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We've now seen how to compute the log
likelihood, but before continuing,

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we're going to have to express
the probability of y = -1

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because we've only so
far expressed the probability y = +1.

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This is still part of that very,

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very optional PhD level only
derivation of the gradient.

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So this is only for
those who are really interested.

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We need to take a moment to derive
what is the probability that y = -1

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if the following up here is
the probability that y = +1?

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So let's just derive that.

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So the probability that y = -1 given x and
w

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is just one minus the probability
that y is equal to +1 given x and w.

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This is just because
probabilities add up to one.

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So, let's plug that in,
that's one minus one times one plus

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E to the minus W transpose h(x).

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I just plugged it in with
the definition of probability of y=+1.

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Now, if we multiply both elements
by 1+e to the -w transpose x,

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we get the probability
that y is equal to -1 is,

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so I'm multiplying both terms
by 1+e to the -w transpose h(x).

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So the one is 1 + e to
the -w transpose h(x) and

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then you get 80 minus one.

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So the whole thing here simplifies to e

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to the -w transpose h(x) divided by

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1+e to the -w transpose h(x).

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Pretty cool.

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Very simple.

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Here we go.

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