I am trying to use a Bayesian approach to carry out model selection and estimate the posterior distributions for parameters in a peak fitting scenario (quasi elastic neutron scattering). The generative model can be described by the following:
$y_{m, calc}(x_m, \Theta) = R(x_m) \circledast \left[f_0(x_m, \theta_0)+f_1(x_m, \theta_1)+...+f_n(x_m, \theta_n)\right]$
i.e. a sum of n functions that are then convolved with the instrument response function, R. Each of the n functions has a set of parameters, $\theta_n$, which go towards making the overall parameter vector, $\Theta$. If R is analytical then it's straightforward to calculate the log-likelihood of the measured data.
$\ln p(y|x, \Theta) = -0.5 * \sum_m [(\frac{y_{m, calc}(x_m, \Theta) - y_{m, obs}}{\sigma_m})^2 + \ln(2\pi \sigma_m^2)]$
where $\sigma_m$ are the uncertainties of the m measured points, $y_{m, obs}$.
So far this is a 'normal' non-linear least squares problem ($f_n$ are non-linear).
However, I don't know how to amend the log-likelihood if the response function, R, is no longer analytical but also has experimental uncertainties because it's measured.
Can anyone provide guidance on this topic? Or is it better suited to e.g. stackoverflow?
