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So, now, let’s look at an example in [an] actual network,

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and try to see what the CPD’s look like,

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what behavior we get,

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and how we might augment the network

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to include additional things.

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Now, let me warn you right upfront

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that this is a baby network;

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it’s not a real network,

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but it’s compact enough to look at, but

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still interesting enough to get some non-trivial behaviors.

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So, to explore the network,

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we’re going to use a system called SAMIAM.

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It was produced by Adnan Darwiche and his group at UCLA,

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and it’s nice

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because it actually works on all sorts of different platforms,

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so it’s usable by pretty much everyone.

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So let’s look at a particular problem:

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Imagine that we’re an insurance company

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and we’re trying to decide

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for a person who comes into the door

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whether to give them insurance or not.

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So the operative aspect of making that decision

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is how much the policy is going to cost us,

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that is, how much we’re going to have to pay

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over the course of a year to insure this person.

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So there is a variable called Cost.

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Let’s click on that to see what properties that variable have.

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And we can see that in this case,

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we’ve decided to only give two values to the Cost variable,

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Low and High.

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This is clearly a very coarse-grained approximation

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and not one that we will use in practice.

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In reality we would probably

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have this be a continuous variable

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whose mean depends on various aspects of the model.

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But for the purposes of our illustration,

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we’re going to use this discrete distribution

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that only has values Low and High.

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Okay.

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So now, let’s build up this network using the technique of

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“expanding the conversation” that we’ve discussed before.

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And so what is most important determining factor

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as to the cost of the insurance company has to pay?

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Well, probably whether the person has accidents

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and how severe they are.

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So here we have a network that has two variables:

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One is Accident and one is Cost.

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And in this case we decided to select

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three possible values for the accident variable,

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None, Mild, and Severe,

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and with the probabilities that you see listed.

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And what you see down below is the Cost variable.

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And let’s open the CPD

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of the Cost variable given the Accident variable.

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And we can see that, in this case,

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we have a conditional probability table

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of Cost given Accident.

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Note that this is actually inverted

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from the notation that we have used in the class before,

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because here the conditioning cases are columns,

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whereas in the examples that we’ve given

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[they] have been rows.

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But that’s okay, it’s the same thing, just inverted.

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And so we see, for example,

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that if the person has no accidents,

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the costs are very likely to be very low;

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mild accidents incur different distributions over cost;

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and severe accidents have

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a probability of 0.9 of having high cost

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and 0.1 of having low cost.

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So now, let’s continue extending the conversation

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and ask what Accident depends on.

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And it seems that one of the obvious factors

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is whether the person is a good driver or not.

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And so we would expect driver quality

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to be a parent of Accident.

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But there is other things

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that also affect not just the presence of an accident,

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but also the severity of the accident.

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So for example, vehicle size would affect

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both the severity of an accident

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because if you are driving a large SUV, then chances are

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you are not likely to be in an accident as severe

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but it might also perhaps increase

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the chance of having an accident overall

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because maybe driving a large car is harder to handle.

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And then vehicle year might affect the chances of an accident

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because of the presence or absence of certain safety features

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like anti-lock brakes and airbags.

90
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So let’s open the CPD of Accident

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and see what that looks like

92
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now that we have all these parents for it.

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And we can see here that we have these,

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in this case, eight conditioning cases,

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correspond[ing] to three variables, two values each.

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And so here just to look at one of the samples,

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just as an example, distribution for example.

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So, if this is a fairly new vehicle—after 2000—

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and it’s an SUV,

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the probability of having a severe accident is quite low.

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and the probability of having a mild accident is moderate

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and the probability of having of no accidents is 0.85

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whereas if you compare that to corresponding entry

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when we keep everything fixed except that now it’s a compact car,

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we see that the probability of having a mild accident is lower,

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but the probability of having no accidents is higher,

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representing different driving patterns, for example.

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Okay, so with this network,

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we can now start asking simple questions.

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So to do some example of causal inference,

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let’s instantiate, for example, driving quality to be good.

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And bad.

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And we can see that for bad driver

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the probability of low cost is 81%.

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And for a good driver the probability of low cost is 87%.

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If we look at the accidents

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we can see that for a good driver

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there is a probability of 87.5 percent of no accidents

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and ten percent of mild accident.

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And the probability of no accident goes down for a bad driver,

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and mild accident goes up

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and severe accidents also goes way up.

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Now note that many of these differences are quite subtle.

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There’s a difference of a couple percent one way or the other.

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And you might think,

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if you were designing a network,

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that you’d like these really extreme probability changes

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when you instantiate values.

