navy_captain#

Series: lecture-python.myst
File: lectures/navy_captain.md
Audit date: 2026-08-26
Corpus snapshot: e25fdf2345
Categories audited: writing, math, code, figures, links (JAX out of scope)
Overall score: 6.0 / 10
Priority: HIGH

Score breakdown#

Category	Score	One-line note
Writing	3/10	`qe-writing-006` ×7; `qe-writing-005` ×5; `qe-writing-003` ×6, +4 more.
Math	6.5/10	`qe-math-010` (proposed) ×4; `qe-math-009` ×5.
Code	7.5/10	`qe-code-001` ×7.
JAX	out of scope	JAX rules target `lecture-jax`.
Figures	3/10	`qe-fig-003` ×20; `qe-fig-005` ×19; `qe-fig-006` ×4, +2 more.
References	N/A	no citations in this lecture.
Links	10/10	no mechanical violations detected.
Admonitions	N/A	no admonitions, exercises or solutions.

Issues#

Critical#

None found.

High severity#

[qe-code-001] (reviewer) — Follow PEP8 unless closer to mathematical notation. Count: 7. Lines: 294, 366, 488, 579, 893, 916, 1070. Example: 294 is l = lambda z: wf.f0(z) / wf.f1(z) - the ambiguous single-character name PEP8 singles out in E741 and a lambda binding (E731) in the same line, and l0_arr/l1_arr at 298-299 inherit the name; 488 binds a second lambda, h_func = lambda p: np.interp(p, π_grid, h). Twenty code lines exceed 79 characters (E501): 366 (93), 893 (101), 911, 942, 943, 949, 950, 1015, 1016, 1017, 1040, 1043, 1078, 1079, 1081, 1082, 1099, 1100, 733, 740. 579 omits the space after a comma, np.column_stack([h_star, cost_L0, cost_L1]),axis=1 (E231). 1070 and 1071 both put two spaces after the operator, (1 - π_star) * A / (...) (E222). 916 ends plt.legend(); with a semicolon (E703), the only one in the file. 558-565 wraps np.searchsorted( so that the closing bracket and - 1] sit alone at 30 and 15 columns of indent, and 1021, 1023, 1038, 1043 and 1097 indent continuations under plt.plot( to a flat 8 spaces instead of aligning with the opening bracket (E128). 688, 690 and 734 write B+0.01, A+0.01, π_optimal+0.05 where 582-583 write B + 0.01.
[qe-fig-001] — Do not set figure size unless necessary. Count: 6. Lines: 446, 573, 723, 762, 816, 940. Example: figsize=.
[qe-fig-003] — No matplotlib embedded titles. Count: 20. Lines: 193, 326, 400, 437, 450, 456, 692, 738, 740, 766, …. Example: plt.title.
[qe-fig-005] — Descriptive figure names for cross-referencing. Count: 19. Lines: 186, 284, 309, 394, 434, 445, 539, 681, 720, 750, …. Example: code-cell figure without mystnb figure metadata.
[qe-fig-008] — Use lw=2 for line charts. Count: 31. Lines: 316, 319, 398, 435, 448, 452, 453, 575, 576, 577, …. Example: plot() without lw=.
[qe-math-009] (reviewer) — Choose simplicity in mathematical notation. Count: 5. Lines: 220, 244, 410, 639, 823. Example: \left/\right is used where plain delimiters would do throughout - \left\{ z_{i}\right\} _{i=0}^{t} (80, 220), L\left(z^{t}\right) (218), \bar{V}_{fre}\left(t,d\right) (244, 268, 336), V^{0}\left(\pi\right) (614, 624), \left(1-\pi\right) (633, 639) - and nothing in the file is taller than a one-line fraction. $PFA$ and $PD$ (244, 250, 251, 257, 259, 262, 263, 307, 443) are multi-letter names set in italic math, so they render as the products $P\cdot F\cdot A$ and $P\cdot D$; \mathrm{PFA} would read correctly. 410 puts a code identifier into math as $t_{\rm optimal}$. The same two objects carry two names and two bar commands: $\bar{V}_{fre}$/$\bar{V}_{Bayes}$ in the prose (244, 336, 639, 699, 831) against $\overline{V}_{baye}$ in the figures (737, 740, 766, 802, 819). And 823 labels the lecture’s punchline series label='$diff$', four italic letters in math mode where the prose two lines later (830-831) names it $\bar{V}_{fre}-\bar{V}_{Bayes}$.
[qe-math-010 (proposed)] — Blackboard \mathbb{P}, \mathbb{E}, \mathbb{V} with braces. Count: 4. Lines: 250, 251, 340, 698. Example: non-blackboard \Pr.
[qe-writing-002] (reviewer) — Keep writing clear, concise, and valuable. Count: 5. Lines: 55, 59, 64, 867, 1000. Example: 64 is one 48-word sentence (301 characters) carrying two bolded terms, a spelled-out cross-reference title and both decision rules; 55 is 33 words that reach their point through “proceeded to try to solve”; 59-62 is 45 words that name two lectures as “this lecture {doc}`Exchangeability and Bayesian Updating`” and “this lecture {doc}`Likelihood Ratio Processes`” in one breath, and the full title “A Bayesian Formulation of Friedman and Wald’s Problem” is spelled out six times over the file (42, 64, 98, 229, 466, 473). 867-868 says it twice and ungrammatically: “the frequency distribution of Bayesian times to decide of Bayesian decision maker”. 1000-1003 is 43 words whose “and similarly it equals $1$ minus the optimal probability of a false alarm under $f_0$” attaches “it” to the wrong probability - the sentence starts from the correct decision under $f_1$ and ends by describing the correct decision under $f_0$.
