Bayes Bayes theorem

The crux of the problem is: do people take into account the big background/priori/premises of the problem to make a rough estimate . This brings us to our discussion of rationality, which does not mean knowing the facts but recognizing which factors are relevant .

x.1 An example of Bayes' theorem

Introduce a steve hypothesis. We know that the ratio of librarians to farmers in the general background is 1:20. After getting a new evidence (a description about Steve), it is believed that 40% of librarians meet the new given evidence, and only 10% of farmers meet the new given evidence. But considering the general background, the probability that we finally get that Steve is a librarian is very low.

The core essence of Bayesian lies in that new evidence cannot determine opinions out of thin air, but only updates prior opinions (the prior here refers to the previous experience, that is, the known background) . As follows, the 4% librarians are updated to 16%.

x.2 What is Bayes Theorem

Bayes' theorem is a method of calculating probabilities after taking into account all the evidence.

The formula is as follows:

x.3 When to use Bayesian

On when to use Bayesian: Bayesian is used when you have multiple pieces of evidence (background and introducing new evidence) and you want to find the probability that your hypothesis still holds after introducing new evidence.

First you have a hypothesis/proposition (e.g. Steve is a librarian), second you know some new evidence (description about Steve => 40% of librarians fit the description, and only 10% of farmers fit the description), now You want to combine the newly obtained evidence, combined with the background/evidence/statistics/priori premise (librarian: farmer=1:20), to find the probability P(H|E) of your hypothesis:

x.4 How to use Bayesian/Bayesian usage process

1. First of all, you have a hypothesis H: Steve is a librarian, and secondly, we know that in the general background, the prior prior probability of this hypothesis is P(H)=1/21

He comes from the ratio of librarians to farmers in the larger context.

2. In the second step, after obtaining a new evidence, our prior probability space is compressed. The key to this step is that the introduction of new evidence has different restrictions on the left and right . In this step, we need to know the probability of the evidence we see (likelihood likelihood) under the condition that the hypothesis is established

This step is actually to facilitate the calculation of the number of librarians who meet the new evidence in the crowd $=sum\ast P(H)\ast P(E|H)$

3. In the third step, you need to know the likelihood of the evidence you see under the condition that the hypothesis is not established

That is, in the population, the number of farmers who meet the new evidence $=sum\ast P(\neg H)\ast P(E|\neg H)$

4. Finally we calculate the total and use the classical probability to calculate the posterior probability

5. Simplify the formula to obtain an abstract expression form with only probability. Note: The denominator part of the fraction is the total probability formula P(E) - that is, the sum of the probability of seeing the evidence, which can be expanded into any form with the idea of ​​mutual exclusion + full set decomposition.

x.6 Fast memory Bayes formula

We chose to memorize Bayes by image.

In the general context, the prior probability divides the probability of event occurrence into two parts

When new evidence is introduced, the event probability is compressed

get the Bayes formula