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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:

enter image description here

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:

  1. 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
  2. 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

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  • $\begingroup$ Do you want intuition or rigour? If intuition, you need to think about how the likelihood changes too -- smaller sample means wider likelihood means wider posterior. $\endgroup$ – Will Jun 5 '17 at 16:13
  • $\begingroup$ The rigour is what I am looking for, since for the intuition I think I am more or less there already! $\endgroup$ – Dave92 Jun 5 '17 at 16:20

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