>> Alright Charles can you work out these numbers?
>> I can. So the first one is pretty easy because
we did that once when we were talking a couple slides back.
>> We did.
>> We just look at the case where a storm is set to
be true. Those are, those two mega rows there and those are .25 and
.4. We add that up and we get .65. We're pointing out that
since we know that S is .65, we know that not S is .35.
>> Good.
>> Okay. Although that table really has two numbers in it, we only need one
of them.
>> Right. Yes. Very good point.
>> because it's constrained by needing to add
up to one. Then we do something similar
with lightning. We look at the cases where
lightning is true. And s is also true.
>> Yep. There's just one case like that. Huh?
>> Huh, there is only one case like that.
>> Right, but what we really want to know is
what's the probability that lightning is true given that storm
is true. So we need to think about both cases where storm is
true and say of these, what's
the probability that storm...that lightning is true.
>> And it's .25 over .65.
>> Right.
>> Which is .385 rounded up.
>> because you're a cowboy.
>> Which means that... The probability of it, of
not L given S is one minus that or .615.
>> That's right.
>> Okay. So we do the same
trick with probability of L given not S and we find the case where lightning
is true but storm is false and that's .05, or we have to do it out of
both cases where S is false and so it's .05. Divided by,
point .05 divided by .35 which is, 1 7th. And 1 7th is
approximately .143, rounded up. And so not L given
not S is .857.
>> [LAUGH] Nicely done.
>> I use subtraction in my head.
>> In your head yeah, but it was like with carries and stuff
that was nice. And right, so let's see. And, does these sorts of things
make sense. Of not a storm, it's kind of unlikely that we'll see
lightening. Or, if there is a storm,
it's moderately common that we'll see lightening.
>> Okay, that makes sense. Okay, good. So, now we
do the same trick again with thunder. Except now, instead of
looking at L and S, we look at Thunder and, and
lighting, so we need to look a case where thunder is true
and lightning is true, so that would be, point, that's all the
cases where lightning is true, so it would be .2 divided by .25
>> Alright and why are we looking at the case where storm is true?
>> Why are we doing it? Because it's conditionally independent of storm.
>> It doesn't matter.
>> [CROSSTALK] Information, so it doesn't matter which rows we look at.
What matters is we look at a case where thunder and
lightening are both true, and we compare that to thunder is false
and lightening is true. So that's this number. Those add up to
the 0.25, we get 0.2, over the 0.25, which is 0.8. Right.
>> So it's very likely to hear thunder if you see lightning.
>> That makes sense. And there's only a 20%
chance that you don't hear thunder when you hear lightning.
>> It's lightning not thunder, yup. Mmhmm.
>> And so we do the same thing in the case
where we have thunder and there's not lightning. So we find that row.
>> Okay. Not lightning and there is thunder. There's one.
>> Right and we do the same trick we did before and we get,
.04 over .4. Which I think we did last time, actually, and we get .1.
>> We did. So, if it's, if there's not
lightening out, it's very unlikely to hear thunder. Alright.
>> Alright and just to drive this point home.
That was great. Just to drive this point home.
What if it was the case that it mattered what's
value storm had, how would we fill in this table.
>> Well we'd have to look at a lot more rows.
>> Well in particular we couldn't draw this kind
of belief network if that were the case, right?
>> Right.
>> Because it wouldn't be conditionally independent. So we'd have to draw
basically another edge. Here, and what that represents is that thunder, to work
out to what the probability of thunder is, you have to look at
storm and lightning, all the joint combinations of those to make it work.
<< And that grows exponentially as you add more and more data. <<
And that's right, and that's something that threw me when I started to look
at this, because the picture looks a lot like a neural net. Right? In
a neural net, you've got these nodes, you've got arrows going into the nodes,
and when you have a bunch of arrows going into the same node,
you just end up like adding all
those different influences together, weighted by what's,
what it has on the weight. This
belief network representation is an entirely different
animal. In particular, now, what we're really saying is, to work out the value
of this node, you need to know what's going on
in all combinations of what the inputs are. And so,
as you pointed out, so astutely, that grows exponentially as
you have more variables coming into the node. Higher in degree.
>> Hm. So this is not just a network. It's
a graph. And so we can talk about parents and children
right? So, basically, the number of numbers you have to
keep track of is exponential in your number in your parents.
>> I mean it's a, yes.
Though it's not exactly a tree. Doesn't have to be a tree so the parents
relationships are kind of weird. Like in particular,
if you use parent terminology in this graph,
what you're saying is that lightning has
one parent which is storm and thunder has
two parents which are storm and lightning. So
it's, storm is it's own grandfather and parent.
>> So let me ask you a quick question, Michael. So earlier on when
you were describing this, this graph, I
noticed you used the word dependencies. You said
we're going to capture the dependencies.
>> Hm.
>> So if you erase the red line between storm and thunder,
>> I'd be happy to.
>> So you erased that, should I read
this as storms cause lightning, and lightning causes thunder.
>> You can do that, but you would be wrong.
>> Oh okay.
>> You can not infer that there is a cause of relationship
just because there is an arrow between them. These arrows are just telling
us about the relationship between the probabilities and
not anything about the physically processes that underlie them.
>> Okay so let me make sure I understand, what you are saying is, it
would be very natural to look at a belief network or a Bayesian net or a
Bayes Nets or graphical model. And read
the arrows as causes, and therefore read them
as talking about dependencies. But actually what's happening
here is that these things represent conditional independencies.
So, it is not true that lightening is
dependent on storm and thunder is dependent on
lightening. So much as is the case that
storm and thunder are conditionally independent given lightning.
>> That's, that is a good point. I guess I never really realized
that dependence. You use the word dependence.
Sometimes it means a physical dependence. Like,
in the real world it's dependent.
Here I'm just talking about statistical dependence.
It's really just talking about the fact that we can derive numbers from other
numbers, and not that You know things cause other things. So yeah, that's a
really good point. It seems like that was an easy place to get slipped up.
>> Okay. Cool.