Dear all,

I have a small understanding problem with the concept of Bayesian priors.
I use a computer program which samples a posterior distribution P(\theta | D) using Markov-Chain Monte-Carlo given some Data D and prior information on a parameter \theta.
P(\theta | D) = P(D | \theta) * \frac{P(\theta)}{P(D)}

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:
P(\theta | D) = P(\theta)

The prior information about theta is that it should have a value of 152\pm 4.

The computer program incorporates priors the following way:

P(\theta)=\frac{1}{2}\left( \frac{log\theta - log\theta^*}{\sigma_{log\theta}} \right)^2

where \theta^* 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 \theta^*=152\pm 4 into the prior function or in other words: what should be the value of
\sigma_{log\theta}?

The manual of the software says, if, for example I want to constrain \theta with 95% probability between \theta / 100 and \theta * 100, \sigma_{log\theta} 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:
\theta / 100 + 4 * \sigma = \theta * 100, solving for \sigma yields
\sigma = log(10)

However, I tried to do the same for my 152+-4 prior as follows:

To constrain \theta between \frac{\theta}{\sqrt{156/148}} and \theta * \sqrt{156/148}, I calculated \sigma=\frac{1}{4} * log(156/148).

But here comes the problem: When I put this value as \sigma_{log\theta} in the program, the sampled distribution of \theta does really have a mean of 152, like I would expect. But the standard deviation of \theta 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...

Greetings,
straussvogel