I have a small understanding problem with the concept of Bayesian priors.
I use a computer program which samples a posterior distribution using Markov-Chain Monte-Carlo given some Data and prior information on a parameter .
Lets assume the the simple case that my data D is missing, so my posterior distribution is determined only by the prior and not the likelihood function:
The prior information about theta is that it should have a value of .
The computer program incorporates priors the following way:
where is my prior knowledge: 152.
The logarithm of the parameters is taken to not get the parameter values get negative.
My question is now: How do I put my prior information into the prior function or in other words: what should be the value of
The manual of the software says, if, for example I want to constrain with 95% probability between and , must be log(10), because I think in the normal distribution, a value lies with 95% probablities within 2 standard deviations of the mean, so:
, solving for yields
However, I tried to do the same for my 152+-4 prior as follows:
To constrain between and , I calculated .
But here comes the problem: When I put this value as in the program, the sampled distribution of does really have a mean of 152, like I would expect. But the standard deviation of is always about 2 and not 4, as I would expect.
Does anybody see, what I got wrong here? I'd appreciate any help, I went over it a lot of times and checked by calculations, but this does not make any sense to me!
By the way, sorry for the long post, but I hope my problem is clear...