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Rewrite clear Bayesian formula with base ratio

2022-06-10 18:48:00 【User 1908973】

Bayes theorem is based on Thomas, a British statistician and philosopher · Bayes (1701-1761) Named after , He formally proved that new evidence can be used to renew beliefs . Pierre, French mathematician · Simon · Laplace (1749-1827) Further developed this formalism , He was in 1812 Year of 《 probability analysis 》 The traditional expression of Bayes theorem is first published in .(9.4):

The traditional expression of Bayes theorem :

Using Bayes Theorem , From the condition p(y|x) Calculate the inverse condition p(x|y). However , This traditional expression of Bayesian theorem hides some subtleties related to the base interest rate , As follows . Those that have basic

Knowledge of probability theory , But people who are not familiar with Bayes Theorem , In the face of it for the first time , It's easy to get confused .

Suppose, for example, buying a lottery ticket with a low probability of winning , Expressed as conditional probability p(y|x)= 0.001, Where the status is x:“ Purchased lottery tickets ” and y:“ Win the prize ”. Suppose you actually bought a ticket , such p(x)= 1.0, Actually won the prize , such p(y)= 1.0. An intuitive but erroneous explanation of Bayes' theorem is p(x|y)= (0.001 × 1.0)/1.0 = 0.001, in other words , In the case of winning the lottery, the probability of buying the lottery is only 0.001, This is obviously wrong . The obvious correct answer is , If you win , Then you must have bought a ticket , use p(x|y)= 1.0 Express .

Anyone familiar with Bayes theorem knows p(x) and p(y) Is the base rate ( Prior probability ), But this is not usually explained in textbooks . Only when practical examples appear , People understand , Bayes theorem requires x and y The basic probability of ( transcendental ), instead of x and y Depends on the probability of the situation

for fear of x The basic rate and x Confusion between probabilities , We use the term a(x) To express x The basic rate of . Similarly , The term a(y) Express y The basic rate of . Use this Convention , Bayesian theorem can be more formalized

Intuitively speaking , Such as the theorem 9.1 Shown .

Theorem 9.1( Bayes theorem with basic interest rate ).

prove . Formally , The conditional probability is defined as follows :

Conditional sentences represent general dependencies between statements , So on the right side of the equation p(x/\y) and p(x) term .(9.6) A general prior probability must be expressed , Not, for example, the probability of a particular target

service . As the first 2.6 Section , The general prior probability is the same as the basic probability . therefore , A more explicit version of the equation .(9.6) It can be expressed as

Bayes theorem can be easily deduced from the definition of conditional probability of equation .(9.7) Expressed in basic interest rate p(y|x) and p(x|y) Conditional probability of :

However , Bayes theorem in equation form .(9.5) Basic rate is hidden a(y) Is the marginal basic rate (MBR) Fact , The marginal base rate must be expressed as the base rate a(x) Function of [56]. This requirement is found in the theorem 9.2 To realize .

(9.4) Is unnecessary ambiguity , Because it doesn't distinguish between basic rates ( transcendental ) And probability ( Posttest ), And because it doesn't show x and y The basic rate is dependent . Of Bayes theorem MBR Use formula (9.9) By way of y Of MBR Expressed as basic rate a(x) To correct this problem .

Let's take another look at the example of lottery , The probability of winning the lottery is p(y|x)= 0.001, Intuition tells us that the probability of winning the lottery must be p(x|y)= 1. We assume that the probability of winning a lottery without a ticket is zero , use p(y|x)= 0 Express . The correct answer appears directly in Eq in .(9.9), Expressed as

in fact , Whether it's the basic winning ratio , Or the basic rate of buying tickets , It has no effect on the results , Because without tickets , The probability of winning the prize is always zero .

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AGI Basics , Uncertainty reasoning , Subjective logic ppt1

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