We now come to parameter learning in Bayesian networks,

which is somewhat related to what we've been doing here.

Only that the hypotheses are really about these little boxes

in the parameters in Bayesian networks.

Eventually, for that you will have to read up in Russell and Norwick,

there's a very good way,

a very good exposition of that.

You can also learn the shape of Bayesian networks.

It's just basically also learning certain probabilities,

conditional independence and all of those things,

are what you have to learn for that.

Okay? So, what's the problem?

So, we're going to do something extremely simple first.

We're assuming that in our candy example,

there's a new manufacturer on the market.

One who thinks it's cool not to tell us about the distribution

of cherry candies in their various bags,

or one bag to make,

they make one bag,

but they don't tell you it could be only limes,

it could be three-quarters, one-quarter,

or it could be half-half.

They say, well, it's always the same,

but we don't tell you what.

That's something where we have essentially a Bayesian network,

namely with one random variable for the flavor,

which has no dependencies,

but we have one conditional probability table,

which is essentially just about the flavor here,

which we can describe by exactly one parameter,

theta, which is a real number.

Of course, we don't know what that parameter is.

Anything between zero and one could be possible.

Of course, we only need one parameter because if we

know the fraction of limes,

we also know the fraction of cherries,

because that's just n minus c.

Just using what we've done before the break,

we know that if we have a set of observations,

we've unwrapped a couple of candies from this new bag,

what we have is that the probability,

the likelihood of the data given a hypothesis.

Now, we have not just one hypothesis,

but for every theta we have a hypothesis,

is actually this product.

After having observed c limes,

we get that this is theta to the power of c,

and one minus theta to the power of l,

where l of course is n minus c,

the number of lines.