Make the posterior-concentration argument visible and concrete#917
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Two fixes to the "Why the posterior concentrates" section of prob_meaning: - The opening referred to "the patterns above", but those plots lived inside the pm_ex3 solution dropdown (collapsed by default). Move the mean/std figure out of the solution into this always-visible section and reword the opening so it stands on its own. - The section gave the generic beta mean/variance in terms of (alpha, beta) — which are the *prior* parameters, so it actually displayed the prior's moments, not the posterior's. Substitute the posterior parameters (alpha+k, beta+n-k) and take the limit explicitly: the mean -> 0.4 since k/n -> 0.4, and the variance ~ theta(1-theta)/n -> 0. Now the stated convergence is shown, not merely asserted. Verified with jupytext export + headless execution. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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- Add a one-line pointer from the pm_ex3 part-f solution to the
"Why the posterior concentrates" section.
- Open that section with a back-reference to the solution of Exercise 3
("In the solution to pm_ex3 we watched ..."), since the concentration
it discusses is shown in a solution box that is collapsed by default.
- Introduce the coverage-interval plot with a lead-in sentence (it
previously followed the mean/std figure with none).
- Replace the 5th/95th-quantile scatter with a box-and-whisker plot
(median, IQR, 5-95% whiskers) built from the analytical posterior
quantiles, with the true theta marked.
Verified with jupytext export + headless execution.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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Addresses two issues in the "Why the posterior concentrates" section of
prob_meaning.md.1. The referenced plots were hidden
The section opened with "How shall we interpret the patterns above?", but the plots it pointed to lived inside the
pm_ex3solution dropdown, which is collapsed by default — so for most readers there was nothing visibly above.Fix: move the mean/std figure out of the solution into this always-visible section, and reword the opening to stand on its own.
2. It showed the prior's moments, not the posterior's
The section gave the generic beta mean
α/(α+β)and variance using the symbolsα, β— but those are the prior parameters, so it literally displayed the prior mean (0.5), never the posterior's. The stated convergence was therefore asserted, not shown.Fix: substitute the posterior parameters
a = α+k,b = β+n−kand take the limit explicitly:(α+k)/(α+β+n) ≈ k/n → 0.4(sincek/n → 0.4by the LLN),≈ θ(1−θ)/n → 0(numerator ~ n², denominator ~ n³).Now a reader can see why the posterior concentrates on the truth.
Verified end-to-end via
jupytext --to py+ headless execution (exit 0).🤖 Generated with Claude Code