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Can Et Jaynes Probability Theory Explain Bayesian Model Selection?
2025-09-03 06:03:41
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4 Answers
Mason
2025-09-07 04:05:55
Quick, human take: Jaynes lays out the why very cleanly — probability as rational belief and maximum entropy priors give a principled rationale for Bayesian model selection — but he doesn’t magically remove computational and pragmatic headaches.
In model selection you rely on marginal likelihoods (evidence) that integrate the likelihood over the prior, so what Jaynes teaches about priors and information is directly relevant: bad priors distort comparisons. At the same time, getting good evidence estimates needs methods like nested sampling or bridge sampling, and I always recommend complementing evidence with predictive checks or cross-validation to see how models behave on new data. If you’re curious, skim 'Probability Theory: The Logic of Science' for the foundations, then try a small case study with prior predictive simulations — it’s a neat exercise that quickly shows the theory in action.
In model selection you rely on marginal likelihoods (evidence) that integrate the likelihood over the prior, so what Jaynes teaches about priors and information is directly relevant: bad priors distort comparisons. At the same time, getting good evidence estimates needs methods like nested sampling or bridge sampling, and I always recommend complementing evidence with predictive checks or cross-validation to see how models behave on new data. If you’re curious, skim 'Probability Theory: The Logic of Science' for the foundations, then try a small case study with prior predictive simulations — it’s a neat exercise that quickly shows the theory in action.
Holden
2025-09-07 15:05:20
My short thesis: yes, Jaynes’ probability-as-logic framework not only explains Bayesian model selection conceptually, it actually illuminates its philosophical underpinnings in a way that other presentations often gloss over.
Starting from Cox’s axioms and Jaynes’ consequent treatment, probability is the unique consistent extension of Boolean logic to uncertain propositions. When you compare models you are literally updating the plausibility of competing hypotheses. The marginal likelihood emerges naturally because you marginalize over nuisance parameters instead of arbitrarily picking point estimates — this enforces parsimony because integrating over large ineffective parameter volumes reduces evidence. Jaynes’ entropy-based prior construction also connects to the information-theoretic view of model complexity: choosing priors that reflect true prior information avoids spurious penalties or rewards.
That philosophical clarity has practical implications: it argues for careful prior elicitation (maximum entropy when information is limited), for hierarchical models when appropriate, and for preferring predictive checks or model averaging where purely comparative metrics might mislead. I often think of model selection as as much about choosing defensible assumptions as about raw numbers from a Bayes factor.
Starting from Cox’s axioms and Jaynes’ consequent treatment, probability is the unique consistent extension of Boolean logic to uncertain propositions. When you compare models you are literally updating the plausibility of competing hypotheses. The marginal likelihood emerges naturally because you marginalize over nuisance parameters instead of arbitrarily picking point estimates — this enforces parsimony because integrating over large ineffective parameter volumes reduces evidence. Jaynes’ entropy-based prior construction also connects to the information-theoretic view of model complexity: choosing priors that reflect true prior information avoids spurious penalties or rewards.
That philosophical clarity has practical implications: it argues for careful prior elicitation (maximum entropy when information is limited), for hierarchical models when appropriate, and for preferring predictive checks or model averaging where purely comparative metrics might mislead. I often think of model selection as as much about choosing defensible assumptions as about raw numbers from a Bayes factor.
Derek
2025-09-07 16:14:45
I like to boil it down: Jaynes explains why Bayesian model selection works at the level of inference and information, but applying it cleanly needs care. He shows that probabilities are degrees of rational belief and that maximizing entropy gives principled priors; when you compare models you’re really comparing their integrated support for the data, not just best-fit parameters. That integrated support embodies an automatic Occam’s razor: models that waste prior mass on poor fits get penalized.
From a practical angle, Bayes factors and evidence are sensitive to priors and can be computationally gnarly. MCMC samples the posterior but doesn’t directly give the marginal likelihood, so people use techniques like nested sampling, bridge sampling, or the Savage–Dickey ratio when applicable. A pragmatic workflow I follow is: justify priors via maximum entropy ideas, check prior predictive simulations, compute evidence with a robust estimator, and always report sensitivity or use predictive criteria like cross-validation in tandem. That balance—Jaynes’ logic plus modern computation—feels right to me.
From a practical angle, Bayes factors and evidence are sensitive to priors and can be computationally gnarly. MCMC samples the posterior but doesn’t directly give the marginal likelihood, so people use techniques like nested sampling, bridge sampling, or the Savage–Dickey ratio when applicable. A pragmatic workflow I follow is: justify priors via maximum entropy ideas, check prior predictive simulations, compute evidence with a robust estimator, and always report sensitivity or use predictive criteria like cross-validation in tandem. That balance—Jaynes’ logic plus modern computation—feels right to me.
Una
2025-09-09 12:48:23
Totally — Jaynes gives you the conceptual scaffolding to understand Bayesian model selection, and I get excited every time I think about it because it ties logic, information, and probability together so cleanly.
In Jaynes' world probability is extended logic: you assign plausibilities to hypotheses and update them with data using Bayes' theorem. For model selection that means comparing posterior probabilities of different models, which collapses to comparing their marginal likelihoods (a.k.a. evidence) when the prior model probabilities are equal. Jaynes' maximum-entropy arguments also give guidance on constructing priors when you want them to encode only the information you actually have — that’s crucial because the marginal likelihood integrates the likelihood across the prior, and the choice of prior can make or break model comparisons.
That said, Jaynes doesn’t hand you a turnkey computational recipe. The philosophical and information-theoretic explanation is beautiful and powerful, but in practice you still wrestle with marginal likelihood estimation, sensitivity to priors, and paradoxes like Lindley’s. I often pair Jaynes’ book 'Probability Theory: The Logic of Science' with modern computational tools (nested sampling, bridge sampling) and predictive checks so the theory and practice reinforce each other.
In Jaynes' world probability is extended logic: you assign plausibilities to hypotheses and update them with data using Bayes' theorem. For model selection that means comparing posterior probabilities of different models, which collapses to comparing their marginal likelihoods (a.k.a. evidence) when the prior model probabilities are equal. Jaynes' maximum-entropy arguments also give guidance on constructing priors when you want them to encode only the information you actually have — that’s crucial because the marginal likelihood integrates the likelihood across the prior, and the choice of prior can make or break model comparisons.
That said, Jaynes doesn’t hand you a turnkey computational recipe. The philosophical and information-theoretic explanation is beautiful and powerful, but in practice you still wrestle with marginal likelihood estimation, sensitivity to priors, and paradoxes like Lindley’s. I often pair Jaynes’ book 'Probability Theory: The Logic of Science' with modern computational tools (nested sampling, bridge sampling) and predictive checks so the theory and practice reinforce each other.
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