talked about that, but one could also 00:46:58.020 --> 00:47:01.005 put an assumption on P of S-- 00:47:04.520 --> 00:47:07.890 S of t given S of t minus 1, which is implicitly 00:47:07.890 --> 00:47:09.870 an assumption on Q, and I gave you 00:47:09.870 --> 00:47:12.333 a hint of how one might do that over here when 00:47:12.333 --> 00:47:15.000 I said that you might put 0's to the left-hand side, meaning you 00:47:15.000 --> 00:47:17.017 can never go to the left. 00:47:17.017 --> 00:47:18.600 And indeed, we did something like that 00:47:18.600 --> 00:47:20.903 here as well, which is another type of constraint. 00:47:24.220 --> 00:47:27.130 And finally, we regularize the learning problem 00:47:27.130 --> 00:47:30.130 by asking about that graph involving 00:47:30.130 --> 00:47:32.890 the conditions, the comorbidities, 00:47:32.890 --> 00:47:35.440 and the diagnosis codes be sparse, 00:47:35.440 --> 00:47:40.090 by putting a beta prior on those edge weights. 00:47:40.090 --> 00:47:42.340 So here's what one learned. 00:47:42.340 --> 00:47:43.900 So the first thing I'm going to do 00:47:43.900 --> 00:47:46.570 is I'm going to show you the-- 00:47:46.570 --> 00:47:48.640 we talked about how we specified anchors, 00:47:48.640 --> 00:47:51.550 but I told you that the anchors weren't the whole story. 00:47:51.550 --> 00:47:54.340 That we were able to infer much more interesting things 00:47:54.340 --> 00:47:57.130 about the hidden variables given all of the observations 00:47:57.130 --> 00:47:58.420 we have. 00:47:58.420 --> 00:48:01.630 So here I'm showing you several of the phenotypes that 00:48:01.630 --> 00:48:03.400 were learned by this unsupervised learning 00:48:03.400 --> 00:48:04.150 algorithm. 00:48:04.150 --> 00:48:06.480 First, the phenotype for kidney disease. 00:48:06.480 --> 00:48:10.660 In red here, I'm showing you the anchor variables 00:48:10.660 --> 00:48:15.010 that we chose for kidney disease, and what you'll notice 00:48:15.010 --> 00:48:16.250 are a couple of things. 00:48:16.250 --> 00:48:18.880 First, the weight, which you should think about 00:48:18.880 --> 00:48:22.270 as being proportional in some way 00:48:22.270 --> 00:48:25.360 to how often you would see that diagnosis code given 00:48:25.360 --> 00:48:27.550 that the patient had kidney disease, 00:48:27.550 --> 00:48:30.730 the weights are all far less than one, all right? 00:48:30.730 --> 00:48:34.690 So there is some noise in this process of when you observe 00:48:34.690 --> 00:48:37.340 a diagnosis code for a patient. 00:48:37.340 --> 00:48:39.090 The second thing you observe is that there 00:48:39.090 --> 00:48:46.080 are a number of other diagnosis codes that are observed to be-- 00:48:46.080 --> 00:48:52.050 which are explained by this kidney disease comorbidity, 00:48:52.050 --> 00:48:57.980 such as anemia, urinary tract infections, and so on, 00:48:57.980 --> 00:49:01.220 and that aligns well with what's known in the medical literature 00:49:01.220 --> 00:49:03.470 about kidney disease. 00:49:03.470 --> 00:49:06.050 Look at another example for lung cancer. 00:49:06.050 --> 00:49:08.240 In red here I'm showing you the anchors 00:49:08.240 --> 00:49:10.250 that we had pre-specified for these, which 00:49:10.250 --> 00:49:14.030 mean that these diagnosis codes could only be explained 00:49:14.030 --> 00:49:19.970 by the lung cancer comorbidity, and these 00:49:19.970 --> 00:49:22.550 are the noise rates that are learned for them, 00:49:22.550 --> 00:49:24.085 and that's everything else. 00:49:24.085 --> 00:49:26.660 And here's one more example of lung infection 00:49:26.660 --> 00:49:29.390 where there was only a signal anchor that we specified 00:49:29.390 --> 00:49:32.900 for pneumonia, and you see all of the other things that 00:49:32.900 --> 00:49:34.640 are automatically associated to that as 00:49:34.640 --> 00:49:36.914 by the unsupervised learning algorithm. 00:49:36.914 --> 00:49:37.808 Yep? 00:49:37.808 --> 00:49:41.180 AUDIENCE: So how do you [INAUDIBLE] 00:49:41.180 --> 00:49:42.282 for the mobidity, right? 00:49:42.282 --> 00:49:42.990 PROFESSOR: Right. 00:49:42.990 --> 00:49:45.490 So that's what the unsupervised learning algorithm is doing. 00:49:45.490 --> 00:49:47.850 So these weights are learned, and I'm showing you 00:49:47.850 --> 00:49:51.180 something like a point estimate of the parameters 00:49:51.180 --> 00:49:53.460 that are learned by the learning algorithm. 00:49:53.460 --> 00:49:54.630 AUDIENCE: And so we-- 00:49:54.630 --> 00:49:56.922 PROFESSOR: Just like if you're learning a Markov model, 00:49:56.922 --> 00:50:00.190 you learned some transition and [INAUDIBLE],, same thing here. 00:50:00.190 --> 00:50:00.690 All right. 