MCMC
Contents
Markov chain
A sequence of random states where each next state depends only on the current one, not on how it got there (the "memoryless" property): $p(x_{t+1} \mid x_t, x_{t-1}, \dots, x_0) = p(x_{t+1} \mid x_t)$. Run for long enough, a well-constructed chain settles into a stationary distribution — the probability of being in any given state stops changing from step to step, even though the chain keeps moving.
MCMC's trick is designing the chain's step rule so that this stationary distribution is exactly the posterior; then running the chain and recording where it visits is sampling from the posterior.
MCMC
Markov Chain Monte Carlo: a general class of algorithms for drawing samples from a probability distribution that can't be sampled from directly, by constructing a Markov chain whose stationary distribution is the target distribution.
Where the posterior has no closed form (or the model has too many interacting parameters to derive one by hand), MCMC draws correlated samples from the posterior this way. Requires convergence diagnostics (trace plots, $\hat{R}$, effective sample size) since the samples are only approximately from the posterior until the chain has mixed.
Posterior draw
One sample from the (approximate) posterior, produced by MCMC: a single joint value for every parameter in the model at once — e.g. one $(\mu_{PC1}, \sigma_{PC1}, \mu_{PC2}, \sigma_{PC2}, \mu_{PC3}, \sigma_{PC3})$ tuple — consistent with both the prior and the observed data at that point in the chain. Running the sampler for many iterations across several chains (Step 3) produces thousands of these draws; taken together, their empirical spread is the posterior (or an approximation of it, pending the convergence checks in Step 4).
A single draw isn't a "best estimate" on its own — it's one plausible parameter configuration among many. What matters is the whole collection: summarising them (mean, HDI) gives Step 5's posterior distribution plots, and feeding each draw's parameters through the likelihood one more time gives Step 6's posterior predictive distribution.
Why MCMC for this project
For a single component's score distribution with a non-informative prior, the conjugate/closed-form route (Normal-Inverse-Gamma) would have been sufficient — there's no modelling reason it couldn't be solved analytically at this stage.
MCMC (via a probabilistic programming library) was chosen instead because the 4th aim (see the Aims page) is a hierarchical model with an informative prior encoding expert/economic judgement on the parameters of the prior predictive distribution. That kind of model — parameters of parameters, potentially with dependence across components — won't have a tidy conjugate form. Building the uninformative-prior pass on an MCMC framework now means the same modelling setup carries forward to the informative/hierarchical version later, rather than needing to be rebuilt when the closed-form approach runs out of road.
The trade-off accepted: more setup work up front (choice of library, sampler, convergence checks) than the closed-form route would have needed for this step alone.