Port worthwhile Bayes and MSE derivation work from rme#1194 and rme#1196

[d-morrison]

Requested by Ezra · project thread

Before: rme#1234 deletes rme's statistics appendices now that sds hosts them. Two open rme PRs, rme#1194 (Bayesian inference subfiles) and rme#1196 (estimation.qmd and proof-mse-bias-variance.qmd), edited those appendices. Their work would be lost along with the files.

After: The parts of those PRs that add something sds lacked are now in sds, rewritten to sds's style (one operation per line with a reason, exercise and solution for derivations, semantic macros, computed rather than hard-coded values). Everything else was checked against sds's current content and left out for the reasons listed under "Skipped".

How

Ported from rme#1194:

• Uniform-prior coin posterior → new #exm-marginal-likelihood-uniform in bayesian-inference.qmd, after #exm-marginal-likelihood. It works out the marginal likelihood as an integral, then the posterior 2θ and its mean 2/3, one step per line. It doesn't name the Beta family, because the Beta family is introduced later on the page.
• Step-by-step normal-normal derivation → #exm-normal-normal. The combined "expanding the square; Σxᵢ = n x̄" step and the combined "adding exponents" and "factoring; definition of m" steps are now split into single operations. The skeptical-prior derivation in sds already had the rme PR's level of detail, so it is unchanged.
• AR(1) Markov chain → new #exm-ar1-chain (states the chain and shows the Markov property) and #exr-ar1-stationary/#sol-ar1-stationary in mcmc.qmd, after #exm-two-state-chain. The rme PR stated the stationary distribution N(0, σ²/(1−ρ²)) without proof. Here it is derived, and the solution shows that no N(0, v) is stationary when |ρ| ≥ 1. A simulation checks the result. The solution adds a \arcoef macro to _macros-sds.qmd.
• Credible-interval interpretation → added to #exm-beta-bernoulli. The endpoints are computed inline: [0.501, 0.699].

Ported in the spirit of rme#1196 (full annotation of the MSE = bias² + variance derivation):

• #sol-bias-sq-plus-var: the single "cancel" step is split into reorder, group, a − a = 0 and a + 0 = a.

Skipped:

• rme#1196's MSE derivation. It adds and subtracts E[θ̂], which sds's derivation rules tell authors to avoid. sds's existing exercise route already has every step annotated.
• rme#1196's annotated "bias = E − truth" proof. #sol-bias-exprs already has the same annotated steps.
• rme#1194's DIC example. Its numbers are hypothetical, and #exm-dic-bernoulli already computes DIC from real posterior draws.
• rme#1194's burn-in example. Its numbers are hypothetical, and #exm-burnin already shows burn-in with simulated chains.
• Slidebreaks, wording churn and \eqdef changes. sds's parent pages already place the slidebreaks. The \eqdef in rme#1194's Markov-chain definition was wrong anyway (see below).

Errors found in the source PRs:

• rme#1194 writes \eqdef in the Markov-chain definition. That equation is a condition, not a definition.
• rme#1194 says Pr(0.50 ≤ π ≤ 0.70 | y) = 0.95 for Beta(56, 37). The equal-tailed interval is [0.501, 0.699]; the rounded interval [0.50, 0.70] has probability 0.953.
• rme#1194's AR(1) example needs ε^(t+1) to be independent of the earlier states, and the PR doesn't say so. It also asserts convergence without a citation.
• rme#1196's MSE derivation writes \eqdef on the step that substitutes Var(θ̂). That equality uses the definition of variance but is not itself a definition. Some of its lines also combine two operations.

Checks:

• estimation.qmd and mcmc.qmd render with no unresolved cross-references.
• bayesian-inference.qmd can't render in full locally because reticulate isn't installed (this predates this PR). I rendered a temporary copy without its Python chunks instead. It had no unresolved references, and I then deleted it.
• lintr reports no lints on the changed files.
• I added antiderivative and exponentials to inst/WORDLIST.
• The [@CaseBerg01, Corollary 4.6.10] locator (linear combinations of independent normals are normal) comes from memory. It is not yet checked against the PDF.

🤖 Generated with Claude Code

https://claude.ai/code/session_01GPxetNyfUNvZoA23DwYCUg

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Generated by Claude Code

[d-morrison]

check-dois failed because publisher DOI endpoints didn't respond: davison1997bootstrap timed out at www.cambridge.org, and `efron1993introduction` returned 503. This PR doesn't touch references.bib, so the failure isn't from its changes, and there's no fix to port. I've re-run the failed job once.

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[github-actions]

Claude finished review — View run

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Commit 6629216 replaces the hardcoded 0.81 and 0.19 in _subfiles/bayesian-inference/_exr-ar1-stationary.qmd:87 with inline R expressions. This confirms the earlier rounds, which ended in Ready for merge. The latest of those is the review of 359a1d5, and it still stands. I did not render the site or run the spell check.

What I checked

• Values: 0.9^2 prints as 0.81, and 1 - 0.9^2 prints as 0.19 with R's default 7 significant digits, which knitr's inline formatting uses. The sentence therefore renders as 1/(1 − 0.9²) = 1/(1 − 0.81) = 1/0.19 ≈ 5.26, the same text as before.
• Syntax: the inline `r ...` spans sit inside $...$ math, as in the neighbouring inline expression. They are balanced and well-formed.
• Scope: only that one line changed. No other file is touched.

Verdict

Ready for merge

Structured Review Data (JSON)

{
  "schema_version": "1.1",
  "reviewer": "claude",
  "commit_sha": "66292168a7c578216b9bd2248895bef42048f645",
  "verdict": "CLEAN",
  "findings": [],
  "detailed_assessment": "The changed line in _exr-ar1-stationary.qmd uses inline R for 0.9^2 and 1-0.9^2, which render as 0.81 and 0.19, matching the prior hardcoded text.",
  "holistic_assessment": "The commit is a one-line, scoped follow-up that leaves the earlier clean verdict intact, with no regression, integration or build-artifact concerns."
}

Reviewed commit: 6629216

[github-actions]

💰 Cost: $0.1639 (review) — run

[d-morrison]

Cross-model adversarial review (OpenCode harness, GPT-6 Luna model) at head 6629216: ALL CLEAR, after fixes for the "approximately 0.95" credible-interval wording, a theorem and proof recording the AR(1) stationary distribution, labelled display equations, and inline-computed intermediate values. The reviewer's suggestion to move the Casella and Berger citation into a Source callout was declined, because it supports one factual step rather than crediting adapted content.

Posted by Claude Code (AI agent) --- not written by a human.

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