I am trying to interpret/explain a result that I obtained while generating a posterior distribution, and maybe add some informations to what I had so far. The environment that I am using is the following:
Where T is my sample size. I am generating this posterior using different initial samples from which I estimate the parameter x and $\sigma$, and that are different in size as well. According to my results, the lower is the sample size higher is the standard deviation of the posterior's mean. I am now trying to explain how this happens, and which is the main driver through which the sample size impacts the posterior.
What I was thinking for now is:
- First of all T is the denomination of the variance in the normal distribution. Thus lower is T, higher I expect the variance to be
- It influences the shape parameter of the inverse gamma. But honestly I cannot figure out how its decrease can cause the posterior's standard deviation of $\mu$ to increase
If somebody has any suggestion or can provide some reading material, is very welcomed!
Thanks in advance