likelihood_bayes#

  • Series: lecture-python.myst

  • File: lectures/likelihood_bayes.md

  • Audit date: 2026-08-26

  • Corpus snapshot: e25fdf2345

  • Categories audited: writing, math, code, figures, references, links, admonitions (JAX out of scope)

  • Overall score: 7.8 / 10

  • Priority: HIGH

Score breakdown#

Category

Score

One-line note

Writing

5.5/10

qe-writing-005 ×4; qe-writing-003 ×3; qe-writing-002 ×4, +2 more.

Math

4/10

qe-math-010 (proposed) ×31; qe-math-015 (proposed) ×4; qe-math-014 (proposed) ×4.

Code

7.5/10

qe-code-001 ×5.

JAX

out of scope

JAX rules target lecture-jax.

Figures

7.5/10

qe-fig-005 ×6; qe-fig-001 ×1.

References

10/10

no mechanical violations detected.

Links

10/10

no mechanical violations detected.

Admonitions

10/10

no mechanical violations detected.

Issues#

Critical#

None found.

High severity#

  • [qe-code-001] (reviewer) — Follow PEP8 unless closer to mathematical notation. Count: 5. Lines: 322, 577, 832, 853, 951. Example: 322 names a parameter l (def update(π, l)), which PEP8 E741 singles out as ambiguous, and uses it again at 326; 577 and 156-157 bind lambdas to names (E731); 832-834, 845-846, 853, 899, 932 and 951-954 index without a space after the comma - np.empty((N,T+1)), π_path[:,0], π_path[:,t], π_path[j,:] (E231) - where the rest of the file writes [i, :]; 853 runs to 98 characters (E501); 951-954 and 370-371 and 410-411 indent continuation lines four spaces from the statement start rather than aligning under the opening bracket (E128); and 848-850 and 854-856 separate top-level defs by one blank line instead of two (E302). Line 844 also recomputes f(w)/g(w) twice in one expression.

  • [qe-fig-005] — Descriptive figure names for cross-referencing. Count: 6. Lines: 50, 874, 896, 929, 948, 1005. Example: code-cell figure without mystnb figure metadata.

  • [qe-math-010 (proposed)] — Blackboard \mathbb{P}, \mathbb{E}, \mathbb{V} with braces. Count: 31. Lines: 202, 217, 218, 226, 229, 230, 233, 247, 288, 300, …. Example: non-blackboard {\rm Prob}.

  • [qe-writing-008] — Remove excessive whitespace between words. Count: 65. Lines: 96, 105, 107, 143, 229, 230, 242, 262, 271, 274, …. Example: 2 spaces.

Medium severity#

  • [qe-math-014 (proposed)] (reviewer) — Braces {…} for events, parentheses (…) for sets. Count: 4. Lines: 247, 288, 446, 669. Example: the lecture states the convention itself and then breaks it. 236-242 defines the events with braces - A = \{q=f\}, B = \{w^{t+1}\} - and says outright “where braces \(\{\cdot\}\) are our shorthand for ‘event’”. Five lines later 247 writes {\rm Prob}(q=f |w^{t+1}) with parentheses around exactly that event, and the same parenthesised form recurs at 288, 300, 446, 472 and 669, plus \pi_0 = {\rm Prob}(q=f|\emptyset) at 217-218. The one place the braces survive is inside {\rm Prob}({a = f | w_0}) at 669, where they are typed inside the parentheses rather than replacing them. Written as \mathbb{P}\{q = f \mid w^{t+1}\} throughout, the notation would match the convention the lecture announces at 242.

  • [qe-math-015 (proposed)] (reviewer) — Lowercase for densities/PMFs, uppercase for CDFs. Count: 4. Lines: 637, 678, 946, 1037. Example: from section 623 onward the lecture switches to uppercase \(F\) and \(G\) for the two objects it has called densities \(f\) and \(g\) for six hundred lines - “successive draws of his wage are drawn from either \(F\) or \(G\)” (637), “the probability that \(w_{t+1}\) is being drawn from distribution \(F\)” (678), “pushes the subjective distribution to draw from \(F\) more frequently” (946), “almost sure that \(w_t\) is drawn from \(F\), or is almost sure it is drawn from \(G\)” (1037) - while the surrounding mathematics keeps using \(f(w_0)\), \(g(w_0)\), \(\ell(w_t) = f(w_t)/g(w_t)\) and the code keeps using F_a, F_b as beta parameters of the density. Neither \(F\) nor \(G\) is ever a CDF anywhere in the lecture, so the uppercase is not the case distinction proposed qe-math-015 (proposed) reserves it for; 669-678 has both forms four lines apart.