00:50:00.690 --> 00:50:02.190 And this should look a lot like what 00:50:02.190 --> 00:50:04.770 you would see when you do topic modeling on a text copora, 00:50:04.770 --> 00:50:05.270 right? 00:50:05.270 --> 00:50:08.880 You would discover a topic-- this is analogous to a topic. 00:50:08.880 --> 00:50:11.390 It's a discrete topic, meaning it either occurs 00:50:11.390 --> 00:50:13.170 or it doesn't occur for a patient. 00:50:13.170 --> 00:50:15.838 And you would discover some word topic distribution. 00:50:15.838 --> 00:50:17.880 This is analogous to that word topic distribution 00:50:17.880 --> 00:50:20.430 for a topic in a topic model. 00:50:24.090 --> 00:50:25.850 So one could then use the model to answer 00:50:25.850 --> 00:50:28.280 a couple of the original questions we set out to solve. 00:50:28.280 --> 00:50:32.750 The first one is given a patient's data, 00:50:32.750 --> 00:50:35.880 which I'm illustrating here on the bottom, 00:50:35.880 --> 00:50:37.910 I have artificially separated out 00:50:37.910 --> 00:50:41.150 into three different comorbidities, 00:50:41.150 --> 00:50:43.280 and a star denotes an observation 00:50:43.280 --> 00:50:45.780 of a data type of that one. 00:50:45.780 --> 00:50:47.910 But this was artificially done by us, 00:50:47.910 --> 00:50:50.180 it was not given to learning algorithm. 00:50:50.180 --> 00:50:54.890 One can infer, when the patient initiated-- 00:50:54.890 --> 00:50:58.610 started with each one of these comorbidites, and also, when-- 00:50:58.610 --> 00:51:00.585 so for the full three-year time range 00:51:00.585 --> 00:51:02.210 that we have data for the patient, what 00:51:02.210 --> 00:51:04.670 stage was the patient in in the disease at any one 00:51:04.670 --> 00:51:05.340 point in time? 00:51:05.340 --> 00:51:08.450 So this model infers that the patient starts out in stage 1, 00:51:08.450 --> 00:51:12.230 and about half a year through the data collection 00:51:12.230 --> 00:51:15.290 process, transitioned into stage 2 of COPD. 00:51:18.400 --> 00:51:20.860 Another thing that one could do using this model 00:51:20.860 --> 00:51:25.240 is to simulate from the model and answer the question of what 00:51:25.240 --> 00:51:27.700 would, let's say, a 20-year trajectory of the disease look 00:51:27.700 --> 00:51:28.986 like? 00:51:28.986 --> 00:51:31.070 And here, I'm showing a 10-year trajectory. 00:51:31.070 --> 00:51:33.280 And again, only one to three years of data 00:51:33.280 --> 00:51:36.700 was used for any one patient during learning. 00:51:36.700 --> 00:51:41.670 So this is the first time we see the those axes, 00:51:41.670 --> 00:51:43.620 those comorbidities really start to show up 00:51:43.620 --> 00:51:46.440 as being important as the way of reading out from the model. 00:51:46.440 --> 00:51:49.690 Here, we've thrown away those O's, those diagnosis codes 00:51:49.690 --> 00:51:53.357 altogether, we only care about what we conjecture is truly 00:51:53.357 --> 00:51:55.440 happening to the patient, those X variables, which 00:51:55.440 --> 00:51:58.300 are unobserved during training. 00:51:58.300 --> 00:52:04.130 So what we conjecture is that kidney disease is very uncommon 00:52:04.130 --> 00:52:08.810 in stage 1 of the disease, and increases slowly as you 00:52:08.810 --> 00:52:13.160 transition from stage 2, stage 3, to stage 4 of the disease, 00:52:13.160 --> 00:52:15.645 and then really bumps up towards stage 5 00:52:15.645 --> 00:52:16.770 and stage 6 of the disease. 00:52:16.770 --> 00:52:18.320 So you should read this as saying 00:52:18.320 --> 00:52:22.280 that in stage 6 of the disease, over 60% 00:52:22.280 --> 00:52:24.650 people have kidney disease. 00:52:24.650 --> 00:52:27.050 Now the time interval is here. 00:52:27.050 --> 00:52:30.860 So how I've chosen these-- 00:52:30.860 --> 00:52:35.420 where to put these triangles, I've 00:52:35.420 --> 00:52:39.050 chosen them based on the average amount of time 00:52:39.050 --> 00:52:42.560 it takes to transition from one stage to the next stage 00:52:42.560 --> 00:52:45.490 according to the learned parameters of the model. 00:52:45.490 --> 00:52:48.160 And so you see that stages 1, 2, and 3, 00:52:48.160 --> 00:52:50.890 and 4 take a long period of-- 00:52:50.890 --> 00:52:54.440 amount of time to transition between those four stages, 00:52:54.440 --> 00:52:56.280 and then there's a very small amount 00:52:56.280 --> 00:52:58.030 of time between transitioning from stage 5 00:52:58.030 --> 00:53:01.162 to stage 6 on average. 00:53:01.162 --> 00:53:02.370 So that's for kidney disease. 