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[MUSIC]

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Now let's talk about a slightly
more advanced concept,

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the concept of conditional probabilities.

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And this is what we're going to use a lot
today, so it's a good thing to review and

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a good thing to really keep in mind and
understand.

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Now a sentence here might say
the probability that a review that has

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three awesomes in it and
one awful is positive is 0.9.

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So how, an awful, what does that mean?

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Let;s interpret it by
looking at the real data.

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So, let's say that I have my data set.

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And just highlighted here awesomes and
awfuls.

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And in fact, I'm highlighting synthesis
that had three awesomes and one awful.

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So if I go and mark all of those,
and I just look at those rows where

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the condition,
this part up here is called condition.

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And that's why it's called conditional
probability, three awesomes and one awful.

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So out of the rows that have
three awesomes, one awful,

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I expect on average that 90% of those
are going to be positive reviews.

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And that makes sense,
if you have three awesomes and one awful,

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very likely to be a positive review.

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So this is how we interpret it,

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I expect 90% of the rows
are going to have a +1 in them.

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See there's one here that have a -1,
but in general, you have a lot of +1s.

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Now we can again interpret
the conditional probabilities

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in terms of degrees of beliefs of
how we feel about the reviews, but

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this is not how we feel
about reviews in general.

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How we feel about reviews specifically.

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So for example, if I have a condition xi,

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an input xi that says,
all the sushi was delicious.

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What's the probability that the review's
positive in that condition?

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So here we have
the probability of the output

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being positive and we use the word given.

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This little bar here means given or
conditioned on

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the input, which in our case was
all the sushi was delicious?

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It was delicious.

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And so if that conditional probability has

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value, I wrote on top of my own labels.

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But I hope it makes sense.

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That's the output label, input sentence.

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And these are all the parts
of the probability.

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Now if I say that that output

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has probability one given this input, all
sushi was delicious, then that means that

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I'm absolutely sure that review all the
sushi was delicious is a positive review.

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I feel pretty good about that.

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So we're going to denote it by the
probability of y equals plus one given xi.

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This is our shorthand notation,
is equal to one.

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What does that imply by the way?

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Is it a probability that
y = -1 given xi is one

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minus the probability of y = +1 given xi,

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which in our case would be zero?

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So what does that mean?

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That if the probability that this
is a positive review given that

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the input sentence is all
sushi was delicious is one,

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then the probability that
it's negative is zero.

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Similarly, if we output to
the probability y = +1 is 0 and

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I'm absolutely sure that their review,

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all the sushi was delicious is negative,
and actually I'm not sure of that.

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But in that case it would mean that and

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that means the probability of y
= +1 given the input xi is zero,

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and that would imply
that we believe that it's

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a negative review given this
input of probability one.

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It could be somewhere in between.

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So for example, let's say
the estimated probability of y = +1,

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given the input all the sushi
was delicious is 0.5.

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And so I'm not sure if that
review is positive or negative.

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So what I'm saying is that
the probability of y is +1 given xi is

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the same as the probability of
y = -1 given xi, which is 0.5.

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So I would believe that it's 50/50.

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In reality, for this input I believe
it will be somewhere around here.

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So the probability would be very,
very high that it's a positive review.

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So let's talk about a few important
properties of conditional probabilities.

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First if we have two classes,

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conditional probabilities
are always between zero and one.

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So the probability that y = +1 given

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xi is somewhere between zero and one.

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Similarly the probably that y = -1 given
xi is somewhere between zero and one.

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Now interestingly and importantly,
the conditions probability, conditional

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probabilities, add up to one over y
over the left side of the conditional.

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So the probability that y = +1 given xi,

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if I add it to the probability of the y =

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-1 given xi, that adds up to one.

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However, but,
it's not true if you sum the right side.

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If you sum over the sentences, or
over xi or anything like that.

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So for example, if you said,
sum over all possible

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sentences, of the probability of y = +1,

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given x, that is not required to be 1.

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Or for example, let's say that
you sum over all the data points.

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And you might have, end of them of
the probability of y = +1 given xi.

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Again, that is not one.

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Not required to be one.

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So the other to one on the,
left side over wise.

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Now, we talked about the binary case,
let's talk about multiple classes.

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So in this case, same example as before so

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y can be dog but
now we conditioned it on the image xi.

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Or y can be cat with
a particular image xi as input,

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or y can be bird with
a particular input xi,

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and we have that these are between zero or
one.

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And that's input, output here.

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Now, similarly with multiple classes,
we have the probabilities add up to one,

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if you sum over positive values of y.

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So, if you have the probability
that y is equal to dog for

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this particular image xi,
and you add that probability

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that y is equal to cat for
this particular image xi, and

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you add the probable to the y is
equal to bird for this particular xi,

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and you add all those together,
and you get one.

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Now here we go.

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Now we've covered, very quickly, a summary
of the key properties of probabilities and

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conditional probabilities and
we're going to use them in a high level

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throughout this particular module and
throughout this specialization.

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[MUSIC]