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ARXIV:2603.05480 · BAYESIAN LEARNING · SUBMITTED 02 APR · 02:30 UTC · FRESHNESS STALE
ARXIV:2603.05480BAYESIAN LEARNINGSUBMITTED 02 APR · 02:30 UTCFRESHNESS STALEarXiv
Analyze and visualize the structural reorganization of posterior geometry in singular Bayesian models using thermodynamic response functions.
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Pain Analyze and visualize the structural reorganization of posterior geometry in singular Bayesian models using thermodynamic response functions.
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Analyze and visualize the structural reorganization of posterior geometry in singular Bayesian models using thermodynamic response functions. Although singular learning theory characterizes marginal likelihood behavior through invariants such as the real log canonical threshold…
Singular statistical models-including mixtures, matrix factorization, and neural networks-violate regular asymptotics due to parameter non-identifiability and degenerate Fisher geometry. Although singular learning theory characterizes marginal likelihood behavior through invariants such as the real log…
ScienceToStartup currently rates this 5.0/10 on the public viability pass. We show that posterior tempering induces a one-parameter deformation of the posterior distribution whose associated observables generate a hierarchy of thermodynamic response functions.
Bayesian Learning moved forward this cycle; last verified April 2026. Public score 5.0/10.
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Analyze and visualize the structural reorganization of posterior geometry in singular Bayesian models using thermodynamic response functions.
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10.48550/arXiv.2603.05480Analyze and visualize the structural reorganization of posterior geometry in singular Bayesian models using thermodynamic response functions.
Abstract
Singular statistical models-including mixtures, matrix factorization, and neural networks-violate regular asymptotics due to parameter non-identifiability and degenerate Fisher geometry. Although singular learning theory characterizes marginal likelihood behavior through invariants such as the real log canonical threshold and singular fluctuation, these quantities remain difficult to interpret operationally. At the same time, widely used criteria such as WAIC and WBIC appear disconnected from underlying singular geometry. We show that posterior tempering induces a one-parameter deformation of the posterior distribution whose associated observables generate a hierarchy of thermodynamic response functions. A universal covariance identity links derivatives of tempered expectations to posterior fluctuations, placing WAIC, WBIC, and singular fluctuation within a unified response framework. Within this framework, classical quantities from singular learning theory acquire natural thermodynamic interpretations: RLCT governs the leading free-energy slope, singular fluctuation corresponds to curvature of the tempered free energy, and WAIC measures predictive fluctuation. We formalize an observable algebra that quotients out non-identifiable directions, allowing structurally meaningful order parameters to be constructed in singular models. Across canonical singular examples-including symmetric Gaussian mixtures, reduced-rank regression, and overparameterized neural networks-we empirically demonstrate phase-transition-like behavior under tempering. Order parameters collapse, susceptibilities peak, and complexity measures align with structural reorganization in posterior geometry. Our results suggest that thermodynamic response theory provides a natural organizing framework for interpreting complexity, predictive variability, and structural reorganization in singular Bayesian learning.
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PROBLEM
Analyze and visualize the structural reorganization of posterior geometry in singular Bayesian models using thermodynamic response functions. Although singular learning theory characterizes marginal likelihood behavior through invariants such as the real log canonical threshold...
METHOD
Singular statistical models-including mixtures, matrix factorization, and neural networks-violate regular asymptotics due to parameter non-identifiability and degenerate Fisher geometry. Although singular learning theory characterizes marginal likelihood behavior through invaria...
RESULT
ScienceToStartup currently rates this 5.0/10 on the public viability pass. We show that posterior tempering induces a one-parameter deformation of the posterior distribution whose associated observables generate a hierarchy of thermodynamic response functions.
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Bayesian Learning moved forward this cycle; last verified April 2026. Public score 5.0/10.
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Analyze and visualize the structural reorganization of posterior geometry in singular Bayesian models using thermodynamic response functions. Although singular learning theory characterizes marginal likelihood behavior through invariants such as the real log canonical threshold and singular fluctuation, these quantities remain difficult to interpret operationally.
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
partial
Singular statistical models-including mixtures, matrix factorization, and neural networks-violate regular asymptotics due to parameter non-identifiability and degenerate Fisher geometry. Although singular learning theory characterizes marginal likelihood behavior through invariants such as the real log canonical threshold and singular fluctuation, these quantities remain difficult to interpret operationally.
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
partial
ScienceToStartup currently rates this 5.0/10 on the public viability pass. We show that posterior tempering induces a one-parameter deformation of the posterior distribution whose associated observables generate a hierarchy of thermodynamic response functions.
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
partial
Bayesian Learning moved forward this cycle; last verified April 2026. Public score 5.0/10.
Abstract-backed fallback claim; anchored extraction has not materialized a public claim row yet.
partial
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Analyze and visualize the structural reorganization of posterior geometry in singular Bayesian models using thermodynamic response functions.
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