00:53:02.370 --> 00:53:06.400 One could also read this out for other comorbidities. 00:53:06.400 --> 00:53:09.120 So in orange here-- in yellow here 00:53:09.120 --> 00:53:11.910 is diabetes, in black here is musculoskeletal conditions, 00:53:11.910 --> 00:53:15.300 and in red here is cardiovascular disease. 00:53:15.300 --> 00:53:18.530 And so one of the interesting inferences 00:53:18.530 --> 00:53:20.030 made by this learning algorithm is 00:53:20.030 --> 00:53:25.700 that even in stage 1 of COPD, very early in the trajectory, 00:53:25.700 --> 00:53:27.800 we are seeing patients with large amounts 00:53:27.800 --> 00:53:29.318 of cardiovascular disease. 00:53:29.318 --> 00:53:30.860 And again, this is something that one 00:53:30.860 --> 00:53:32.277 can look at the medical literature 00:53:32.277 --> 00:53:35.220 to see does it align with what we expect? 00:53:35.220 --> 00:53:39.740 And it does, so even in patients with mild to moderate COPD, 00:53:39.740 --> 00:53:43.310 the leading cause of morbidity is cardiovascular disease. 00:53:43.310 --> 00:53:45.770 Again, this is just a sanity check 00:53:45.770 --> 00:53:49.610 that what this model is learning for a common disease actually 00:53:49.610 --> 00:53:54.020 aligns with the medical knowledge. 00:53:54.020 --> 00:53:58.970 So that's all I want to say about this probabilistic model 00:53:58.970 --> 00:54:00.680 approach to disease progression modeling 00:54:00.680 --> 00:54:02.060 from cross-sectional data. 00:54:02.060 --> 00:54:03.560 I want you to hold your questions 00:54:03.560 --> 00:54:05.435 so I can get through the rest of the material 00:54:05.435 --> 00:54:07.760 and you can ask me after class. 00:54:07.760 --> 00:54:10.740 So next I want to talk about these pseudo-time methods, 00:54:10.740 --> 00:54:13.500 which are a very different approach for trying 00:54:13.500 --> 00:54:17.300 to align patients into early-to-late disease stage. 00:54:17.300 --> 00:54:19.700 These approaches were really popularized in the last five 00:54:19.700 --> 00:54:25.340 years due to the explosion in single-cell sequencing 00:54:25.340 --> 00:54:28.660 experiments in the biology community. 00:54:28.660 --> 00:54:31.900 Single-cell sequencing is a way to really understand 00:54:31.900 --> 00:54:36.010 not just what is the average gene expression, 00:54:36.010 --> 00:54:39.520 but on a cell-by-cell basis can we 00:54:39.520 --> 00:54:43.270 understand what is expressed in each cell. 00:54:43.270 --> 00:54:45.690 So at a very high level, the way this works is 00:54:45.690 --> 00:54:51.270 you take a solid tissue, you do a number of procedures 00:54:51.270 --> 00:54:54.090 in order to isolate out individual cells 00:54:54.090 --> 00:54:56.670 from that tissue, then you're going 00:54:56.670 --> 00:55:00.960 to extract the RNA from those individual cells, 00:55:00.960 --> 00:55:03.360 you go through another complex procedure 00:55:03.360 --> 00:55:07.350 which somehow barcodes each of the RNA 00:55:07.350 --> 00:55:10.920 from each individual cell, mixes them all together, 00:55:10.920 --> 00:55:14.070 does sequencing of it, and then deconvolves 00:55:14.070 --> 00:55:17.610 it so that you can see what was the original RNA 00:55:17.610 --> 00:55:20.355 expression for each of the individual cells. 00:55:23.442 --> 00:55:28.450 Now the goal of these pseudo-time algorithms 00:55:28.450 --> 00:55:31.420 is to take that type of data and then 00:55:31.420 --> 00:55:37.510 to attempt to align cells to some trajectory. 00:55:37.510 --> 00:55:40.780 So if you look at the very top of this figure part figure 00:55:40.780 --> 00:55:45.100 a that's the picture that you should have in your mind. 00:55:45.100 --> 00:55:50.180 In the real world, cells are evolving with time-- 00:55:50.180 --> 00:56:00.940 for example, B cells will have a well-characterized evolution 00:56:00.940 --> 00:56:04.300 between different cellular states, 00:56:04.300 --> 00:56:07.010 and what we'd like to be understand, 00:56:07.010 --> 00:56:09.342 given that you have cross-sectional data-- so you 00:56:09.342 --> 00:56:11.800 can imagine-- imagine you have a whole collection of cells, 00:56:11.800 --> 00:56:14.080 each one in a different part, a different stage 00:56:14.080 --> 00:56:18.350 of differentiation, could you somehow 00:56:18.350 --> 00:56:21.140 order them into where they were in different stages 00:56:21.140 --> 00:56:22.448 of differentiation? 00:56:22.448 --> 00:56:23.240 So that's the goal. 00:56:23.240 --> 00:56:25.040 We want to take this-- 00:56:25.040 --> 00:56:27.350 so there exists some true ordering that I'm 00:56:27.350 --> 00:56:29.960 showing from dark to light. 