To change this example even further.

How did we get this?

Here are my answers.

We now apply Bayes rule.

And here's my code, this implements Bayes rule.

Suppose we multiply two Gaussians as in Bayes rule a prior and a measurement probability.

And see what happens. It multiplies.

The Bayes network is composed of 2 variables, A and B.

We apply the same trick as before where we use the exact same prior of 0.01.

So this is the essence of Bayes Rule, which I'll give to you to you in a second.

In the M step we now figure out where these parameters should have been.

And what we care about is what's our prior probability of spam that maximizes the likelihood of this data?

Here is our sequence. There's a couple of sunny days 5 in total a rainy day, 3 sunny days, 2 rainy days.

This makes Bayes Rule really simple.

This expression is called the likelihood.

So, the probability of given that we don't have cancer is 0.2, but the prior here is 0.99.

You remember that we cared about a grid cell xi, and we asked what is the chance of being in xi after robot motion?

So therefore the answer to this question would be 0.5, or half.

We normalize this by the same expression plus the probability for the non spam case.

This gives us 0.0009.

And just like before, we multiply the prior, this guy over here, that gives you 0.3.

To make this correct, we also have to compute the posterior for the non cancer option, which there is no cancer given a positive test.

Thrun So we just learned a number of things.

This was exactly the same as in our diagram in the beginning.

Use Laplacian smoothing with k 1 to compute the probability of a movie being old this is a prior probability, which is just based on class counts the probability of the word top as a title word in the class of old movies, and the probability that a new movie that we look at by new I mean a movie we've never seen before that is called top, the probability this movie that corresponds to the old movie class with the new movie class.

This form is easily transformed into this expression over here, the probability of the message given spam times the prior probability of spam over the normalizer over here.

In maximum likelihood estimation, we assign towards our probability the quotient of the count of this specific event over all events in our data set.

However, if I now divide, that is, I normalize those non normalized probabilities over here by this factor over here,

You remember this because that's what you programmed.

You might say OK, that's the probably of getting exactly 1 times the probability of getting 2 out of 3 plus the probability of getting 3 out of 3.

If I now change some parameters say the robot knows the probability that it's red, and therefore, the probability 1 is under the green cell as a prior.

So for example, we observed that we always have a single first day, and this becomes our prior probability.

For example, for the prior probability, we found that 3 8 messages are spam.

Then the value of the state for the action go up would be obtained as follows.

And when you work this all out, it is 0.376 as indicated over here.

Probability

Phil, the odds against

The sum or set of all those vectors together form the belief space.

Why is this?

Our likelihood is the probability of seeing a positive test result given that you have cancer multiplied by the prior probability of having cancer over the probability of the positive test result, and that is according to the tables we looked at before 0.9 times a prior of 0.01 over now we're going to expand this right over here according to total probability which gives us 0.9 times 0.01.

A very low probability event is that this might happen, but it's much more likely that Oh, wait.

I suppose so, but it might mean that you It means that, statistically speaking, one of the most impressive records of failure is destined to be broken.

So suppose we know we are looking at faces.

So if you get a positive test result you're going to raise the probability of having cancer relative to the prior probability.

But to correct for this inversion, we have to multiply by the prior of the cause to be the case in the first place, in this case, having cancer or not, and divide it by the probability of the evidence, P(B), which often is expanded using the theorem of total probability as follows.