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function and the priors. The ratio s = P ( M $^{∗}$| D ) /P ( M | D ) is then calculated. If s > 1, M ∗ is accepted; otherwise, M ∗ is accepted or rejected with a uniform random probability ϱ . Over the course of many such iterations, the chain populates the posterior PDFs, in the process supplying the PDFs for each parameter by marginalizing over the full PDF. 

Initially, the chain quickly travels to regions of ‘good’ fits from any set of initial param- eters, and eventually settles into the best fit region, a process known as ‘burn-in’. It can be proven that the resulting density of points in parameter space samples the real probability density distribution of the parameter space (Gregory 2005). We aim towards an acceptance rate of 23% in our chains (Roberts et al. 1997). A detailed description of the technique may be found in Gregory (2005). 

The benefits of MCMC are numerous: it fully samples the PDFs for all fitted parameters, meaning that formal errors and error contours are easily determined; it is easily extended to include more parameters if desired; it tends to move to a region of high likelihood fairly rapidly; other quantities, such as the stability parameters Q and X , are easily calculated from the PDFs; and there is some evidence that it provides more realistic error constraints than maximum likelihood methods (Kelly et al. 2008). 

The primary disadvantages are twofold. First, there is no guarantee of uniqueness to the final solutions – multimodal PDFs are possible in any given parameter space. However, a judicious choice of the step size and proper tweaking to obtain the correct acceptance rate mitigates this problem. Second, although MCMC chains tend to converge quickly, it is not so simple to verify that the converged chains have fully populated the joint posterior PDF. Moreover, the chain may ‘spread around’ the parameter space, reflecting either the real structure of the joint PDF or the fact that the chain has not run long enough to fully populate the PDF. Formally ensuring convergence may require prohibitively long computing time. However, in practice we are able to obtain good constraints for most parameters in a reasonable time. This does not preclude the possibility that a MCMC chain passed through multiple PDF peaks after burn-in, but there would be no reason to prefer any one such peak over another. 

The likelihood function is given by 

## 4.2. The likelihood function and the priors 

<img src='content_image/361902.jpg'>

