Let's first look at the traditional way of writing the total probability formula ：

P(B)=∑i=1nP(B|Ai)P(Ai)

To the left of the equal sign P(B) Is a state quantity , dexter ∑ Is a process quantity ! This representation is consistent with the integral representation ：

F(n)=∫n0f(x)dx

actually , It's so common that process variables are added to state variables , So that you can give a few examples , Let's not talk about Newton - Leibniz formula , Stokes formula , I use the representation of information entropy as an example ：

H=−∑0npi×logapi

Look at the integral with the above , Is the total probability formula very consistent ?

Forget entropy , Otherwise, this paper will be infinitely extended , Let me talk about total probability and definite integral for the rest of the time .

In fact, the process quantity on the right side of the total probability formula is the integral of the result to the cause ! Although it's not rigorous , But I still write it in the form of definite integral ：

P(B)=∫n0P(B|x)dP(x)

Um. , It's reasonable . Let's see what we can conclude from this pretty good idea . first , There are the following conclusions ：

∫n0P(B|x)dP(x)=−∫0nP(B|x)dP(x)

Namely ：

∫n0P(B|x)dP(x)+∫0nP(B|x)dP(x)=0 – Formula (1)

If ∑ Symbol , How can the sum of two positive probabilities be 0, however , When we use it more widely ∫ After symbol , You'll find that it's an integral loop , The result is 0 Of course, it's true .

here , I mainly generalize the total probability formula in the sense of integral according to the similarity of form , And then we export the above formula (1)
, But it's just that you get this, except that it looks like it's shaking , Nothing else , Let's see what's best .

Above formula (1) The physical meaning of , If the event x
It happened , Let it go back down the timeline , Time is back to the beginning , Nothing happened . The previous integral is a sequence of events that occur along time , The latter integral is equivalent to the inversion of an event .

that , If we record the events in the process one by one , For example, the sequence of events in the first integral is A0,A1,A2...An, So the sequence of events in the second integral is obviously An,An−1,An−
2..A0, Did you see? ? Cause and effect are reversed , Final zeroing .

what is it? ? Isn't that another explanation of Bayesian formula ?

Bayes formula is not important , It is important that , It's going to regularize the way we've been thinking of approaching the cause with certain results , It allows machines to learn step by step turn
, Make a fuzzy thing Howto, For the hot AI The algorithm provides a manual .

In fact, we can say more about this article , About Richard . Feynman's path integral . We know that quantum physics is about probability , What state is a particle in , Not for sure , And then path integration provides a very useful way , I'm here 2013 Studied in the winter of , Almost into theoretical physics , however , There will be no future …

When I have time, I will talk about Feynman's path integral and the story of Dirac's operator .