[Probability theory] Bayesian Law

Basic knowledge description:

Joint probability:

Definition: refers to the probability that multiple random variables in a probability distribution satisfy their respective conditions at the same time.

For example, if both X and Y are normal distributions, P {x <4, Y <0} is a joint probability, which indicates x <4, the probability that Y <0 conditions are established at the same time. The Union probability between x and y is expressed as P (xy), P (x, y), or p (x ∩ Y)

 

Conditional Probability:

Definition: event a's probability of occurrence when another event B has occurred.

Example: Under Condition B, the probability of Condition A is expressed as P (A | B)

 

Bayes rule:

Generally, the probability of event a under event B is different from that of Event B Under event a. However, there is a definite relationship between the two, and Bayesian law is the statement of this relationship. As a norm principle, the Bayesian law is effective in interpreting all probabilities. However, the frequency and Bayesian have different opinions on how probabilities are assigned values in applications: A frequent operator assigns values based on the frequency of a random event or the number of samples in the population. A Bayesian operator assigns values based on an unknown proposition. One result is that Bayes has more opportunities to use Bayesian rules.

Therefore, if there are only two events a and B, the following can be obtained:

P (A | B) = P (AB)/P (B), P (B | A) = P (AB)/P () => P (A | B) = P (B | A) * P (A)/P (B)

Therefore, the Bayesian Rule is about the conditional probability and edge probability of random events A and B.

P (A | B) = P (B | A) * P (A)/P (B) ≈ L (A | B) * P ()

Here, l (a | B) is the possibility that a will occur when B occurs.

In Bayes's law, each term has a common name:

1. P (A) is the prior probability or edge probability of. It is called "A prior" because it does not consider any factors of B. 2. P (A | B) is the conditional probability of a after B is known. It is also called the posterior probability of a because of the value of B.

3. P (B | A) is the conditional probability of B after occurrence of A. It is also called the posterior probability of B because of the value of.

4. P (B) is the prior probability or edge probability of B, and is also a standard constant (normalized constant ).

Bayesian formula:

Posterior Probability = (likelihood * prior probability)/standard constant

That is, the posterior probability is proportional to the product of the prior probability and likelihood.

Therefore, posterior probability = standard likelihood * prior probability

Example:

For events a and B, the posterior probability of a indicates P (A | B), the prior probability of a indicates P (A), and the likelihood is P (B | ), standardized constant P (B), so the standard likelihood is P (B | A)/P (B)

[Probability theory] Bayesian Law