learning_approximation#

  • Series: lecture-python.myst

  • File: lectures/learning_approximation.md

  • Audit date: 2026-08-26

  • Corpus snapshot: e25fdf2345

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

  • Overall score: 7.9 / 10

  • Priority: LOW

Score breakdown#

Category

Score

One-line note

Writing

5.5/10

qe-writing-005 ×4; qe-writing-003 ×2; qe-writing-002 ×3, +1 more.

Math

8/10

qe-math-010 (proposed) ×1; qe-math-009 ×3.

Code

5.5/10

qe-code-001 ×20; qe-code-002 ×6.

JAX

out of scope

JAX rules target lecture-jax.

Figures

6.5/10

qe-fig-003 ×2; qe-fig-005 ×1; qe-fig-004 ×1, +1 more.

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: 20. Lines: 192, 220, 222, 235, 239, 312, 314, 319, 327, 353, …. Example: eighteen lines inside code cells run past 79 characters (PEP8 E501): 192, 220, 235, 239, 314, 319, 327, 353, 386, 460, 473, 582, 594, 701, 752, 753, 754 and 830, with 460 at 93 and 220 at 90 - and in several cases the overrun is a trailing comment (460 pushes # date t-1 Euler realizes now out to column 57) or a nested generator expression that would read better split (220, 239). 222 binds a lambda to a name inside solve_benchmark (E731) where a two-line def would do. Blank-line separation between top-level defs is inconsistent within the same cells: one line at 302-312, two at 316-318, one at 173 after the assignment.

  • [qe-code-002] — Use Unicode symbols for Greek letters in code. Count: 6. Lines: 173, 326, 334, 454, 585, 761. Example: spelled-out psi.

  • [qe-fig-001] — Do not set figure size unless necessary. Count: 5. Lines: 257, 351, 484, 605, 828. Example: figsize=.

  • [qe-math-010 (proposed)] — Blackboard \mathbb{P}, \mathbb{E}, \mathbb{V} with braces. Count: 1. Lines: 112. Example: non-blackboard \operatorname{Prob}.

Medium severity#

  • [qe-fig-003] — No matplotlib embedded titles. Count: 2. Lines: 492, 499. Example: .set_title.

  • [qe-math-009] (reviewer) — Choose simplicity in mathematical notation. Count: 3. Lines: 203, 425, 549. Example: \(R\) is the gross return on currency at 120, 128, 131 and 140 - the object the whole functional equation turns on - and then at 425-431 becomes the second-moment matrix of the regressors in the recursive least squares display, with 430 explaining “\(R_t\) tracks the second moment of the regressors” three hundred lines after \(R_t\) was defined as a return; the code carries the same clash (R = ridge * np.eye(...) at 450 in a lecture whose s_from_psi implements \(f = w_1\psi/(1+\psi)\) derived from returns). \(F\) is the transition kernel \(F(G', G)\) at 112 and a kernel distribution estimate \(\hat F_t(x)\) at 549. And \(T\) is “the \(T\) map” at 203, cited from another lecture and never defined here, while T is the simulation length in every function signature in the file (318, 440, 568, 695).

  • [qe-writing-002] (reviewer) — Keep writing clear, concise, and valuable. Count: 3. Lines: 73, 555, 563. Example: the epigraph at 73 (“Learning algorithms and equilibrium computation algorithms look like each other”) is the opening sentence of the longer Sargent quotation at 520-526, so the same sentence is quoted twice, and the claim is additionally restated in the author’s own words at 81-82, 91, 411-417, 528-531 and 650-654 - seven statements of one thesis. 555-558 is a 46-word sentence carrying a definition, a construction and an iteration scheme. And 560-565 states the batch-versus-recursive correspondence a third time in six lines, after 546-553 and 555 have each stated it.

  • [qe-writing-003] (reviewer) — Maintain logical flow. Count: 2. Lines: 46, 546. Example: “non-parametric” names two different things in the same lecture: at 46 and 92 it is the one-number-per-state rule of the previous lecture, the approach the Overview rejects as hopeless in a continuous state, and at 540-544 it is the kernel smoother offered as the alternative to the parametric route - so the term labels both the method being dismissed and the method being recommended, with no note that the two senses differ. Separately, 546-553 displays a recursion for \(\hat F_t(x)\), a kernel estimate of a distribution, in a section whose subject and whose implementation are the conditional expectation \(\psi(G)\); 563-565 then claims that display “stands to this smoother exactly as {eq}`recursive_pea` stands to the batch algorithm”, but a Nadaraya-Watson conditional mean is not the batch counterpart of a recursive density estimator, so the correspondence the sentence asserts is not the one the display shows.