00:56:29.960 --> 00:56:32.343 The capture process is going to ignore 00:56:32.343 --> 00:56:34.760 what the ordering information was, because all we're doing 00:56:34.760 --> 00:56:37.790 is getting a collection of cells that are in different stages. 00:56:37.790 --> 00:56:40.910 And then we're going to use this pseudo-time method 00:56:40.910 --> 00:56:44.720 to try to re-sort them so that you could figure out, 00:56:44.720 --> 00:56:46.670 oh, these were the cells in the early stage 00:56:46.670 --> 00:56:48.740 and these are the cells in the late stage. 00:56:48.740 --> 00:56:50.730 And of course, there's an analogy 00:56:50.730 --> 00:56:52.730 here to the pictures I showed you in the earlier 00:56:52.730 --> 00:56:54.520 part of the lecture. 00:56:54.520 --> 00:56:58.270 Once you have this alignment of cells into stages, 00:56:58.270 --> 00:57:00.880 then you can answer some really exciting scientific questions. 00:57:00.880 --> 00:57:05.380 For example, you could ask a variety of different genes 00:57:05.380 --> 00:57:08.650 which genes are expressed at which point in time. 00:57:08.650 --> 00:57:11.050 So you might see that gene a is very highly 00:57:11.050 --> 00:57:15.700 expressed very early in this cell's differentiation 00:57:15.700 --> 00:57:18.875 and is not expressed very much towards the end, 00:57:18.875 --> 00:57:20.875 and that might give you new biological insights. 00:57:23.570 --> 00:57:26.120 So these methods could immediately 00:57:26.120 --> 00:57:28.850 be applied, I believe, to disease progression modeling 00:57:28.850 --> 00:57:32.870 where I want you to think about each cell as now a patient, 00:57:32.870 --> 00:57:38.360 and that patient has a number of observations for this data. 00:57:38.360 --> 00:57:41.660 The observations are an expression for that cell, 00:57:41.660 --> 00:57:43.580 but in our data, the observations 00:57:43.580 --> 00:57:46.640 might be symptoms that we observe for the patient, 00:57:46.640 --> 00:57:48.618 for example. 00:57:48.618 --> 00:57:50.660 And then the goal is, given those cross-sectional 00:57:50.660 --> 00:57:54.050 observations, to sort them. 00:57:54.050 --> 00:57:56.060 And once you have that sorting, then you 00:57:56.060 --> 00:57:58.400 could answer scientific questions, 00:57:58.400 --> 00:58:01.820 such as, I mentioned, of a number of different genes, 00:58:01.820 --> 00:58:04.430 which genes are expressed when. 00:58:04.430 --> 00:58:08.060 So here, I'm showing you sort of the density of when 00:58:08.060 --> 00:58:09.860 this particular gene is expressed 00:58:09.860 --> 00:58:12.610 as a function of pseudo-time. 00:58:12.610 --> 00:58:14.517 Analogously for disease progression modeling, 00:58:14.517 --> 00:58:16.350 you should think of that as being a symptom. 00:58:16.350 --> 00:58:19.340 You could ask, OK, suppose there are some true progression 00:58:19.340 --> 00:58:25.580 of the disease, when do patients typically develop diabetes 00:58:25.580 --> 00:58:27.360 or cardiovascular symptoms? 00:58:27.360 --> 00:58:29.825 And so for cardiovascular, going back to the COPD example, 00:58:29.825 --> 00:58:32.450 you might imagine that there's a peak very early in the disease 00:58:32.450 --> 00:58:36.510 stage; for diabetes, it might be in a later disease stage. 00:58:36.510 --> 00:58:37.210 All right? 00:58:37.210 --> 00:58:39.220 So that-- is the analogy clear? 00:58:42.060 --> 00:58:47.970 So this community, which has been developing methods 00:58:47.970 --> 00:58:51.090 for studying single-cell gene expression data, 00:58:51.090 --> 00:58:55.770 has just exploded in the last 10 years. 00:58:55.770 --> 00:58:58.140 So I lost count of how many different methods there 00:58:58.140 --> 00:59:03.420 are, but if I had to guess, I'd say 50 to 200 different methods 00:59:03.420 --> 00:59:06.047 for this problem. 00:59:06.047 --> 00:59:08.380 There was a paper, which is one of the optional readings 00:59:08.380 --> 00:59:11.980 for today's lecture, that just came out earlier this month 00:59:11.980 --> 00:59:14.860 which looks at a comparison of these different trajectory 00:59:14.860 --> 00:59:18.220 inference methods, and this picture 00:59:18.220 --> 00:59:19.943 gives a really interesting illustration 00:59:19.943 --> 00:59:21.610 of what are some of the assumptions made 00:59:21.610 --> 00:59:23.030 by these algorithms. 00:59:23.030 --> 00:59:25.990 So for example, the first question, 00:59:25.990 --> 00:59:28.660 when you try to figure out which method of these tons of methods 00:59:28.660 --> 00:59:31.450 to use is, do you expect multiple disconnected 00:59:31.450 --> 00:59:33.808 trajectories? 