1 00:00:01,940 --> 00:00:05,900 Conditional probabilities are like ordinary probabilities, 2 00:00:05,900 --> 00:00:08,680 except that they apply to a new situation where some 3 00:00:08,680 --> 00:00:11,210 additional information is available. 4 00:00:11,210 --> 00:00:14,850 For this reason, any concept relevant to probability models 5 00:00:14,850 --> 00:00:17,520 has a counterpart that applies to 6 00:00:17,520 --> 00:00:20,050 conditional probability models. 7 00:00:20,050 --> 00:00:23,230 In this spirit, we can define a notion of conditional 8 00:00:23,230 --> 00:00:27,440 independence, which is nothing but the notion of independence 9 00:00:27,440 --> 00:00:29,840 applied to a conditional model. 10 00:00:29,840 --> 00:00:31,720 Let us be more specific. 11 00:00:31,720 --> 00:00:34,590 Suppose that we have a probability model and two 12 00:00:34,590 --> 00:00:42,190 events, A and B. We are then told that event C occurred, 13 00:00:42,190 --> 00:00:44,690 and we construct a conditional model. 14 00:00:44,690 --> 00:00:47,250 Conditional independence is defined as ordinary 15 00:00:47,250 --> 00:00:50,100 independence but with respect to the conditional 16 00:00:50,100 --> 00:00:51,430 probabilities. 17 00:00:51,430 --> 00:00:56,420 To be more precise, remember that independence is defined 18 00:00:56,420 --> 00:00:59,710 in terms of this relation, that the probability of two 19 00:00:59,710 --> 00:01:03,840 events happening is the product of the probabilities 20 00:01:03,840 --> 00:01:07,780 that one of them is happening times the probability that the 21 00:01:07,780 --> 00:01:10,890 other one is happening. 22 00:01:10,890 --> 00:01:15,240 This is the definition of independence in the original 23 00:01:15,240 --> 00:01:17,350 unconditional model. 24 00:01:17,350 --> 00:01:21,950 Now, in the conditional model we just use the same relation, 25 00:01:21,950 --> 00:01:26,960 but with conditional probabilities instead of 26 00:01:26,960 --> 00:01:29,050 ordinary probabilities. 27 00:01:29,050 --> 00:01:34,190 So this is the definition of conditional independence. 28 00:01:34,190 --> 00:01:38,170 We may now ask, is there a relation between independence 29 00:01:38,170 --> 00:01:40,100 and conditional independence? 30 00:01:40,100 --> 00:01:42,120 Does one imply the other? 31 00:01:42,120 --> 00:01:44,789 Let us look at an example. 32 00:01:44,789 --> 00:01:48,160 Suppose that we have two events and these two events 33 00:01:48,160 --> 00:01:50,320 are independent. 34 00:01:50,320 --> 00:01:54,440 We then condition on another event, C. And suppose that the 35 00:01:54,440 --> 00:01:59,220 picture is like the one shown here. 36 00:01:59,220 --> 00:02:02,620 Are A and B conditionally independent? 37 00:02:05,460 --> 00:02:10,850 Well, in the new universe where C has happened, events A 38 00:02:10,850 --> 00:02:13,860 and B have no intersection. 39 00:02:13,860 --> 00:02:17,290 As we discussed earlier this means that events A and B are 40 00:02:17,290 --> 00:02:19,180 extremely dependent. 41 00:02:19,180 --> 00:02:27,240 Within G, if A occurs, this tells us that B did not occur. 42 00:02:27,240 --> 00:02:31,440 The conclusion from this example is that independence 43 00:02:31,440 --> 00:02:35,090 does not imply conditional independence. 44 00:02:35,090 --> 00:02:38,090 So in this particular example, we saw that the 45 00:02:38,090 --> 00:02:40,160 answer here is no. 46 00:02:40,160 --> 00:02:43,070 Given C, A and B are not independent.