What is Sequential Bayesian Estimation and Bayesian Updating

This is an explanation of Bayes' theorem, and explains how to perform sequential Bayesian estimation using this theorem. Sequential Bayesian estimation is a method of updating estimated values in accordance with information that is updated continuously, to improve the accuracy of the estimation. This is a very effective method when data processing is required in real time. This method is also called Bayesian updating.

I will explain the specific method using an example.

■Example
There are two boxes containing many white and black balls, and these are called Y₁ and Y₂. The ratio of white and black balls in Y₁ is 2 to 1, and Y₂ is 1 to 2. You are blindfolded and you choose one of them, and when you pull one ball from the box three times, the result is black, black, and white. What is the probability that you pulled this from box Y₂?

■Answer
After each ball is drawn, estimate the probability of choosing from Y₂. The formula is as follows.

＜1st time＞(draw black)

Substituting the above into equation (1), we get:

This shows that the probability (posterior probability) of getting the information that you drew black is higher than the initial probability (prior probability).

＜Second Time＞ (Black drawn)
The calculation is the same as the first time, but the difference from the first time is that the posterior probability "probability of being Y2" obtained in the first time is treated as the prior probability in the second calculation. (The prior probability of choosing Y in the first time was treated as 1/2).

Substituting the above into formula (1), we get:

The probability that the box is Y2 has increased even more compared to the first time.

＜Third Time＞ (White drawn)
The calculation is the same as the second time, but this time white is drawn, so the calculation changes slightly.

Substituting this into equation (2), we get

The probability of Y2 has decreased since the second round. This is how sequential estimation is done each time information is updated This is called sequential Bayesian estimation. Note that the final answer will not change even if the balls are drawn in the order white, black, and black. This is called sequential rationality.