[qe-writing-003] (reviewer) — Maintain logical flow. Count: 6. Lines: 228, 259, 265, 835, 925, 1092. Example: six places where the text does not join up with what surrounds it. 259-260 defines $PD$ as “the probability of a detection error, i.e., not rejecting $H_0$ when $H_1$ is true”, but 251 sets $PD = \Pr\{L(z^t) < d \mid q = f_1\}$ and {eq}`val1` at 244 charges $\bar L_0$ on $(1-PD)$ - so $PD$ is the probability of detection and the sentence describes $1-PD$; a reader checking the equation against the definition is stopped cold. 228-229 says the loss parameters are “$\bar L_1$ and $\bar L_2$” - there is no $\bar L_2$ in the lecture, the pair defined at 89-92 is $\bar L_0$ and $\bar L_1$. 265-266 defers receiver operating characteristic curves to another lecture (“please see this lecture {doc}`Likelihood Ratio Processes`”) and then 305-328 draws them here, forty lines later. The H2 “More Details” at 835 promises “more insights by focusing on the case in which $\pi^*=0.5=\pi_0$” and contains no insight at all - three code cells that set π_star = 0.5, print t_optimal and define t_idx. 925-926 promises “Later we’ll figure out how these distributions ultimately affect objective expected values under the Neyman-Pearson and Bayesian decision rules”, but that comparison was already made at 809-833 and nothing after 925 returns to it. And 1092-1094 says “The next graph plots the unconditional distribution of Bayesian times to decide” when the cell at 1096-1107 plots the unconditional distribution of $\log L$ at the frequentist $t$ - the times-to-decide graph it describes is back at 981-992.
[qe-writing-005] (reviewer) — Use bold for definitions, italic for emphasis. Count: 5. Lines: 64, 306, 845, 875, 876. Example: definitions are bolded correctly - frequentist decision rule (51), type I error (89), type II error (91), false alarm (257), detection error (259), receiver operating characteristic curve (262) - but bold is then reused for plain emphasis in four places the rule assigns to italic: frequentist and Bayesian at 64 (contrastive emphasis, and 51 has already defined the first of them), ex ante at 845, and earlier/later at 875-876, which carry the lecture’s headline result. 306 re-bolds receiver operating characteristic curves as though defining it again, 44 lines after the definition at 262. The one correct italic emphasis in the file is and at 1007, so the two markers are doing each other’s jobs.
[qe-writing-006] — Capitalize lecture titles properly. Count: 7. Lines: 208, 461, 779, 835, 860, 995, 1054. Example: H2 Title Case: ‘Frequentist Decision Rule’ (Decision, Rule).
[qe-writing-008] — Remove excessive whitespace between words. Count: 32. Lines: 46, 51, 64, 70, 80, 86, 91, 94, 95, 102, …. Example: 3 spaces.

Medium severity#

[qe-fig-006] — Lowercase axis labels. Count: 4. Lines: 323, 324, 1028, 1048. Example: axis label Probability of false alarm.
[qe-writing-001] — Use one sentence per paragraph. Count: 1. Lines: 73. Example: 3 sentences in one paragraph.
[qe-writing-007] (reviewer) — Use visual elements to enhance understanding. Count: 3. Lines: 823, 875, 935. Example: the figure that answers the question in the section title “Was the Navy Captain’s Hunch Correct?” (815-828) is the least labelled in the lecture - neither panel gets a title, the left panel’s two series are distinguished only by legend, and the right panel’s sole label is '$diff$' (823). Second, the continuation region $B \leq \pi \leq A$ is the whole content of the Bayesian rule (600-605), but it is only ever shown as two annotations on the value-function plot (582-586) and two vlines at 687-690; the belief paths that actually wander inside it are computed at 935-936 and then immediately collapsed to a mean and a variance (942-956), so the reader never sees one $\pi_t$ path crossing a cutoff even though the data is in hand. Third, the file contains no admonition of any kind, and the two results it exists to deliver - 833 “It is always positive” and 875-876 “the Bayesian rule decides earlier … and later” - are bare paragraphs using bold to do the work a {note} would do.

