0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

I have since worked out an approach. One option is to simply take,

$$\chi^2=[\vec D - \mathbb E(\vec D)]^T \Sigma^{-1}[\vec D - \mathbb{E}(\vec D)]$$

where $\vec D$ would be the measured values of $K_i$ and $\mathbb E(\vec D)$ would be the actual $K_i$ values. Then we can define a likelyhood based on $\exp(-\chi^2)$.

MCMC then proceeds as usual, picking a proposal to generate $K_i$ and a prior on $K_i$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.