00:59:33.808 --> 00:59:35.600 What might be a reason why you would expect 00:59:35.600 --> 00:59:38.315 multiple disconnected trajectories for disease 00:59:38.315 --> 00:59:39.190 progression modeling? 00:59:41.890 --> 00:59:43.160 TAs should not answer. 00:59:46.770 --> 00:59:48.695 AUDIENCE: [INAUDIBLE] 00:59:48.695 --> 00:59:53.120 PROFESSOR: Different subtexts would be an example. 00:59:53.120 --> 00:59:56.360 So suppose the answer is no, as we've 00:59:56.360 --> 01:00:00.395 been assuming for most of this lecture, then you might ask, 01:00:00.395 --> 01:00:03.020 OK, there might only be a single trajectory, because we're only 01:00:03.020 --> 01:00:04.940 assuming that a single disease subtype, 01:00:04.940 --> 01:00:09.090 but do we expect a particular topology? 01:00:09.090 --> 01:00:12.290 Now everything that we've been talking about up until now 01:00:12.290 --> 01:00:14.750 has been a linear topology, meaning 01:00:14.750 --> 01:00:16.520 there's a linear projection, there's 01:00:16.520 --> 01:00:19.260 such notion of early and late to stage, 01:00:19.260 --> 01:00:21.020 but in fact, the linear trajectory 01:00:21.020 --> 01:00:23.120 may not be realistic. 01:00:23.120 --> 01:00:25.130 Maybe the trajectory looks a little bit more 01:00:25.130 --> 01:00:27.420 like this bifurcation. 01:00:27.420 --> 01:00:31.370 Maybe patients look the same very early in the disease 01:00:31.370 --> 01:00:34.280 stage, but then suddenly something 01:00:34.280 --> 01:00:36.470 might happen which causes some patients to go 01:00:36.470 --> 01:00:39.350 this way and some patients to go that way. 01:00:39.350 --> 01:00:44.260 Any idea what that might be in a clinical setting? 01:00:44.260 --> 01:00:45.177 AUDIENCE: A treatment? 01:00:45.177 --> 01:00:46.677 PROFESSOR: Treatments, that's great. 01:00:46.677 --> 01:00:47.550 All right? 01:00:47.550 --> 01:00:50.340 So maybe these patients got t equals 0, 01:00:50.340 --> 01:00:52.875 and maybe these patients got t equal as 1, and maybe 01:00:52.875 --> 01:00:54.250 for whatever reason wouldn't even 01:00:54.250 --> 01:00:56.208 have good data on what treatments patients got, 01:00:56.208 --> 01:00:58.600 so we don't actually observe the treatment, right? 01:00:58.600 --> 01:01:01.630 Then you might want to be able to discover that bifurcation 01:01:01.630 --> 01:01:05.410 directly from the data, then that might suggest, going back 01:01:05.410 --> 01:01:07.270 to the original source of the data, to ask, 01:01:07.270 --> 01:01:09.970 what differentiated these patients at this point in time? 01:01:09.970 --> 01:01:11.530 And you might discover, oh, there 01:01:11.530 --> 01:01:13.530 was something in the data that we didn't record, 01:01:13.530 --> 01:01:18.420 such as treatment, all right? 01:01:18.420 --> 01:01:19.810 So there are a variety of methods 01:01:19.810 --> 01:01:23.980 to try to infer these pseudo-times under a variety 01:01:23.980 --> 01:01:25.240 of different assumptions. 01:01:25.240 --> 01:01:26.740 What I'll do in the next few minutes 01:01:26.740 --> 01:01:31.470 is just give you an inkling of how two of the methods work. 01:01:31.470 --> 01:01:35.130 And I chose these to be representative examples. 01:01:35.130 --> 01:01:38.330 The first example is an approach based 01:01:38.330 --> 01:01:42.530 on building a minimum spanning tree. 01:01:42.530 --> 01:01:45.140 And this algorithm I'm going to describe 01:01:45.140 --> 01:01:46.910 goes by the name of Monocle. 01:01:46.910 --> 01:01:50.870 It was published in 2014 in this paper by Trapnell et al, 01:01:50.870 --> 01:01:53.750 but it builds very heavily on an earlier published 01:01:53.750 --> 01:01:58.210 paper from 2003 that I mostly citing here. 01:01:58.210 --> 01:02:00.580 So the way that this algorithm works is as follows. 01:02:00.580 --> 01:02:03.670 It starts with, as we've been assuming 01:02:03.670 --> 01:02:06.310 all along, cross-sectional data, which lives 01:02:06.310 --> 01:02:08.140 in some high-dimensional space. 01:02:08.140 --> 01:02:10.480 I'm drawing that in the top-left here. 01:02:10.480 --> 01:02:16.323 Each data point corresponds to some patient or some cell. 01:02:16.323 --> 01:02:17.740 The first step of the algorithm is 01:02:17.740 --> 01:02:19.270 to do dimensionality reduction. 01:02:19.270 --> 01:02:21.010 And there are many ways to do dimensionality reduction. 01:02:21.010 --> 01:02:22.900 You could do principal components analysis, 01:02:22.900 --> 01:02:25.067 or, for example, you could do independent components 01:02:25.067 --> 01:02:25.570 analysis. 01:02:25.570 --> 01:02:27.927 This paper uses the independent component analysis. 