Low severity#

None found.

Strengths#

The value-function figure at 573-592 makes the cutoff rule readable rather than asserted: it overlays the two terminal payoffs cost_L0 and cost_L1 (570-571, 576-577) on the continuation value, draws their pointwise minimum as a wide translucent band (578-580), and then annotates $A$ and $B$ exactly where the band switches branch (582-586) - so the rule stated at 600-605 can be checked against the picture.
The claim $\pi_0^* = \pi^*$ is established at three increasing strengths instead of once: four worked cases at 720-742, then all 20 values of $\pi^*$ at 750-777 with an explicit 45-degree line drawn for comparison (769-771), then used at 838 to fix the parameterisation of the comparison that follows.
959-971 supplies the mechanism, not just the plot: it points out at 962-964 that the two conditional mean-belief paths lie on top of each other, and then attributes the Bayesian’s longer wait under $f_1$ to the lower posterior variance visible in the right panel of 940-956.
Events are written with braces throughout - $\Pr\{L(z^t)<d \mid q=f_0\}$ (250-251), $\pi^*=\Pr\{\text{nature selects }f_0\}$ (340, 698) - which is exactly what proposed qe-math-014 (proposed) asks for and is not common in this corpus.
The two rules are scored on equal terms: the same $c$, $\bar L_0$, $\bar L_1$ (98-102), the frequentist minimising over $(t,d)$ (330-337) and the Bayesian over $\pi_0$ (701-702), both evaluated on the same simulated likelihood-ratio panels L0_arr/L1_arr (301-302) - which is what makes the comparison at 815-833 mean anything.
97-100 states which parameters were changed from the source lecture and why - “we increase both $\bar L_0$ and $\bar L_1$ from $25$ to $100$ to encourage the Bayesian decision rule to take more draws before deciding” - so the reader can reproduce the difference from wald_friedman_2.

Recommended actions#

Fix the definition of $PD$ at 259-260: 251 and {eq}`val1` make it the probability of detection, and the loss term is $(1-PD)\bar L_0$, so “the probability of a detection error, i.e., not rejecting $H_0$ when $H_1$ is true” describes $1-PD$. In the same pass rename the non-existent $\bar L_2$ at 229 to $\bar L_0$.
Correct 1092-1094 to describe the graph that follows it - the unconditional distribution of $\log L$ at the frequentist $t$, not “the unconditional distribution of Bayesian times to decide”, which is the graph at 981-992.
Label the headline figure at 815-828: give both panels titles and replace label='$diff$' (823) with the difference the prose names at 830-831. Then work through the figure backlog - mystnb: figure: caption/name metadata for the 19 un-named figure cells, the 20 plt.title/set_title calls moved into those captions, the six figsize overrides dropped (446, 573, 723, 762, 816, 940), lw=2 added to the 31 bare plot() calls, and the four capitalised axis labels lowercased (323, 324, 1028, 1048).
Settle one spelling per object: $\bar{V}_{Bayes}$ (639, 699, 831) against $\overline{V}_{baye}$ (737, 740, 766, 802, 819), and \bar against \overline; write $PFA$ and $PD$ as \mathrm{PFA} and \mathrm{PD} so they stop rendering as products; and replace $t_{\rm optimal}$ at 410 with the code name t_optimal in backticks.
Add a figure that shows a handful of $\pi_t$ paths from π0_arr and π1_arr (935-936) against horizontal lines at $A$ and $B$, so the stopping rule of 600-605 is visible; the paths are already computed and currently only their mean and variance are plotted (940-956).
Either deliver the content “More Details” (835-858) promises or fold its three cells into the section that follows, and drop the forward reference at 925-926 whose comparison has already been made at 809-833.
Rename l at 294 (E741) and convert both lambdas (294, 488) to defs; wrap the twenty over-length code lines (366, 733, 740, 893, 911, 942, 943, 949, 950, 1015-1017, 1040, 1043, 1078, 1079, 1081, 1082, 1099, 1100); drop the trailing semicolon at 916, the double operator spacing at 1070-1071 and the missing comma space at 579.
Sweep the mechanical remainder: the 32 double spaces (46, 51, 64, 70, 80, 86, 91, 94, 95, 102 and 22 more), the seven title-case H2 headings (208, 461, 779, 835, 860, 995, 1054), the three-sentence bullet at 72-78, and the missing full stop at 206.