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 signP(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. Fairly 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 so∑ Symbol, How can the sum of two positive probabilities be0, however, When we use it more widely∫ After symbol, You'll find that it's an integral loop, The result is0 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.

Top formula(1) The physical meaning of, If eventsx
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 isA0,A1,A2...An, So the sequence of events in the second integral is obviouslyAn,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 regularizing the way of thinking that we've been using to approach the cause with certain results, It allows machines to learnstep by step turn
, Make a fuzzy thingHowto, HottestAI 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 here2013 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.

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