Reduce repetition of the beta posterior derivation (#912)
* State beta posterior in prob_meaning and reference it from bayes_nonconj
Move the closed-form beta posterior into the main text of prob_meaning
(before the exercise, which now asks the reader to recover it), and
replace the duplicated derivation in bayes_nonconj's "Analytical
posterior" section with a back-reference. Reduces repetition across the
two lectures.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
* Replace Bayesian class with functions in prob_meaning
The class only bundled a few parameters with three short methods; its
state (draws, posterior_list, prior) reads more clearly as plain values
produced by small functions. Refactor into simulate_flips and
form_posterior, update the call sites and the exercise prompt.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
* Replace Frequentist class with functions in prob_meaning
Like the Bayesian class, this only bundled parameters with methods that
stashed results on self. Refactor into simulate_head_counts and
compare_frequencies, and update the comparison loops and exercise prompt
to call them directly.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
* Use binomial coefficient notation in the binomial pmf
Replace the n!/(k!(n-k)!) form with the standard \binom{n}{k} in the
binomial pmf definition (and its restatements) across prob_meaning and
bayes_nonconj.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
* Link to the two videos instead of embedding them
Replace the {youtube} embeds with a bullet list of links in prob_meaning.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
* Link to Wikipedia where the binomial distribution is first defined
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
* Use leading zero in decimals (0.x not .x) in prob_meaning
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
* Tidy exercises: drop 'please', fix stale class refs, indicator notation
- Drop "please" from each exercise prompt in pm_ex1 and pm_ex2.
- Update the remaining "Python class" references (part h prompt and
solution) to match the function-based refactor.
- Define f_k^I with an indicator-function sum instead of words.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
* Use proper subsections instead of bold pseudo-headings
Convert the "Comparison with different theta/n/I" headings and "Code for
answering questions" from bold paragraphs to ### subsections.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
* Replace rng-or-default boilerplate with a seed parameter
Give simulate_flips, simulate_head_counts, and compare_frequencies a
seed=1234 default and build the generator with default_rng(seed),
dropping the `rng = rng or np.random.default_rng()` guard. Call sites
pass seed (seed=i in the comparison loops) instead of constructing and
threading an rng object. Also fixes a stray undefined-rng line left in
simulate_head_counts.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
* Split pm_ex2 into a derivation exercise and a coding exercise
The lead-in promised a derivation that the exercise never asked for.
Now pm_ex2 keeps parts (a) and (b) and adds a new (c) asking the reader
to derive the closed-form Beta(alpha+k, beta+n-k) posterior; the
solution's existing n-flip generalization becomes solution part (c).
The coding problems move to a new exercise pm_ex3 (parts a-f) with its
own solution.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
* Align Overview with delivered content; add synthesis and subsections
(a) Reword the Overview to frame the lecture around the two meanings of
probability (dropping the unfulfilled frequentist-confidence-interval
promise), and add a "Comparing the two interpretations" synthesis
section that contrasts the two answers before the conjugate-prior
discussion.
(c) Add ### subsections to the Bayesian half for parity with the
Frequentist half, and add a caveat explaining the shape of the
"different n" comparison (zero for n<10, peak near n=14).
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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Co-authored-by: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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