How to understand a priori probability and a posterior probability

Prior probability. It refers to the probability obtained based on past experience and analysis, such as the full probability formula.

It tends to be"Seek effect from cause"in question"Cause"Probability of occurrence.

In Bayesian statistical inference, a priori probability distribution of uncertain quantities.

is a probability distribution that expresses the degree of confidence in this amount before considering a number of factors. For example, a priori probability distribution might represent a probability distribution of the relative proportion of voters who voted for a particular politician in a future election. Unknown quantities can be parameters of the model or latent variables.

Posterior probability. It is one of the basic concepts of information theory. In a communication system, after receiving a message, the probability that the receiving end knows that the message will be sent is called the posterior probability.

The calculation of the posterior probability is based on the prior probability. The posterior probability can be determined according to the Bayesian formula.

with prior probability and likelihood functions.

Calculate it. <>

These are two concepts related to Bayesian probability updates. If the subjective probability of an uncertain event occurring changes because of the emergence of a new situation, then the probability before the change is called the prior probability, and the probability after the change is called the posterior probability.

Let's take a simple example of update probability.

Imagine three opaque bowls A, B, and C on the tabletop, and one of them (and only) a porcelain bowl with an egg under it. At this point, what is the probability that the egg will be under bowl A? Answer: 1 3.

Now something happens: someone uncovers bowl C and finds no eggs underneath bowl C. At this point, ask again:

What is the probability of an egg being under bowl A? Answer: 1 2. Note that due to the new situation of "uncovering bowl C and finding that the egg is not under bowl C", the subjective probability of "egg under bowl A" has increased from 1 3 to 1 2.

The prior probability here is 1 3, and the posterior probability is 1 2.

That is, "first" and "after" are relative to the new situation that causes the change in subjective probability.

Prior probability: the probability that is summarized based on previous experience before the event occurs; (If you're doing a question, the question will tell you).

Posterior probability: A correction (which can be understood in this way) to the prior probability after the event has occurred, depending on the circumstances in which the event occurred

Prior probability is the probability obtained based on previous experience and analysis, such as in the full probability formula, it often appears as the "cause" in the "cause to effect" problem. The posterior probability refers to the probability of re-correction after obtaining the information of the "result", which is the "cause" in the problem of "finding the cause of the failure to attain the effect". The posterior probability is based on the new information, and the original prior probability is modified to obtain a probability estimate that is closer to the actual situation.

Prior probabilities and posterior probabilities are relative. If new information is introduced in the future, the so-called posterior probability is updated, and a new probability value is obtained, then this new probability value is called a posterior probability.

Difference Between Prior Probability and Posterior Probability:

A priori probabilities are not determined from the totality of the data about the state of nature, but only from the available materials (mainly historical data); The posterior probability uses more comprehensive information about the state of nature, both a priori probability data and supplementary data.

The calculation of prior probability is relatively simple and does not use the Bayesian formula; The calculation of posterior probability should use Bayesian formula, and when calculating logical probability using sample data, theoretical probability distribution should be used, which requires more knowledge of mathematical statistics.

The prior and posterior probabilities are as follows:

Prior probability: It is the probability obtained based on previous experience, which belongs to objective probability. Probability in statistical history. For example, the probability of rain in a certain place based on statistics (experience) or climate (common sense) over a number of years, this method is a priori probability.

Posterior probability: It is the probability of cause and effect in the present moment. For example, according to the probability that there are dark clouds in the sky (cause or evidence observation data), rain (result), this method is a posterior probability.

In fact, the concepts of prior probability and posterior probability are relative, and the posterior probability is usually the probability obtained after adding new information on the basis of the prior probability, so it is also commonly called conditional probability.

Characteristics of probability:

Probability is a numerical measure of the likelihood of a chance event occurring. Suppose after many repeated experiments (represented by x) and several times by chance (represented by a) occur several times (represented by y).

With x as the denominator and y as the numerator, a numerical value (represented by p) is formed. In multiple experiments, p is relatively stable at a certain value, and p is called the probability of a occurrence. If the probability of a chance event is determined by long-term observation or a large number of repeated experiments, then such probability is statistical or empirical.

The discipline that studies the internal laws that govern accidental events is called probability theory. It belongs to a branch of mathematics. Probability theory reveals the manifestations of the internal laws contained in accidental phenomena.

Posterior probability. Formula: p(a|b)=p(b|a)*p(a)/p(b)。

Posterior probability: Posterior probability is one of the basic concepts of information theory, which refers to the probability that the message is sent that the receiver knows after receiving a message in a communication system is called posterior probability.

Let a priori probability.

is p(a), and assuming that the probability of getting b from a is p(b|).a), then re-amending a from b, the result is the posterior probability of a(a|b）。From the above, we can see that whether it is a priori probability or a posterior probability, the object we are talking about is a, which has never changed.

As the saying goes, the posterior probability is to correct the prior probability, that is, to correct the "cause a" after obtaining the information of "result b".

Prior probability refers to the probability obtained based on past experience and analysis, such as the full probability formula.

It tends to act as the causal probability of occurrence in a causation problem. In Bayesian statistical inference, the prior probability distribution of the indeterminate number of cautions.

is a probability distribution that expresses the degree of confidence in this amount before considering some of the apronic elements.

A priori probabilities are not determined from the totality of the data about the state of nature, but only from the available materials (mainly historical data); A posterior probability uses more comprehensive information about the state of nature, both a priori probability and supplemental data.

The calculation of prior probability is relatively simple and does not use the Bayesian formula;

The calculation of the posterior probability can be based on the Bayesian formula, using the prior probability and likelihood functions. Knowledge.

The difference and connection between prior probability and posterior probability are: the pre-test probability is the commonly said probability, and the chain travel post-test probability is a conditional probability, but the conditional probability is not necessarily the post-test probability. The Bayesian formula is a formula for finding the post-test probability from the pre-test probability.

These are two concepts related to Bayesian probability updates. If the subjective probability of the occurrence of an uncertain event changes because of the emergence of a new situation, then the probability before the change is called the prior probability, and the probability after the change is called the posterior probability.

Let's take a simple example of update probability.

Imagine three opaque bowls A, B, and C on the tabletop, and one of them (and only) a porcelain bowl with an egg under it. At this point, what is the probability that the egg will be under bowl A? Answer: 1 3.

Now something happens: someone uncovers bowl C and finds no eggs underneath bowl C. At this point, ask again: What is the probability that the egg will be under bowl A? Answer: 1 2.

Note that due to the new situation of "uncovering bowl C and finding that the egg is not under bowl C", the subjective probability of "egg under bowl A" has increased from 1 3 to 1 2. The prior probability here is 1 3, and the posterior probability is 1 2.

That is to say, "first" and "after" are the new situation that causes the change in subjective probability.