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But in many cases that’s not actually true,

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and these subtle differences

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are actually quite significant for an insurance company

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that insures hundreds of thousands of people—

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a couple of percentage points in the probability of an accident

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can make a very big difference to one’s profitability.

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So now let’s think about

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how we would expand this network even further.

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Vehicle size and vehicle year are things

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that we’re likely to observe in the insurance forum.

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But driver quality is something that’s very difficult to observe.

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You can’t go ask somebody, “Oh, are you a good driver?”

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Because everyone’s going to say,

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“Sure, I’m the best driver ever!”

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And so that’s not going to be a very useful question.

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So what evidence do we have that we can observe

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that might indicate to us the value of the driver quality?

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One obvious one is the person’s driving record.

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That is, whether they’ve had previous accidents

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or previous moving violations.

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So let’s think about adding a variable

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that represents driving history.

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And so let’s go ahead and introduce that variable.

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So we can click on this button

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that allows us to create a node.

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The node is now called variable1

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so we’d have to give it a name.

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So for example we’re going to call it DrivingHistory.

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And that’s its identifier,

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and we also have the name of the variable,

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which is usually the same.

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And let’s make that two values,

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say PreviousAccident and NoPreviousAccident.

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Now where will we place this variable in the network?

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One might initially think that the right thing to do

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is to place DrivingHistory as a parent of Driver_quality

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because driving history can influence

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our beliefs about driver quality.

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Now it’s true that observing driving history

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changes our probability within driver quality,

169
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but if you think about the actual causal structure of this scenario,

170
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what we actually have is that driver quality is a causal factor

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of both a previous accident

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as well as a subsequent accident.

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And so if we want to maintain

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the intuitive causal structure of the domain,

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a more appropriate thing is to add DrivingHistory as a child

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rather than parent of Driver_quality.

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[You] might question why it matters

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and in this very simple example

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the two models are in some sense equivalent

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and we could have placed it either way

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except that the CPD for driver quality given driving history

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might be a little bit less intuitive.

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But if we had other indicators of driver quality,

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for example a previous moving violation,

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then it actually makes a lot more sense

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to have all of these be children of driver quality

187
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as opposed to parents of driver quality.

188
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Okay.

189
00:09:02,680 --> 00:09:07,481
So that shows us how we would add a variable into the network.

190
00:09:07,481 --> 00:09:09,764
And now let’s go and open up a much larger network

191
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that includes these variables as well as others.

192
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So let’s look now at this larger network.

193
00:09:15,971 --> 00:09:17,347
And we can see

194
00:09:17,347 --> 00:09:20,237
that we’ve added several different variables to the network.

195
00:09:20,237 --> 00:09:23,211
We’ve added attributes of the vehicle,

196
00:09:23,211 --> 00:09:27,284
for example whether the vehicle had antilock brakes and an airbag,

197
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which is going to allow us to give

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more informative probabilities regarding the accident.

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00:09:31,483 --> 00:09:35,227
We’ve also introduced aspects of the driver,

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00:09:35,227 --> 00:09:38,243
for example, whether they’ve had extra-track training,

201
00:09:38,243 --> 00:09:40,084
which is going to increase driving quality,

202
00:09:40,084 --> 00:09:41,684
whether they’re young or old,

203
00:09:41,684 --> 00:09:42,952
where the presumption is

204
00:09:42,952 --> 00:09:45,883
that younger people tend to be more reckless drivers,

205
00:09:45,883 --> 00:09:50,373
and whether the driver is focused or more easily distracted,

206
00:09:50,373 --> 00:09:52,848
which again is going to affect driving quality.

207
00:09:53,648 --> 00:09:58,681
Now since personality type is hard to observe,

208
00:09:58,681 --> 00:10:03,155
we added another variable which is Good_student

209
00:10:03,155 --> 00:10:05,654
which might indicate one’s personality type.

210
00:10:05,654 --> 00:10:08,862
So let’s open [the] CPD for that one.

211
00:10:11,293 --> 00:10:14,071
And so we can see here that, for example,

212
00:10:14,071 --> 00:10:20,999
if you are a focused person who is young,

213
00:10:20,999 --> 00:10:23,720
you’re much more likely to be a good student,

214
00:10:23,720 --> 00:10:28,439
much more so than if you are not a focused person who is young.

215
00:10:28,439 --> 00:10:32,303
If you’re old, you’re just not very likely to be a student,

216
00:10:32,303 --> 00:10:38,021
and so this probability basically says that if you’re old,

217
00:10:38,021 --> 00:10:39,883
you’re just not very likely to be a student,

218
00:10:39,883 --> 00:10:41,322
and therefore not likely to be a good student.