01:02:27.927 --> 01:02:29.385 What ICA is going to do, it's going 01:02:29.385 --> 01:02:31.718 to attempt to find a number of different components that 01:02:31.718 --> 01:02:35.260 seem to be as independent from one another as possible. 01:02:35.260 --> 01:02:37.420 Then you're going to represent the data now 01:02:37.420 --> 01:02:41.140 in this low-dimensional space, and in many of these papers, 01:02:41.140 --> 01:02:44.328 it's quite astonishing to me, they actually use dimension 2. 01:02:44.328 --> 01:02:46.620 So they'll go all the way down to two-dimensional space 01:02:46.620 --> 01:02:49.490 where you can actually plot all of the data. 01:02:49.490 --> 01:02:51.990 It's not at all obvious to me why you would want to do that, 01:02:51.990 --> 01:02:53.698 and for clinical data, I think that might 01:02:53.698 --> 01:02:56.610 be a very poor choice. 01:02:56.610 --> 01:03:02.070 Then what they do is they build a minimum spanning tree on all 01:03:02.070 --> 01:03:04.920 of the patient or cells. 01:03:04.920 --> 01:03:06.750 So the way that one does that is you 01:03:06.750 --> 01:03:11.220 create a graph by drawing an edge between every pair 01:03:11.220 --> 01:03:14.070 of nodes where the weight of the edge 01:03:14.070 --> 01:03:17.625 is the Euclidean distance between those two points. 01:03:20.130 --> 01:03:23.190 And then-- so for example, there is this edge from here to here, 01:03:23.190 --> 01:03:26.130 there's an edge from here to here and so on. 01:03:26.130 --> 01:03:28.740 And then given that weighted graph, 01:03:28.740 --> 01:03:32.050 we're going to find the minimum spanning tree of that graph, 01:03:32.050 --> 01:03:34.590 and what I'm showing you here is the minimum spanning tree 01:03:34.590 --> 01:03:38.100 of the corresponding graph, OK? 01:03:38.100 --> 01:03:39.920 Next, what one will do is go look 01:03:39.920 --> 01:03:45.440 for the longest path in that tree. 01:03:45.440 --> 01:03:48.670 Remember, finding the longest path in a graph-- 01:03:48.670 --> 01:03:50.610 in an arbitrary graph has a name, 01:03:50.610 --> 01:03:53.030 it's called the traveling salesman problem 01:03:53.030 --> 01:03:54.897 and it's the NP-hard problem. 01:03:54.897 --> 01:03:56.230 How has that gotten around here? 01:03:56.230 --> 01:03:58.230 Well we're not-- this is not an arbitrary graph, 01:03:58.230 --> 01:04:01.190 this is actually a tree. 01:04:01.190 --> 01:04:03.460 So here's that poor-- 01:04:03.460 --> 01:04:08.420 here's a algorithm for finding the longest path. 01:04:08.420 --> 01:04:11.360 I won't talk about that. 01:04:11.360 --> 01:04:14.990 So one finds along this path in a tree-- 01:04:14.990 --> 01:04:17.820 in the tree, and then what one does is one says, 01:04:17.820 --> 01:04:21.200 OK, one side of the path corresponds to, 01:04:21.200 --> 01:04:23.750 let's say, early disease stage and the other side of the path 01:04:23.750 --> 01:04:25.958 corresponds to late disease stage, 01:04:25.958 --> 01:04:27.500 and it allows for the fact that there 01:04:27.500 --> 01:04:28.820 might be some bifurcation. 01:04:28.820 --> 01:04:30.480 So for example, you see here that there 01:04:30.480 --> 01:04:33.530 is a bifurcation over here. 01:04:33.530 --> 01:04:35.120 And as we discussed earlier, you have 01:04:35.120 --> 01:04:37.158 to have some way of differentiating 01:04:37.158 --> 01:04:39.200 what the beginning is and what the end should be, 01:04:39.200 --> 01:04:41.325 and that's where some side information might become 01:04:41.325 --> 01:04:43.280 useful. 01:04:43.280 --> 01:04:45.590 So here's an illustration of applying that method 01:04:45.590 --> 01:04:47.490 to some real data. 01:04:47.490 --> 01:04:52.640 So every point here is a cell after doing 01:04:52.640 --> 01:04:54.200 dimensionality reduction. 01:04:54.200 --> 01:04:56.510 The edges between those points correspond 01:04:56.510 --> 01:04:59.780 to the edges of the minimum spanning tree, 01:04:59.780 --> 01:05:01.947 and now what the authors have done 01:05:01.947 --> 01:05:04.280 is they've actually used some side information that they 01:05:04.280 --> 01:05:07.490 had in order to color each of the nodes 01:05:07.490 --> 01:05:11.390 based on what part of the cell differentiation process 01:05:11.390 --> 01:05:14.480 is believed-- that cell is believed to be in. 01:05:14.480 --> 01:05:18.140 And what one discovers, that in fact, this 01:05:18.140 --> 01:05:21.410 is very sensible, that all of these points 01:05:21.410 --> 01:05:25.160 are in a much earlier disease stage 01:05:25.160 --> 01:05:27.110 than [INAUDIBLE] than these points, 01:05:27.110 --> 01:05:33.050 and this is a sensible bifurcation 01:05:33.050 --> 01:05:35.600 Next I want to talk about a slightly different approach 01:05:35.600 --> 01:05:36.298 to-- 01:05:36.298 --> 01:05:38.090 this is the whole story, by the way, right? 