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Jeffreys-Lindley Paradox

The result that a frequentist significance test and a Bayesian posterior-odds comparison of the same data against the same point null can reach opposite verdicts, with the disagreement growing without bound as sample size increases — because the two answer different questions.

Core Idea

The Jeffreys-Lindley paradox (Jeffreys 1939; Lindley 1957) is the result that a frequentist significance test and a Bayesian posterior-odds comparison of the same data against the same point-null hypothesis can give opposite verdicts, and that this disagreement grows without bound as sample size increases. With a result just past threshold (p ≈ 0.05), the p-value rejects the null while the Bayes factor favors it, because the diffuse prior on the alternative pays a heavy Occam penalty on concentrated data. The disagreement is structural: the two procedures answer different questions.

Scope of Application

The paradox surfaces in every field doing null-hypothesis testing with large samples, but each imports the same inferential machinery — one substrate.

  • Bayesian statistics — the canonical motivating example for the p-value crisis literature.
  • Frequentist statistics — the standard warning that p ≈ 0.05 at large n carries little weight.
  • Model selection — small point null versus parameter-rich alternative under diffuse priors.
  • Psychology and the replication crisis — many "failed" replications are paradoxical originals.
  • Particle physics — the 5σ threshold as a Lindley-avoidance device.
  • Genomics — multiple-testing corrections and empirical-Bayes shrinkage as responses at scale.

Clarity

Naming the paradox makes legible that statistical significance and evidential weight are not the same quantity and at large n can point opposite ways. It converts prior specification from an invisible default into a consequential decision, sharpens the strict-point-null versus interval-null distinction, and exposes the frequentist-versus-Bayesian choice as a substantive question about which question is being asked.

Manages Complexity

A scatter of separate puzzles — negligible-effect significant trials, "failed" replications, Bayes-versus-BIC disagreements — compresses into one structural signature checked by three parameters: large sample, diffuse alternative prior, rejection just past threshold. When all hold, the divergence is predictable behavior, and sample size alone predicts its direction.

Abstract Reasoning

The paradox licenses a diagnostic move inferring a paradoxical artifact from the joint signature; a predictive monotone-in-n move (more data widens the divergence); interventionist moves on its two levers (tighten the prior, replace the point null with an interval null, pre-register); and a boundary-drawing move treating "frequentist or Bayesian" as a choice of which question is asked.

Knowledge Transfer

Within statistical inference the paradox transfers as the full result — the signature, the monotone-in-n divergence, and the levers carry across biomedicine, physics, genomics, and psychology, all importing the same machinery. It does not transfer outside that substrate: it is a result about inference, not a phenomenon in any domain. What generalizes is the parent incommensurability — two valid procedures answering different questions can diverge, and more data magnifies it.

Relationships to Other Abstractions

Local relationship map for Jeffreys-Lindley ParadoxParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Jeffreys-LindleyParadoxDOMAINPrime abstraction: Bayesian Updating — is part ofBayesianUpdatingPRIMEPrime abstraction: Hypothesis Testing (Null vs. Alternative) — is part ofHypothesis Test…PRIME

Current abstraction Jeffreys-Lindley Paradox Domain-specific

Parents (2) — more general patterns this builds on

  • Jeffreys-Lindley Paradox is part of Bayesian Updating Prime

    The Jeffreys-Lindley construction contains a Bayesian posterior-odds update whose prior-averaged likelihood supplies the opposed verdict.

  • Jeffreys-Lindley Paradox is part of Hypothesis Testing (Null vs. Alternative) Prime

    The Jeffreys-Lindley construction contains a frequentist point-null hypothesis test whose tail-area verdict supplies one side of the disagreement.

Hierarchy paths (10) — routes to 5 parentless roots

Neighborhood in Abstraction Space

Jeffreys-Lindley Paradox sits in a moderately populated region (44th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Statistical Inference & Model Failure Modes (16 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-07-12