Why the plug-in P50 ≠ the Bayesian P50

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Why the plug-in P50 ≠ the Bayesian P50#

A common surprise when moving from a frequentist to a Bayesian TEXAS reconstruction: inverting the calibration at the posterior-median parameters — the “plug-in” estimate — does not reproduce the P50 of the full Bayesian inverse posterior returned by predict_T_from_proxyObs.

This is expected, not a bug. Three distinct effects separate the two, and which one dominates depends on where the sample sits on the S-curve:

  1. Asymptote breakdown (dominant at the extremes) — the generalized-logistic inverse explodes as the proxy approaches the lower asymptote b or the upper asymptote 1. For a proxy below the median b, the plug-in is literally undefined, while the marginalized posterior stays finite because most ensemble draws have a lower b.

  2. Mixture / nonlinearity (dominant on the steep flanks) — TEXAS marginalizes the likelihood over the forward ensemble. Because the inverse is steeply nonlinear, median_m[inv(y; θ_m)] ≠ inv(y; θ̂). This shifts the median, not just the mean.

  3. Temperature prior (the systematic mid-curve offset) — the truncated-Normal prior on T shrinks the posterior toward its mean. The plug-in uses none of it.

The interactive explainer below lets you vary the proxy value, the prior mean and σ, and the calibration spread, and watch the gap decompose in real time. It grid-integrates the same marginalized likelihood × truncated-Normal prior that invT_gen_logi_fixed_univ_marginal_truncated_prior.stan samples, using an embedded ensemble drawn from the real forward posterior tx.GHPU.sst.sri03.p0.

Note

The sandbox is for building intuition. Production numbers should still come from predict_T_from_proxyObs, which runs the Stan model rather than a grid approximation.

Open the explainer full-page →

Takeaway#

The mismatch you see down-core is worst at the temperature extremes, for two compounding reasons at once: the S-curve flattens, so the inverse becomes ill-conditioned, and the prior pulls hardest exactly where the data are least informative. Near the middle of the calibration range the two estimates agree to within a couple of degrees.

See also#

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