When a Fixed Forecasting Rule Should Become a Distribution

Edited by Avi Yashchin, CEO and Co-Founder · August 25, 2026

A forecasting pipeline that hard-codes its shrinkage constant either over-regresses every subject or under-regresses every subject, since a single mis-set constant pushes all of them in the same direction. A single point projection compounds that problem: it hides how much confidence a business should place in the number before allocating budget, roster spots, or pricing decisions against it. A case study rebuilding a well-known baseball projection system in a Bayesian framework shows what changes when fixed rules become estimated parameters with distributions attached.

The Baseline: MARCEL

MARCEL is a deliberately simple system for forecasting Major League Baseball player performance, developed by Tom Tango. The name is a nod to Marcel the Monkey, and the underlying idea is a bar: whatever a forecaster builds should beat this baseline of three years of historical data, with recent seasons weighted more heavily, regression toward the league mean, and an age adjustment.

MARCEL rests on three hard-coded rules:

1. Fixed recency weights. Each season is weighted 5/4/3: the most recent season counts as 5, the one before as 4, the oldest as 3.
2. Fixed mean regression. A constant total of league-average plate appearances is added to a player's own data, regardless of how much individual data that player has produced.
3. Fixed age adjustment. A linear age curve applies a constant slope before and after an assumed peak age of 29.

None of the three is estimated from the data it forecasts.

Recasting the Same Structure as a Bayesian Model

The original model is deterministic: one point projection per player, no attached uncertainty. A Bayesian version preserves MARCEL's three-component structure while replacing each fixed constant with a parameter the data estimates directly.

Applied to hard hit rate (the share of batted balls hit at an exit velocity above 95 miles per hour, a leading indicator tracked because it correlates with later outcomes like batting average on balls in play). The original MARCEL specification supplies the deterministic baseline this rebuild departs from.

The three substitutions:

• A hierarchical beta-binomial model replaces the fixed regression constant. Instead of forcing every player's average toward the mean by the same fixed amount, the pooling strength scales itself to each player's own data volume. Players with few batted balls get pulled harder toward the league average; players with large samples keep more of their own signal.
• A Dirichlet distribution replaces the fixed 5/4/3 weighting scheme. The model estimates season weights directly from the evidence rather than assuming Tango's ratio is correct for this metric.
• A triangular aging function with an estimated peak age replaces the fixed age curve. Both the slope of the aging effect and the peak age itself become free parameters instead of assumed constants.

What Did the Estimated Parameters Show?

Historical planning example, not a current benchmark: for hard hit rate, the estimated season weights corresponded to roughly a 6/2/1 split, which leaned much harder on the most recent season than the original 5/4/3 scheme did. The estimated peak age for hard hit rate came out near 28 years, close to but not identical to Tango's fixed assumption of 29.

Why Is Convergence Not Optional?

An estimated distribution is only as trustworthy as the sampling process that produced it, but convergence alone does not confirm the model is correctly specified. Before treating any of these parameters as decision-ready, the case study checks two things:

• Energy plots, which show whether the sampler explored the full parameter space or got stuck in part of it.
• R-hat statistics, which compare variance across independent sampling chains; values at or near 1.0 indicate the chains agree on the same distribution.

A model that reports a tight-looking credible interval without passing these checks is not more trustworthy than the fixed-rule baseline it replaced: it is just a distribution nobody has verified.

Where Does This Generalize?

The same three-part structure (weighted past performance, mean regression inversely proportional to sample size, and an age or lifecycle adjustment) applies to any repeated-measurement forecasting problem: customer lifetime value, renewal propensity, demand projection, or performance tracking outside baseball. The specific weights and peak age are properties of hard hit rate; the modeling discipline of replacing an assumed constant with an estimated, checked distribution is the transferable part.

That discipline is the same modeling logic behind how Subconscious's causal behavioral platform reports uncertainty in decision-specific studies. Subconscious runs controlled experiments on simulated markets and returns causal effects with confidence intervals rather than a single deterministic estimate. This is a parallel in modeling philosophy, not a claim that Subconscious runs this baseball model or has replicated this specific analysis. Where a team's decision depends on validating a simulated result, Subconscious can also test or validate studies with real-human participants without changing the underlying causal question. See how we work.

Limitations

This rebuild is a starting point, not a finished system. A triangular aging curve is a simplification: most real aging effects are not perfectly linear before and after a peak, and older subjects in any population are typically survivors (the ones still performing well enough to keep being measured), which biases a naive aging curve unless the model accounts for that selection. The same caveat applies outside baseball: a fixed constant replaced by an estimated distribution is progress, but the distribution still needs domain-specific checks before a business treats it as ground truth. For teams evaluating this kind of shift on their own forecasting pipeline, research covers how Subconscious approaches uncertainty and validation in more depth.