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.