01:05:38.090 --> 01:05:41.542 It's conceptually a very, very simple approach. 01:05:41.542 --> 01:05:43.750 Next I want to talk about a different approach, which 01:05:43.750 --> 01:05:48.430 now tries to return back to the probabilistic approaches 01:05:48.430 --> 01:05:51.490 that we had earlier in the lecture. 01:05:51.490 --> 01:05:57.293 This new approach is going to be based on Gaussian processes. 01:05:57.293 --> 01:05:59.460 So Gaussian processes have come up a couple of times 01:05:59.460 --> 01:06:01.020 in lecture, but I've never actually 01:06:01.020 --> 01:06:02.280 formally defined them for you. 01:06:02.280 --> 01:06:04.655 So in order for what I'm going to say next to make sense, 01:06:04.655 --> 01:06:08.230 I'm going to formally define for you what a Gaussian process is. 01:06:08.230 --> 01:06:14.120 A Gaussian process mu for a collection of time points, 01:06:14.120 --> 01:06:17.850 T1 through T capital N, is defined 01:06:17.850 --> 01:06:23.310 by a joint distribution, mu, of those time points, 01:06:23.310 --> 01:06:26.125 which is a Gaussian distribution. 01:06:26.125 --> 01:06:27.750 So we're going to say that the function 01:06:27.750 --> 01:06:31.290 value for these T different time points is just 01:06:31.290 --> 01:06:33.880 a Gaussian, which for the purpose of today's lecture 01:06:33.880 --> 01:06:35.910 I'm going to assume is zero mean, 01:06:35.910 --> 01:06:38.190 and where the covariance function is given 01:06:38.190 --> 01:06:43.170 to you by this capital K, it's a covariance function of the time 01:06:43.170 --> 01:06:46.180 points-- of the input points. 01:06:46.180 --> 01:06:47.940 And so if you look-- 01:06:47.940 --> 01:06:52.050 this has to be a matrix of dimension capital N 01:06:52.050 --> 01:06:56.830 by capital N. And if you look at the I1 and I2 of the N tree, 01:06:56.830 --> 01:07:00.090 if you look at the N tree of that matrix, 01:07:00.090 --> 01:07:03.660 we're defining it to be given to by the following kernel 01:07:03.660 --> 01:07:04.620 function. 01:07:04.620 --> 01:07:09.510 It looks at the exponential of the negative Euclidean 01:07:09.510 --> 01:07:13.770 distance squared between those two time points. 01:07:13.770 --> 01:07:15.450 Intuitively what this is saying is 01:07:15.450 --> 01:07:17.760 that if you have two time points that 01:07:17.760 --> 01:07:20.700 are very close to one another, then 01:07:20.700 --> 01:07:23.730 this kernel function is going to be very large. 01:07:23.730 --> 01:07:28.710 If you have two time points that are very far from one another, 01:07:28.710 --> 01:07:32.140 then this is very large-- it's a very large negative number, 01:07:32.140 --> 01:07:34.260 and so this is going to be very small. 01:07:34.260 --> 01:07:36.010 So the kernel function for two inputs that 01:07:36.010 --> 01:07:37.900 are very far from another are very small; 01:07:37.900 --> 01:07:39.358 the kernel function for inputs that 01:07:39.358 --> 01:07:41.020 are very close to each other is large; 01:07:41.020 --> 01:07:42.910 and thus, what we're saying here is 01:07:42.910 --> 01:07:46.960 that there's going to be some correlation between nearby data 01:07:46.960 --> 01:07:48.702 points. 01:07:48.702 --> 01:07:50.660 And that's the way which we're going to specify 01:07:50.660 --> 01:07:53.480 a distribution of functions. 01:07:53.480 --> 01:07:55.820 If one were to sample from this Gaussian 01:07:55.820 --> 01:07:58.760 with a covariance function specified in the way I did, 01:07:58.760 --> 01:08:03.130 what one gets out is something that looks like this. 01:08:03.130 --> 01:08:07.560 So I'm assuming here that every-- 01:08:07.560 --> 01:08:11.227 that these curves look really dense, 01:08:11.227 --> 01:08:13.810 and that's because I'm assuming the N is extremely large here. 01:08:13.810 --> 01:08:15.643 If and what small, let's say 3, there'd only 01:08:15.643 --> 01:08:18.660 be three time points here, OK? 01:08:18.660 --> 01:08:23.340 And so if you can make this distribution of functions 01:08:23.340 --> 01:08:28.229 be arbitrarily complex by playing with this little l-- 01:08:28.229 --> 01:08:34.950 so for example, if you made a little l be very small, 01:08:34.950 --> 01:08:37.319 then what you get are these really spiky functions 01:08:37.319 --> 01:08:39.630 that I'm showing you in a very light color. 01:08:39.630 --> 01:08:42.210 If you make a little l be very large, 01:08:42.210 --> 01:08:45.920 you get these very smooth functions, right? 