The difference between Bayesian confidence interval and frequency confidence interval

Preface

 Recently, in the process of learning to use Bootstrap to calculate the confidence interval, the difference between the Bayesian school and the frequency school's confidence interval is involved. I write down my opinion here for reference only.

The difference between Bayesian school and frequency school

 You should know a lot about the Bayesian school and the frequency school. There are also a lot of information on the Internet, so I won't go into too much detail. Here I will first talk about the difference between them.

 The difference between them is: Bayesian that the true value is not a fixed parameter , is a random variable , the observed data is fixed, which is the focus parameter space, attention distribution parameters, the fixed mode of operation is The posterior distribution of the parameter is obtained by combining the prior distribution of the parameter with the sample information; while the frequency school believes that the true value of the parameter is a fixed and unknown constant , and the observed data is random. The focus is on the sample space, and the related probability calculation Both are for the distribution of the sample.

The difference between Bayesian confidence interval and frequency confidence interval

 In many cases, the confidence interval calculated from the perspective of Bayesian or frequency is the same in value, but the meaning is different.

 With $95\%$ confidence interval, for example. We mentioned earlier that the Bayesian school believes that the true value of the parameter is not fixed, but a random variable, sofor a given Bayesian$95\%$ confidence interval, which means that the true value of the parameter is$95\%$ probability falls within our range, which is very intuitive understanding. For example, we calculated from the current sample$95\%$ Bayesian confidence interval is$[-2.34,.\; 4..\; 8.\; 7]$ , the parameter values are really$95\%$ probability falls within this range.

 The frequency school believes that the true value of the parameter is a fixed unknown constant, so for a given frequency $95\%$ confidence interval, thetrue value of the parameter is either within this interval or not, that is to say, the probability that the true value of the parameter is within this interval is either$1$ or$0$ . Similarly, we assume that95% 95\%calculated from the current sample$.\; 9.\; 5\%$ frequency confidence interval$[-2.34,4.87]$ , if we know God from the perspective of the true value of the parameter is$5$ , then the probability that the true value of the parameter falls within this interval is$0$ , and we know from God’s perspective that the true value of the parameter is$If\; 0$ , then the probability that the true value of the parameter falls within this interval is$1$。

 Then 95% 95\% under the frequency school$95\%$ confidence interval What does it mean? We mentioned earlier that the frequency school believes that sample data is random, that is, sample data can be obtained multiple times. For each sample data, use our structure$95\%$ confidence interval method, such as$[\bar{X}-{FROM}_{\frac{a}{2}}\ast Se(X),\bar{X}+{FROM}_{\frac{a}{2}}\ast Se(X\left)\right]$ , can get a new confidence interval (maybe the difference between them is small). Among these confidence intervals,$.\; 9.\; 5\%$ confidence interval containing the true parameter values. In other words,$.\; 9.\; 5\%$ is a description of a method for constructing a confidence interval, an interval is not per se. For example, suppose the true value of the parameter is$5.24$ , we repeat sampling$.\; 100$ times, the method will be able to construct confidence intervals to give$.\; 100$ th confidence intervals, this resulting in100 100Before$1$$0$$0$ confidence interval, we can say that this$.\; 100$ th confidence interval, there are$95$ or so contains the true value of the parameter$5.24$。

 You can also understand 95% 95\% under the frequency school from another angle$.\; 9.\; 5\%$ confidence interval. Suppose we haven’t sampled yet, but the post-sampling structure has been formulated$95\%$ confidence interval method. We can say that after sampling once, the probability that the confidence interval obtained (not yet known) contains the true value is$95\%$ . However, after sampling and obtaining a specific interval, the probability that this interval contains a true value cannot be discussed under the framework of the frequency school. That is, they can be discussed before sampling, but they cannot be discussed after sampling.

to sum up

 The difference between the Bayesian confidence interval and the frequency confidence interval is mainly due to the different views of the two schools on the true value of the parameter: the Bayesian school believes that the true value of the parameter is a random variable, and the frequency school believes that the true value of the parameter is an unknown constant. . And this is also the most fundamental difference between Bayesian school and frequency school.

Reference: How to understand the 95% confidence interval?