[unemployment_shocks] Rigorous LOO derivation + jax-only computation (#941)

* [unemployment_shocks] Rigorous LOO derivation + jax-only computation

Rewrite the "Computational methods" section as a formal
assumption/proposition/proof: state the conditional-independence
assumption (noting it is false for the time series and revisited
below), prove the exact Gelfand identity for the LOO predictive
density, and obtain the harmonic-mean estimator as a Monte Carlo
average that becomes exact a.s. as S -> infinity. Use explicit
integral notation throughout and follow the blank-line-per-sentence
convention.

Keep the pointwise LOO computation entirely in jax
(jax.scipy.special.logsumexp, jnp.log, jnp.sqrt) instead of bouncing
between jax and numpy.

Verified end-to-end on GPU: r_hat=1.00, 0 divergences, elpd diff 11.8
vs SE 5.7, p_loo 3.6/6.5, all Pareto k < 0.7.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

* [unemployment_shocks] Slim down cross-validation section

Trim the LOO exposition: drop the infinite-variance/importance-weights
caveat and the Pareto-k/PSIS and AIC/BIC discussion (kept consistent,
no dangling references). Remove the posterior-predictive asymmetry
section, and adjust the plan and conclusion so the lecture no longer
promises a check it no longer performs. Fix a few typos.

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>

Author: jstac