The day-one bout: PYMC-Forecast
PyMC Labs released a Bayesian forecasting toolkit on 2026-07-13. This statement was filed the next morning. Four pre-registered hypotheses, including the first challenger arm ever given cross-series information.
Why a Bayesian challenger is different
PYMC-Forecast is batch-refit, posterior-predictive, and lets the user choose the generative model and its priors. That makes it the sharpest test yet of the sandwich claim, because the z stream hands a Bayesian model its two favourite conditions. The marginal is N(0,1) by construction, so priors centred at no structure and unit scale are exactly right rather than guesswork. And z is nearly stationary, so the cost of batch refits, the method's structural weakness in a prequential race, mostly vanishes. The shared model in every arm: a Student-t regression on 8 lagged values, mean-field ADVI, 500 posterior draws, every predictive density the exact posterior-draw mixture.
The four hypotheses, as filed
| hypothesis | result on 226 series | verdict |
|---|---|---|
| H1: raw model vs laplace | laplace wins 82%, +0.45 nats median (p = 5e-23) | the usual story |
| H2: does the sandwich lift it? | sandwiched beats raw on 86%, +0.63 nats median (p = 3e-30) | yes, again |
| H3: convergence to laplace | median absolute gap collapses 0.62 → 0.05 nats; sandwiched wins 65%, +0.04 median (p = 4e-06) | converged, epsilon positive |
| H4: does one pooled fit beat per-series fits? | pooling loses on 72% (p = 2e-11); wins the repeat-heavy stratum by +2.1 nats median | no, except where series share structure |
The positive epsilon
Previous sandwiched challengers converged to laplace minus a hair, except the discrete TabFM head, whose gain was confined to repeat-heavy series. This one finished ahead across the whole universe: 65% of series, +0.04 nats median, under the pre-registered convergence hypothesis. The gain is not spread evenly. It concentrates on repeat-heavy series, +2.1 nats median on the near-martingale repeat stratum, exactly where the discrete TabFM head found structure a week earlier. Three independent detectors have now found predictable structure in the z stream of repeat-heavy series, and it remains a standing library work item. On clean continuous strata laplace still edges the sandwich slightly.
Pooling, the first of its kind
The parade reduces every series to the same coordinate system, which makes thousands of heterogeneous series exchangeable and partial pooling well-posed. The hier arm exploited that: one joint hierarchical fit across all 226 series' z streams, against an identical model fit per series. Same window, same freeze, only the priors differ, so the contrast isolates pooling exactly. Verdict: the shrinkage tax beats the borrowed strength on 72% of series, but pooling wins 79% of the repeat-heavy stratum with a +2.1-nat median. Cross-series information helps precisely where series share an exploitable signature, and nowhere else.
Staleness, priced
The quiet result: a single ADVI fit on pre-test data, frozen with no refits through all 150 test steps, ties laplace overall (wins 54% of series, median gap 0.005 nats). In raw coordinates batch models pay heavily for going stale between refits; in z coordinates there is almost no drift left to track. The sandwich is what makes batch Bayesian machinery usable in an online race.
The verdict
Raw, PYMC-Forecast joins the record like every challenger before it. Sandwiched, it is the first to finish ahead, and its hierarchical machinery asked a question no univariate method can: what do these series know about each other? The answer, on this universe, is measured in nats and concentrated in one stratum. Cooperation beat competition, one day after the package existed.