computing the posterior of two Gaussian probability distributions

Question

I am a bit confused how to solve a Bayesian statistics problem. I have a parameter $\epsilon^s$ which is defined as following:

$$\epsilon^s=\frac{\epsilon-g(\pi,z)}{1-g^*(\pi,z)\epsilon}$$ where $g(\pi,z)$ has an analytical description (model) and it is a function of $z$ and $\pi$, but the empirical discrete probability of $z$ can be estimated, $p(z)$. The probability of $\epsilon^s$ is given by $$p_{\epsilon^s}(\epsilon^s)=\frac{1}{2\pi\sigma^2[1-\exp(-\frac{1}{2\sigma^2})]}\exp\left(-\frac{|\epsilon^s|^2}{2\sigma^2}\right)$$ and I assume $p_{\epsilon}(\epsilon|g)\equiv p_{\epsilon^s}(\epsilon^s|g)$. Now I would like to compute the likelihood of probability of measured $\epsilon$, i.e. $\epsilon^{m}=\epsilon+\epsilon^{err}$, for the given $g$ $$p_{\epsilon^m}(\epsilon^m|g)=\int p_{\epsilon}(\epsilon|g)p_{\epsilon^{err}}(\epsilon-\epsilon^m)d\epsilon$$

assuming a Gaussian probability distribution for the measurement error of $\epsilon$ ($\epsilon^{err}$). The likelihood for a set of $n$ measured $\epsilon$ and for a $g$ quantity with $\pi$ free parameters of the model is given by

$$L(\pi)=\prod_{i=1}^np_{\epsilon^m}(\epsilon^m_i|g_i)$$

My questions:

1. What is the best way to compute $p_{\epsilon^m}(\epsilon^m_i|g_i)$ assuming both probabilities are Gaussian?
2. how could I include the probability of $p(z)$ in the calculations and marginalize over?
3. Can MCMC be a good method to solve this problem?

I will appreciate for any help!

Answer 1

If $\epsilon$ is a scalar (which it seems to be), then your integral is a 1D convolution of $p_\epsilon(\epsilon|g)$ with a Gaussian. If $p_\epsilon$ is fairly well behaved, you could evaluate it on a 1D grid and use any old signal processing toolbox to calculate the convolution (you could also quite easily code it by hand).

You could also include $p(z)$ in the same way. The neat thing about working in one or two dimensions is that brute force is often a feasible approach.