Bayes formula and Information (II)

In fact, there is no complete statement of the problem in my previous post.

To apply Bayes formula, we need some another facts.

If the prior probability is the probability of been on the crime scene and murdered, and the likelihood is probability of been with the girl, than we should know if he was with girl, or not.

And another thing, that we need to apply the formula, is the knowledge that the information used to calculate likelihood is not that, used for the calculation of prior probability.

If we knew this, than the answer would be 2/3 * 0.1 = 1/15 (the formula is simpler in the case, due to zero probability of murder, if he was with girl).

If we could not rely on this, the answers can differ.

For example, the fact that could be used for the calculation of both probabilities could be simple:

If he was in bad mood, he would go on scene and kill any friend (with probability of 1/5), else he could meet girl (with probability of 1/3rd) or with probability of 7/15 he would go from the city in the opposite direction.

In this case probability would be 1/5.

And another extreme case is the following nature of events:

Boris could go only on the scene, always if he was with his girlfriend before and with 1/6th probability, if he was not with girlfriend.

The final probability in this case is 1/6 * 1/5 = 1/30 .

In real life we are having billions events that form another, more big events that could be analyzed with formulae. We have to had as much data as possible, and we should understand nature of events before making formal calculations.

Bayes formula and Information

The basis for the machine learning is the Bayes formula.

The idea of the formula is simple and can be described in one statement:
new information will change probability.
It seems a little mystical, but it works, and the most machine learning algorithms based on this formula.

Here I will introduce some example on how the prior information on the event affect the posterior probability.

For example, we are trying to solve the criminal case about murdering of Boris, a bank clerk.

We are considering some versions of the murder. To understand, which version should be thrashed out first, we are making the study about probabilities of each version.

The rough version, considered in this study first, states, that Victor, colleague of victim was on the crime scene with the probability 50%, (denote it P1).

Another thing, we know about Victor, that he murdered Boris with the 1/5 probability, only if he was on the place.

But after the consideration, we founded, that he could be murderer, only if he was not with his girlfriend in this day.

And he was with the girl with probability of 1/3rd (denote it P2).

So the matter stays more complicated in the case.

What is the probability of the fact that Victor is the murderer of Boris?

Using the information about Victor being on the crime scene, we are having different figures from that, got from using information about being him with his girlfriend.

So, we are having one event and two different probabilities of this event in this case, that can not be.

Let's hold the solution over until the next post.