219
00:10:42,014 --> 00:10:47,608
So, now that we’ve added all these variables to the network,

220
00:10:47,608 --> 00:10:51,161
let’s go ahead and run a few queries to see what happens.

221
00:10:51,161 --> 00:10:56,918
And let’s start by looking at the prior probability of Accident

222
00:10:56,918 --> 00:10:59,942
before we observe anything.

223
00:10:59,942 --> 00:11:04,299
So we can see that the probability of no accident is about 79.5%.

224
00:11:04,299 --> 00:11:07,065
The probability of severe accident is about 3%.

225
00:11:07,065 --> 00:11:10,077
Now let’s go ahead and tell the system

226
00:11:10,077 --> 00:11:11,837
that we have a good student at hand.

227
00:11:11,837 --> 00:11:13,608
So we’re going to observe

228
00:11:13,608 --> 00:11:15,978
that the student is a good student,

229
00:11:15,978 --> 00:11:17,567
and let’s see what happens.

230
00:11:18,059 --> 00:11:19,695
We can see, surprisingly,

231
00:11:19,695 --> 00:11:20,807
that even though we observe

232
00:11:20,807 --> 00:11:21,887
that somebody is a good student,

233
00:11:21,887 --> 00:11:23,954
the probability of no accidents

234
00:11:23,954 --> 00:11:27,699
went down from 79.5% to 78%,

235
00:11:27,699 --> 00:11:29,582
and the probability of severe accidents

236
00:11:29,582 --> 00:11:32,819
went up to 3.5 to 3.67 percent.

237
00:11:32,819 --> 00:11:33,880
You might say,

238
00:11:33,880 --> 00:11:35,986
“Well, but I <i>told</i> you that it’s a good student.

239
00:11:35,986 --> 00:11:38,138
Shouldn’t the probability of accidents go down?”

240
00:11:38,307 --> 00:11:41,972
So let’s look at some active trails in this graph.

241
00:11:41,972 --> 00:11:46,461
One active trail goes from Good_student to Focused,

242
00:11:46,461 --> 00:11:48,928
to Driver_quality,

243
00:11:48,928 --> 00:11:49,939
to Accident.

244
00:11:49,939 --> 00:11:53,602
And sure enough, if we consider that trail in isolation,

245
00:11:53,602 --> 00:11:58,378
it’s probably going to make the probability of no accident be higher.

246
00:11:58,378 --> 00:12:00,204
But, we have another active trail.

247
00:12:00,204 --> 00:12:04,058
We have the active trail that goes from good student up to age,

248
00:12:04,058 --> 00:12:07,085
and then back down, through [to] driver quality.

249
00:12:07,085 --> 00:12:09,921
So, to see that, let’s unclick on good student

250
00:12:09,921 --> 00:12:11,281
and see what happens.

251
00:12:11,281 --> 00:12:15,767
Note that the probability initially that the driver is young was 25%,

252
00:12:15,767 --> 00:12:17,710
but when I observed a good student,

253
00:12:17,710 --> 00:12:20,538
it went up to close to 95%.

254
00:12:20,538 --> 00:12:23,143
And that was enough to counteract the influence

255
00:12:23,143 --> 00:12:27,446
along this more obvious active trail.

256
00:12:27,831 --> 00:12:31,551
So, to demonstrate that this is indeed what’s going on,

257
00:12:31,551 --> 00:12:35,783
let’s click on the fact

258
00:12:35,783 --> 00:12:38,415
and instantiate the fact that the student is young,

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and we can see that the probability of severe accident went up to 3.7%

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and no accident went down to a little bit shy of 77%.

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And now let’s observe good student and see what happens.

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So now we observed good student,

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and the probability of no accidents went down to 78%,

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as opposed to before when it was 77%.

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And the reason for that

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is that we’ve now blocked this trail

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that goes from good student, through age, to driver quality

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by observing this variable which blocks the trail.

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So we can see the reasoning patterns

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in a Bayesian network are sometimes subtle.

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And there are different trails that can affect things

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and interact with each other in different ways.

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And so it’s useful to take the model

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and play around with different queries

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and different combinations of evidence

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to understand the behavior of a network.

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And especially if you’re designing

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such a network for a particular application,

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it’s useful to try out these different queries

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and see if the behavior that you get

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is the behavior that you want to get.

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And if not, then you need to thing about

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how do I modify this network to get behavior

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that’s more analogous to the desired behavior.

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This network is available for you to play with

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and you can try out different things

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and see what behaviors you get.