I have a non-normalized posterior distribution of which I'd like to calculate the expected value:
$$P(\mu|\{x_i\})={\Large\Pi_i}\Big{[} \frac{a^v\Gamma(v+\frac{1}{2})}{\sqrt {2\pi}\Gamma(v)} (\frac{1}{2}(x_i-\mu)^2+a)^{-v-\frac{1}{2}}\Big{]}*P(\mu)$$
where:
the prior $P(\mu)=\frac{1}{200}$ is a uniform distribution from $-100$ to $100$
$a$,$v$ are positive constants
- $\{x_i\}$ is a set of $N$ real valued independently drawn datapoints
To compute the expected value I tried to evaluate the following integral: $$E[\mu|\{x_i\}] = \frac{1}{200}{\Large[}\frac{a^v\Gamma(v+\frac{1}{2})}{\sqrt {2\pi}\Gamma(v)}{\Large]}^N \int_{-100}^{100} \Pi_i [\frac{1}{2}(x_i-\mu)^2+a)^{-v-\frac{1}{2}}] *\mu *d\mu$$
I approached this by integration by parts but this doesn't get me anywhere. Using wolfram alpha on just one of the $i$ factors yields very complex expressions.
Is there an elegant solution to this integral? If so, how does the solution look like? Is there maybe a different prior that would make solving this easier?