Chapter 8: MCMC – Part II (Advanced Topics)

Running MCMC is just the start. We must diagnose whether the sampler worked correctly.

8.1 Advanced MCMC: HMC and NUTS

Modern libraries like PyMC use advanced samplers:
Hamiltonian Monte Carlo (HMC): Uses the gradient (slope) of the posterior to propose efficient, long-distance jumps.
No-U-Turn Sampler (NUTS): An adaptive version of HMC that automates tuning. It's the reason modern Bayesian modeling is so powerful and easy.

8.2 Convergence Diagnostics

Trace Plots: A plot of the sample values over time. Should look like a "fuzzy caterpillar" with no trends.
Effective Sample Size (ESS): An estimate of the number of independent samples. We want this to be high.
Gelman-Rubin Statistic (hatR or R-hat): The most critical diagnostic. It compares the variance between multiple chains to the variance within each chain. If the chains have converged, R-hat should be very close to 1.0 (e.g., < 1.01).

8.3 Code Example: Checking Diagnostics with ArviZ

ArviZ makes diagnostics easy. Let's check the R-hat and ESS from our previous model.
Python

# Continuing from the non_conjugate_model...

# The summary table automatically includes r_hat and ess
summary = az.summary(trace_mcmc)
print(summary)

# You can also create diagnostic plots
az.plot_trace(trace_mcmc)
plt.show()

az.plot_ess(trace_mcmc, kind='evolution')
plt.show()
The summary table will show r_hat values very close to 1.0 and high ess_bulk and ess_tail values, indicating our sampler converged successfully.