[unemployment_shocks] Rigorous LOO derivation + jax-only computation

[jstac]

Tightens the Comparing models with cross-validation → Computational methods section of unemployment_shocks, in response to a review that the importance-sampling argument was too heuristic for an advanced-analysis audience.

What changed

Prose. The derivation of the leave-one-out predictive density is rewritten as a formal assumption / proposition / proof (using sphinx_proof):

• The conditional-independence assumption is stated up front — and plainly flagged as false for the autoregression, adopted for tractability, with consequences deferred to the time-series-structure section.
• The proof is a clean chain of equalities establishing the exact Gelfand identity
  $$\tfrac{1}{p(u_i\mid u_{-i})} = \int \tfrac{1}{p(u_i\mid\theta)},p(\theta\mid u),d\theta,$$
  with explicit integral notation (no E[·]) so the integrating density is visible at every step.
• The harmonic-mean estimator then follows as an ordinary Monte Carlo average; since the integrand has finite mean, the law of large numbers makes the approximation exact with probability one as $S\to\infty$, while a possibly-infinite variance motivates the PSIS refinement that follows.
• Blank-line-per-sentence convention applied throughout.

Code. The pointwise LOO computation now stays entirely in jax (jax.scipy.special.logsumexp, jnp.log, jnp.sqrt) instead of round-tripping through numpy/scipy.

COVID remains excluded: at annual end-of-year frequency the episode is a single 6.7% point with a V-shaped (fast-up-fast-down) recovery, which cuts against the asymmetry thesis rather than illustrating it.

Testing

Ran end-to-end on GPU (jupytext --to py):

• r_hat = 1.00, 0 divergences
• elpd difference 11.8 vs SE 5.7 (~2 SEs)
• effective parameters p = 3.6 (linear), 6.5 (jump)
• all Pareto $\hat k$ < 0.7; jump model ranked first by az.compare
• observed skewness 0.77, inside the posterior-predictive bulk

All numerical claims in the prose match the run.

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Commit: e844b40

📚 Changed Lectures

• unemployment_shocks

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Build Info

• Workflow: Build Project [using jupyter-book]