Full probability formula and definite integral (new solution of Bayesian formula)

First, look at the formula of full probability in the traditional sense:

P (B) =∑I=1NP (b| AI) P (AI)

The P (B) on the left side of the equal sign is a state amount, and the ∑ on the right is a process amount. This representation is consistent with the representation of the integral:

F (n) =∫n0f (x) dx

In fact, this process volume superposition into the state of the way is so common, so that you can give a few examples, do not say what Newton-Leibniz formula, Stokes formula, I use the representation of information entropy to illustrate:

H=−∑0npixlogapi

See if the full probability formula is consistent with the above definite integral.

Do not pull information entropy, otherwise this article will be infinitely elongated, the rest of the length of my to pull the full probability and definite integral bar.

In fact, the process of the right-hand side of the full probability formula is the integral of the result to the cause. Despite the rigor, I still write it in the form of a definite integral:

P (B) =∫n0p (b|x) DP (x)

Well, that's reasonable. Let's see what conclusion we can get from this fairly good idea. First, there are the following conclusions:

∫n0p (B|X) DP (x) =−∫0NP (B|X) DP (x)

That

∫n0p (B|X) DP (x) +∫0NP (B|X) DP (x) =0– equation (1)

If according to the ∑ symbol, two constant positive probability and add how can be 0, however, when we use the ∫ symbol, we will find that this is an integral loop, the result is 0 of course is correct.

Here, I mainly based on the similarity of the form of the full probability formula in the integral meaning of the promotion, and then derived the above equation (1), but only to get this in addition to appear to be clever, nothing else, we look at the most exciting is what.

The physical meaning of the above equation (1) is that if an event x happens, let it go back along the timeline, time goes back to the origin, nothing happens. The previous integral is a sequence of events that occurs along time, followed by an integral equivalent to the inverted drop of an event.

So, if we put the event one by one in the process, for example, the sequence of events in the first integral is a0,a1,a2 ... An, then the sequence of events in the second integral is obviously an,an−1,an−2. A0, did you see that? Cause and effect upside down, end to zero.

What is it. Isn't that another explanation for the Bayesian formula?

The Bayes formula is not important, the important thing is that it will hold the result of the determination to approximate the reason that we have been using the way of thinking to the rule, it can let machine learning step by step, let a vague thing into a howto, for the hot AI algorithm provides a manual.

In fact, this article can also say a little more about Richard Feynman's path integral. We know that quantum physics is playing the probability, a particle in what state, not certain, and then the path integral provides a very useful method, this thing I studied in the winter of 2013, almost into theoretical physics, however, there is no later ...

Have time I must speak well about Feynman's path integral as well as Dirac's operator's story.