NEW: Markov Chain Monte Carlo lecture#332
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Translation of mh_notes.tex (Bayesian inference via Metropolis-
Hastings) into a MyST lecture, placed after stationary_densities
in the TOC.
- Theorems, definitions, examples, proofs and the algorithm use
sphinx-proof (prf:) directives with mcmc_-prefixed labels
- One sentence per paragraph throughout, per the style manual
- Bold for defined terms, sentence-case headings, \mathrm{Uniform},
US spelling, no \tag equation numbering
- Cross-link to the stationary_densities lecture in the Markov
kernels section
- Cites MeynTweedie2009 (already in quant-econ.bib)
Verified with a standalone jupyter-book build: no warnings, all
eq/prf/ref/cite targets resolve, all 11 directives render with
correct titles and numbering.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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…njugate priors Extends the MCMC lecture with worked numerical sections: - A numerical example: Gaussian model with conjugate prior; MH sampler in JAX (lax.scan, jit, log acceptance ratio, explicit PRNG keys); draws validated against the closed-form posterior - Ergodicity in action: running mean of one trajectory (LLN part) and a vmap ensemble of 10,000 chains from a bad initial condition showing total variation convergence (echoing the density-sequence figures in stationary_densities) - Losing conjugacy: quadrature ground truth on a grid, then (i) Student-t prior under prior-data conflict (heavy tails let the data overrule the prior), and (ii) a bimodal Gaussian mixture prior with ambiguous data (posterior inherits the two-regime structure, modes pulled toward the evidence) - Exercises: proposal-scale tuning with trace plots; sample size sweep resolving the bimodal posterior Seeds chosen so figures match the narrative (sample mean near the true value; first two mixture observations genuinely ambiguous). Verified by executing the full lecture via jupytext --to py (all figures inspected) and a standalone jupyter-book build. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Editorial edits by John: new introductory passage, tightened Overview and roadmap, dθ' notation alongside μ(dθ') with explanatory note, TV-convergence remark moved out of the ergodic theorem statement, softened phrasing in the non-conjugate sections. Plus typo fixes on top (Some/essence/produces/case, heavy-tailed double hyphen) and trailing whitespace removal. Ensemble figure: replace dates (10,...,499) with (1,5,15,25,40,80) so the cross-section densities show the full journey to the posterior instead of stacking on it; trim T_short to 100. Parallel-chains restructure: scalar mh_chain is sequential and wastes accelerators, so add a vmap-based mh_ensemble helper and run all sampling workloads as 4,096 parallel chains (500 burn-in, pooled to ~4M draws). Burn-in is now discussed explicitly and justified by the TV-convergence result. Running mean uses a single ensemble row, as the ergodic theorem requires; exercise 1 keeps single chains (mixing is the point) and is self-contained. New passage ties the pattern to NumPyro/BlackJAX: this is what runs under the hood of the libraries used in later lectures. Side effects: total runtime ~9s on CPU (was ~30s), posterior moments match to 4 decimals, smoother histograms. Verified by full execution via jupytext --to py with all seven figures inspected; all references resolve. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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Summary
Adds a new lecture, Markov Chain Monte Carlo (
mcmc.md), placed immediately afterstationary_densitiesin the Tools and Techniques section of the TOC.The first part is a translation of an existing set of LaTeX notes on Bayesian inference via Metropolis-Hastings sampling; the second part adds a worked numerical example in JAX.
Theory (translation)
Numerical example (JAX)
jax.lax.scan+jitwith explicit PRNG keys and the log acceptance ratio; draws validated against the closed-form posterior (moments agree to 3 decimals)vmapensemble of 10,000 chains launched from a deliberately bad initial condition, whose cross-section densities march to the posterior (the TV-convergence conclusion) — visually echoing the density-sequence figures instationary_densitiesConventions
prf:definition, 4prf:theorem(withprf:proof), 1prf:example, 1prf:algorithm— all labels prefixedmcmc_\mathrm{Uniform}, no\tagnumberingmystnbfigure captions on main-text figures; none inside exercise/solution directives (PDF constraint)!pip install jaxfollowing the precedent indovis_accounting_mf.md, with a note admonition saying CPU is sufficient (the examples are small — no GPU needed)stationary_densitieslecture; citesMeynTweedie2009(already inquant-econ.bib)Verification
!piplexer notice; every{eq}/{prf:ref}/{ref}/{cite}target resolves; all 11 numbered directives render correctlyjupytext --to py: clean run, all 7 figures inspected; seeds chosen so the figures match the narrative (sample mean lands near the true value; the first two mixture observations are genuinely ambiguous so the n=2 posterior is visibly bimodal)🤖 Generated with Claude Code