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We've now seen how to compute the log likelihood, but before continuing,
we're going to have to express the probability of y = -1
because we've only so far expressed the probability y = +1.
This is still part of that very,
very optional PhD level only derivation of the gradient.
So this is only for those who are really interested.
We need to take a moment to derive what is the probability that y = -1
if the following up here is the probability that y = +1?
So let's just derive that.
So the probability that y = -1 given x and w
is just one minus the probability that y is equal to +1 given x and w.
This is just because probabilities add up to one.
So, let's plug that in, that's one minus one times one plus
E to the minus W transpose h(x).
I just plugged it in with the definition of probability of y=+1.
Now, if we multiply both elements by 1+e to the -w transpose x,
we get the probability that y is equal to -1 is,
so I'm multiplying both terms by 1+e to the -w transpose h(x).
So the one is 1 + e to the -w transpose h(x) and
then you get 80 minus one.
So the whole thing here simplifies to e
to the -w transpose h(x) divided by
1+e to the -w transpose h(x).
Pretty cool.
Very simple.
Here we go.
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