  • [qe-writing-002] (reviewer) — Keep writing clear, concise, and valuable. Count: 4. Lines: 331, 467, 618, 640. Example: 331-334 repeats 274-279 almost word for word and says so (“As mentioned above, formula {eq}`eq_Bayeslaw1033` shows the key role that the likelihood ratio process … plays in determining the posterior probability \(\pi_{t+1}\)”), so two consecutive sentences appear twice fifty lines apart; 618-621 states the same forward reference twice in a row (“This topic is taken up in {doc}`mix_model`” then “We explore how to learn the true mixing parameter \(x\) in the exercise of {doc}`mix_model`”); 640 announces its own redundancy (“We’ll review and reiterate and rearrange some formulas that we have encountered above”); and 467 is a single 47-word sentence carrying a negation, an equation reference and a restatement of the alternative timing protocol. The closing section at 1039-1042 also repeats line 46 verbatim.

  • [qe-writing-003] (reviewer) — Maintain logical flow. Count: 3. Lines: 636, 648, 938. Example: 636 introduces “a McCall worker” and wage draws with no transition, three sections into a lecture about a statistician observing \(w_t\) - the worker never appears again except at 657-678 and 1037; 648 and 657 silently change the prior’s index from \(\pi_0\), used from 193 to 612, to \(\pi_{-1}\), and both then coexist for the rest of the lecture while the code keeps calling the same quantity π0 (856, 869, 924), so 973-977 describes a table of \(\pi_{-1}\) values produced by a function whose parameter is π0s; and the histogram at 929-937 for \(\pi_0 = 0.3\) is plotted and then never mentioned, with 940 returning to the \(\pi_0 = 0.5\) ensemble.

  • [qe-writing-005] (reviewer) — Use bold for definitions, italic for emphasis. Count: 4. Lines: 29, 33, 710, 718. Example: the lecture bolds two definitions - likelihood ratio process (109) and recursion / multiplicative decomposition (128-129) - and italicises at least six others: likelihood ratio processes and Bayesian learning in the opening sentence (29), prior and posterior at 33, martingale at 710, bounded martingale at 712, sufficient statistic at 718, martingale convergence theorem at 723. All six are terms being introduced, which the rule assigns to bold; the file’s many other italics (some 142, same and different 339, permanent 451, not 467, before 657, subjective 720) are correct emphasis, so the marker is doing both jobs at once.

Low severity#

  • [qe-fig-001] — Do not set figure size unless necessary. Count: 1. Lines: 538. Example: figsize=.

  • [qe-writing-007] (reviewer) — Use visual elements to enhance understanding. Count: 1. Lines: 971. Example: 971 promises “We’ll use our simulations to generate a histogram of this distribution” and 979-982 delivers a pandas DataFrame instead - eleven rows of \(\pi_{-1}\) against the fraction of paths converging to 0 and to 1. The claim being checked (985: “The fraction of simulations for which \(\pi_t\) had converged to \(1\) is indeed always close to \(\pi_{-1}\)”) is that one column equals the index, which is a 45-degree line: plotting the realized fraction against \(\pi_{-1}\) with the identity line drawn would verify the martingale prediction of 810-812 in one picture, where the table asks the reader to compare eleven pairs of rounded numbers by eye.

Strengths#

  • The lecture states the braces-for-events convention explicitly at 236-242 - A = \{q=f\}, B = \{w^{t+1}\}, “where braces \(\{\cdot\}\) are our shorthand for ‘event’” - which is more than most lectures in the corpus do, and it is exactly the distinction proposed qe-math-014 (proposed) is about.

  • Both Bayes formulas are derived rather than quoted: 229-268 gets the batch form {eq}`eq_Bayeslaw1033` from the product rule, 285-310 gets the recursion {eq}`eq_recur1` the same way, and 426-437 then checks numerically that iterating the recursion reproduces the closed form to within 1e-10 - the algebra is verified in code, not asserted.

  • The martingale argument at 700-812 is carried all the way through: \(E(\pi_t \mid \pi_{t-1}) = \pi_{t-1}\) is computed explicitly, boundedness is noted, the convergence theorem is applied, the only admissible limits are shown to be 0 and 1, and iterated expectations then pins \(\mathbb{P}\{\pi_\infty = 1\} = \pi_{-1}\) - and 968-985 tests that prediction across eleven priors.

  • The mixture section (444-621) is a real misspecification experiment with a full explanation: \(\pi_t \to 1\) under an \(x\)-mixture is displayed (531-559), then attributed to \(KL(m,f) < KL(m,g)\) with the divergences actually computed (572-597), then closed with \(\lim_{z\to\infty} h(z) = 1\) at 606-612 - three steps, none of them hand-waved.

  • The {note} at 480-482 names the object that will explain the coming surprise (a KL divergence with respect to the mixture) before the surprise is shown, and the {note} at 509-511 flags what the same simulated path will be reused for later.

  • The twin-axis figures at 367-381 and 407-421 put the two posterior paths and \(\log L(w^t)\) on one set of axes so the reader can see the prior being overwhelmed, and 386 warns explicitly that the two \(y\) scales differ.

  • 943-946 supplies the mechanism behind the split limits rather than only the picture: early luck of the draw makes the subjective distribution sample from \(F\) more often, which pushes \(\pi_t\) further up - a self-reinforcing loop the reader would not infer from the plot alone.