01:08:45.920 --> 01:08:48.620 So this is a way to get a function-- this is a way 01:08:48.620 --> 01:08:50.810 to get a distribution over functions 01:08:50.810 --> 01:08:55.130 just by sampling from this Gaussian process. 01:08:55.130 --> 01:08:57.770 What this paper does from Campbell and Yau published 01:08:57.770 --> 01:09:01.870 two years ago in Computational Biology 01:09:01.870 --> 01:09:07.790 is they assume that the observations that you have 01:09:07.790 --> 01:09:13.700 are drawn from a Gaussian distribution whose mean is 01:09:13.700 --> 01:09:16.740 given to you by the Gaussian process. 01:09:16.740 --> 01:09:27.130 So if you think back to the story that we drew earlier, 01:09:27.130 --> 01:09:33.229 suppose that the data lived in just one dimension, 01:09:33.229 --> 01:09:36.149 and suppose we actually knew the sorting of patients. 01:09:36.149 --> 01:09:39.130 So we actually knew which patients 01:09:39.130 --> 01:09:45.200 are very early in time, which patients are very late in time. 01:09:45.200 --> 01:09:51.380 You might imagine that that single biomarker, biomarker A, 01:09:51.380 --> 01:09:55.820 you might imagine that the function which tells you 01:09:55.820 --> 01:09:58.820 what the biomarker's value is as a function of time 01:09:58.820 --> 01:10:01.640 might be something like this, right? 01:10:01.640 --> 01:10:04.970 Or maybe it's something like that, OK? 01:10:04.970 --> 01:10:06.770 So it might be a function that's increasing 01:10:06.770 --> 01:10:09.920 or a function that's decreasing. 01:10:09.920 --> 01:10:12.740 And this function is precisely what 01:10:12.740 --> 01:10:17.330 this mu, this Gaussian process is meant to model. 01:10:17.330 --> 01:10:20.120 The only difference is that now, one can model the Gaussian 01:10:20.120 --> 01:10:23.250 process-- instead of just being a single dimension, 01:10:23.250 --> 01:10:25.500 one could imagine having several different dimensions. 01:10:25.500 --> 01:10:29.990 So this P denotes the number of dimensions, right? 01:10:29.990 --> 01:10:31.490 Which corresponds to, in some sense, 01:10:31.490 --> 01:10:33.650 to the number of synthetic biomarkers 01:10:33.650 --> 01:10:36.860 that you might conjecture exist. 01:10:36.860 --> 01:10:40.370 Now here, we truly don't know the sorting of patients 01:10:40.370 --> 01:10:43.020 into early versus late stage. 01:10:43.020 --> 01:10:47.270 And so the time points T are themselves 01:10:47.270 --> 01:10:49.610 assumed to be latent variables that 01:10:49.610 --> 01:10:52.130 are drawn from a truncated normal distribution that 01:10:52.130 --> 01:10:53.490 looks like this. 01:10:53.490 --> 01:10:55.580 So you might make some assumption 01:10:55.580 --> 01:10:58.970 that the time intervals for when a patient comes in 01:10:58.970 --> 01:11:02.022 might be, maybe patients come in really typically very 01:11:02.022 --> 01:11:03.480 in the middle of the disease stage, 01:11:03.480 --> 01:11:05.355 or maybe you're assuming it's something flat, 01:11:05.355 --> 01:11:07.750 an so patients come in throughout the disease stage. 01:11:07.750 --> 01:11:10.760 But the time point itself is latent. 01:11:10.760 --> 01:11:13.200 So now the generative process for the data is as follows. 01:11:13.200 --> 01:11:15.860 You first sample a time point from 01:11:15.860 --> 01:11:18.620 this truncated normal distribution, then 01:11:18.620 --> 01:11:21.280 you look to see-- 01:11:21.280 --> 01:11:23.030 oh, and you sample from the very beginning 01:11:23.030 --> 01:11:27.500 your sample this curve mu, and then you 01:11:27.500 --> 01:11:30.260 look to see, what is the value of mu for the sample time 01:11:30.260 --> 01:11:33.440 point, and that gives you the expected value you should 01:11:33.440 --> 01:11:36.740 expect to see for that patient. 01:11:36.740 --> 01:11:41.670 And the one, then, jointly optimizes 01:11:41.670 --> 01:11:44.730 this to try to find the most-- 01:11:44.730 --> 01:11:48.450 the curve, the curve mu which has highest posterior 01:11:48.450 --> 01:11:52.470 probability, and that is how you read out from the model 01:11:52.470 --> 01:11:57.815 both what the latent progression looks 01:11:57.815 --> 01:11:59.940 like, and if you look at the posterior distribution 01:11:59.940 --> 01:12:02.065 over the T's that are inferred for each individual, 01:12:02.065 --> 01:12:05.070 you get the inferred location along the trajectory 01:12:05.070 --> 01:12:07.550 for each individual. 01:12:07.550 --> 01:12:09.200 And I'll stop there. 01:12:09.200 --> 01:12:11.200 I'll post the slides online for this last piece, 01:12:11.200 --> 01:12:13.890 but I'll let you read the paper on your own.