Two understandings of Bayes formula

Bayes' formula ：P(A|B) = P(B|A) * P(A) / P(B)

The meaning is ： In known B When it happens A Probability of occurrence .

Bayes' formula It's machine learning More basic , It is also the first thing you will contact when learning , Let's take a look , Two ways to understand it .

There is a saying that ：
The full probability formula describes For the same result , There are many reasons for this , Ask what is the probability of such an outcome ?     It can be understood as Cause and effect Smooth process
Bayesian formula describes When the results are known , Ask about the possibility of some cause leading to this result How much is the ? Cling to the fruit .   It can be understood as Cause and effect The process of finding the inverse
such , In the above formula , Sure A It's the reason B It is the result. ,   But in the above formula ,A and B It can also be an independent event .

Put the denominator in the formula Move to the other side ,P(A|B) * P(B) = P(B|A) * P(A)   The meaning is ：  happen B And happen A Probability = happen A And happen B Probability   That is to say AB Intersection . That is, both things happen . But the order of occurrence can be different , It can correspond to two How to understand Bayes formula ：

If B First , Just focus on B What happened after , The following picture shows   shadow Occupy B Proportion of events   Is the probability
If A First , Prior probability * Adjustment factor     The adjustment factor refers to ( Conditional probability / Total probability )

 

There are three common examples , Such as ：

The result of an examination of a disease ( Suppose the incidence rate of this disease in a region   One thousandth . Some kind of test method For people who really have this disease The test result is 100% positive , There are no mistakes ; But the results of tests on people without the disease There may be 5% The possibility of Misdiagnosis as positive . One person was detected The disease is positive , So what is the probability that this person is really sick ?)
or Examples of machine status and production of qualified products  ( The machine is in good condition The probability of producing qualified products is 90%, In the fault state The probability of producing qualified products is 30%, The probability of a good machine is 75%. If the first product in a day is qualified , So today What is the probability that the machine is good .)
or Examples of cloudy and rainy ：(50% On a rainy morning It's cloudy , about 40% The day of It's cloudy in the morning , This area averages 30 There are usually only 3 It will rain ,10% Rain probability . If it's cloudy today , What is the probability of rain .)

	shadow Occupy B Proportion of events	Prior probability * Adjustment factor
Disease detection	intersection ( Both Yang and illness ) / Check Yang	Be ill * Adjustment factor ( Really sick positive / All positive )
The state of the machine	intersection ( The product is qualified and the machine is good ) / The product is qualified	The machine is in good condition * Adjustment factor ( Good and qualified / All qualified )
Cloud and rain probability	intersection ( It's cloudy and rainy ) / cloudy	It's raining * Adjustment factor ( Rain cloud / All clouds )