Bayes factors with (overly) informative priors
Pith reviewed 2026-05-25 08:38 UTC · model grok-4.3
The pith
Priors assuming independence among many parameters make Bayes factors insensitive to data.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Priors in which a large number of parameters are specified to be independent are dangerous; they make it hard to learn from data. This is demonstrated by examples drawn from the literature together with large-sample theory that tracks the behavior of the Bayes factor under these priors.
What carries the argument
Large-sample asymptotic analysis of the Bayes factor, which tracks how the factor behaves when parameters are modeled as independent.
If this is right
- Bayes factors may remain near 1 even when data strongly favor one model over another.
- Increasing the number of independently specified parameters reduces the effective sample size for model comparison.
- Model selection conclusions can become independent of the observed data in the large-sample limit.
- Standard default priors in high-dimensional problems require explicit dependence structure to remain useful for Bayes factor calculations.
Where Pith is reading between the lines
- The result implies that default independent priors should be avoided in any high-dimensional Bayesian model selection task.
- One could test the claim by replacing independent priors with a joint prior that induces dependence and checking whether the Bayes factor then tracks the data.
- The same mechanism may affect posterior predictive checks or marginal likelihood estimates in related Bayesian workflows.
Load-bearing premise
The large-sample asymptotic analysis of the Bayes factor under the stated priors accurately reflects finite-sample behavior and the cited literature examples are representative of typical modeling practice.
What would settle it
A finite-sample simulation or real-data example in which the Bayes factor under independent priors responds strongly to data, contrary to the asymptotic prediction that its value stabilizes away from the data-driven value.
read the original abstract
Priors in which a large number of parameters are specified to be independent are dangerous; they make it hard to learn from data. I present a couple of examples from the literature and work through a bit of large sample theory to show what happens.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript argues that priors specifying a large number of parameters as independent are dangerous because they render Bayes factors insensitive to data, making it difficult to learn from observations. It supports the claim by referencing examples from the literature and developing large-sample asymptotic theory for the behavior of Bayes factors under such priors.
Significance. If the asymptotic analysis is shown to apply under the relevant regularity conditions and the cited examples are representative, the result would provide a useful cautionary note on prior specification for Bayes factor model comparison in high-dimensional settings. The paper correctly identifies a potential mechanism by which independence assumptions across many parameters can dominate the marginal likelihood.
major comments (2)
- [large sample theory development (abstract and main text)] The central claim rests on large-sample asymptotics for the Bayes factor, but the manuscript provides no finite-sample analysis, simulations, or explicit verification that the asymptotic regime governs the behavior in the cited literature examples at typical sample sizes (as flagged in the stress-test note). This is load-bearing because the abstract itself notes the argument is illustrated via 'a bit of large sample theory' without bridging to finite n.
- [large sample theory section] Regularity conditions required for the asymptotic results (e.g., on likelihood smoothness, prior tail behavior, and model identifiability) are not stated, preventing assessment of whether they hold for the literature examples referenced.
minor comments (1)
- The title is somewhat informal; a more precise phrasing such as 'On the effect of independent priors on Bayes factors in high dimensions' would better reflect the content.
Simulated Author's Rebuttal
We thank the referee for the detailed comments. Our manuscript is a short note highlighting a cautionary mechanism rather than a comprehensive study; we respond to each major comment below.
read point-by-point responses
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Referee: [large sample theory development (abstract and main text)] The central claim rests on large-sample asymptotics for the Bayes factor, but the manuscript provides no finite-sample analysis, simulations, or explicit verification that the asymptotic regime governs the behavior in the cited literature examples at typical sample sizes (as flagged in the stress-test note). This is load-bearing because the abstract itself notes the argument is illustrated via 'a bit of large sample theory' without bridging to finite n.
Authors: The manuscript intentionally uses a light asymptotic argument plus literature examples to illustrate the mechanism, consistent with its short-note format and abstract wording. We agree that explicit finite-sample verification or simulations are absent and would strengthen the bridge to practice. In revision we will add a short paragraph discussing the sample sizes appearing in the cited examples and noting that the asymptotic divergence provides qualitative insight even for moderate n; this constitutes a partial revision without adding new simulations. revision: partial
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Referee: [large sample theory section] Regularity conditions required for the asymptotic results (e.g., on likelihood smoothness, prior tail behavior, and model identifiability) are not stated, preventing assessment of whether they hold for the literature examples referenced.
Authors: We accept the point. The asymptotics rely on standard Laplace-type approximations to the marginal likelihood. In the revised manuscript we will insert an explicit paragraph listing the regularity conditions (twice continuous differentiability of the log-likelihood, positive-definite Fisher information, and standard prior tail decay) under which the stated large-sample behavior holds, allowing readers to judge applicability to the examples. revision: yes
Circularity Check
No circularity: claims rest on external literature examples and standard large-sample asymptotics
full rationale
The paper presents examples from the literature and applies standard large-sample theory to illustrate the effect of independent priors on Bayes factors. No load-bearing step reduces by construction to a fitted parameter, self-definition, or a self-citation chain; the asymptotics are invoked as external results rather than derived from the paper's own inputs. The derivation is therefore self-contained against external benchmarks, with no evidence of the enumerated circularity patterns.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Standard large-sample asymptotic theory for Bayes factors applies under the stated prior constructions
discussion (0)
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