# Bayesian vs. frequentist estimation

I don't really understand the connection between bayesian to "normal" frequentist estimation.

Suppose we want to estimate the expected value of a population given a sample.

In frequentist statisics i'd calculate the sample mean $\overline{x}=\frac{1}{n}\sum_{i=1}^nx_i$ which is, as we know unbiased, consistent and efficent - and therefore the "best" estimator for the true mean we can use. This estimator has a standard deviation of $\text{SE}_\bar{x}\ = \frac{s}{\sqrt{n}}$.

In contrast in bayesian statistics i'd have some a-priori distribution, plug in data and calculate with the bayesian formula an a-posteriori distribution.

• So how does this a-posteriori distribution compare to the sample mean point estimate?
• Is the expected value of the a-posteriori distribution equal to the sample mean?
• Is the standard deviation of the a-posteriori distribution equal to the standard deviation of the sample mean?
• Which one of the two methods is the better one - or does the bayesian approach just give you more information (it gives you a distribution whereas the frequentist estimation only gives you point values)
• Is it possible to get point estimates like in the frequentist estimation from an a-posterio distibution (e.g. the expected value for an point estimation for the true mean)?

Edit:

Ok, I got the idea but what really bothers me is for example in https://en.wikipedia.org/wiki/German_tank_problem#Example the german tank problem.

We want to estimate the total number of tanks where an intelligence officer has spotted k = 4 tanks with serial numbers, 2, 6, 7, and 14.

The frequentist estimate to the tank count is $16.5$ whereas the bayesian is $19.5 \pm 10$ (although the frequentist answer is in the sd. of the bayesesian).

How can two different mathematical (scientific) approaches for the same question lead to two different answers? Intuitively there must be one "more" correct answer, or not?

• I understand where you are going with this, but I think to make it clearer, what are you calculating the expected value of under the frequentist paradigm? You say the expected value of a population but what characteristic of the population? The mean doesn't always work well or isn't always appropriate unless you define what you are trying to estimate. – user95564 Nov 22 '15 at 0:43
• "Which one of the two methods is the better one - or does the bayesian approach just give you more information (it gives you a distribution whereas the frequentist estimation only gives you point values)" -- Let's not start a civil war here.... – user95564 Nov 22 '15 at 0:44
• – Tim Nov 22 '15 at 10:01
• "as we know unbiased, consistent and efficent" -- we do? I don't think I know this. What makes it efficient? – Glen_b Nov 22 '15 at 10:44
• @Glen_b you're correct, it isn't efficient in every case, thanks – lukstei Nov 22 '15 at 10:54

$$\text{posterior} \propto \text{likelihood} \times \text{prior}$$