  • [qe-writing-005] (reviewer) — Use bold for definitions, italic for emphasis. Count: 4. Lines: 91, 111, 362, 840. Example: the lecture uses bold and italic interchangeably for emphasis, sometimes on the same word. Italic is used correctly and often - separate (45), continuous (53), computing (77), realized (288), slow (509), happens (619), whole (787) - but bold does the same job at 91 (“is a learning economy”, a bolded copula), 111 (“a continuous Markov process”, the same word italicised at 53), 362, 368 and 370 (“the constant family”, “linear”, “quadratic”), 415 (“recursively”), 643 (“generalize”), 660 (“discover”), and 840, 843, 845, 848 (“small”, “variance”, “large”, “bias”). Bold is meanwhile doing genuine definition work elsewhere (approximate equilibrium 69, functional equation 87, Nadaraya-Watson 556), so the marker is carrying two jobs at once.

  • [qe-writing-007] (reviewer) — Use visual elements to enhance understanding. Count: 2. Lines: 700, 822. Example: two of the lecture’s central claims are about the shape of a curve that is only ever tabulated. 700-712 asks the reader to see that the sup-norm error “falls at first but then rises again” across degrees 0 to 5 while the ergodic RMS “falls monotonically and then flattens” - two error paths, six points each, presented as a rounded DataFrame - and 822-838 does the same across five bandwidths and then asserts “There is an interior optimum”. Plotting the two error measures against degree, and against \(\log h\), would make both claims visible at a glance and would show the U-shape the bias-variance discussion at 840-852 describes. The lecture is otherwise willing to plot: exercise lae_ex3 even draws the fitted rules at 827-835 without drawing the error curve that is the point of the exercise.

Low severity#

  • [qe-fig-004] — Caption formatting conventions. Count: 1. Lines: 598. Example: caption of 7 words.

  • [qe-fig-005] — Descriptive figure names for cross-referencing. Count: 1. Lines: 827. Example: code-cell figure without mystnb figure metadata.

Strengths#

  • The five-item plan at 84-93 maps one-to-one onto the five H2 sections (105, 198, 273, 409, 533) and every item is actually delivered - including the hardest one, showing that the recursive form of Marcet’s algorithm is a learning economy, which 415-437 and the code at 439-475 carry out rather than assert.

  • The lecture builds ground truth before measuring anything against it (198-262) and then judges every scheme by two numbers, one of which is the theoretically right one: foc_residual (234-242) evaluates the first-order condition under the true kernel, so it is exactly zero at a rational expectations equilibrium and positive at an approximate one - and 391-393 says precisely that.

  • The batch-versus-online comparison is reported honestly: 505-514 states that the online scheme is slower, attributes the residual gap to incomplete convergence rather than to the method, and quantifies the difference (hundreds of thousands of periods against a handful of passes) instead of presenting the two as equivalent.

  • Four of the five figures carry mystnb: figure: caption and name metadata, and every one of them overlays s_benchmark as the same black lw=2.2 line (258, 352, 495, 606, 831), so the four separate comparisons are visually calibrated against each other and against the benchmark figure.

  • The {note} at 395-407 pre-empts the inference a reader would naturally draw from the degree-0/1/2 figure - that richer is always better - explains the mechanism by which it fails with finite data, ties it back to the rare-state problem of the previous lecture, and points to the exercise that verifies it.

  • Exercise lae_ex2 asks a question with a real answer: whether the degree-0 approximate equilibrium coincides with the economy in which the deficit is fixed at its mean. The solution’s Jensen argument at 779-788 yields a conceptual result - “an approximate equilibrium is not a naive certainty-equivalent object” - rather than a number, and lae_ex3 then closes the loop by identifying the bandwidth with the polynomial degree (850-852).

  • Transposes are written ^\top wherever the lecture actually transposes something (279, 425, 426); the three qe-math-002 hits reported against this file are derivative primes and next-period primes.