I'm having a conceptual problem applying Markov Chain Monte Carlo for Bayesian parameter estimation; I know how MCMC works in principle but it's applying it to this problem that troubles me.
The theory has a function $K_1 \sin^2\theta + K_2 \cos^2\theta$ with parameters $K_1,K_2$ which predicts something (a particle decay width) based on an angle.
What I am given is the result of 10 experiments. Each experiment contains the following data:
- Exact $K_1,K_2$ that were used to simulate experiment, as well as number of particle decays
- Extracted $K_1$ and $K_2$ using moment analysis
- Covariance matrix of $K_1$ and $K_2$
The goal is to estimate $K_1$ and $K_2$, the extracted ones being unnormalised, so they are independent and in a way there is a third parameter, the normalisation. Now, for MCMC I need a probability $P(D|x)$ of finding data $D$ given parameters $x$, and it's in determining this that I'm at a loss.
I see this is a trivial example where we could just compute a mean or something else and be done with it, but the goal is to use MCMC. And I'm not sure what $D$ in particular is in this case.