A Bayesian proof of the lower bound in Talagrand's majorizing measure theorem is obtained by equating Gaussian width to integrated MLE error, bounding integrated MMSE above the Fernique-Talagrand functional, and comparing MLE to Bayes-optimal estimation via a duality argument.
On the subgaussian comparison theorem
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Maximum expected inner product under mutual information constraint equals truncated rate-distortion integral up to multiplicative constants.
Upper and lower bounds are derived for expected soft maxima of Gaussian processes that recover the majorizing measure theorem as temperature goes to zero and produce a finite-size Parisi formula for the Sherrington-Kirkpatrick model.
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A Bayesian Proof and Interpretation of Talagrand's Majorizing Measure Theorem
A Bayesian proof of the lower bound in Talagrand's majorizing measure theorem is obtained by equating Gaussian width to integrated MLE error, bounding integrated MMSE above the Fernique-Talagrand functional, and comparing MLE to Bayes-optimal estimation via a duality argument.
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Two-Sided Bounds for Entropic Optimal Transport via a Rate-Distortion Integral
Maximum expected inner product under mutual information constraint equals truncated rate-distortion integral up to multiplicative constants.
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Upper and Lower Bounds on Expected Soft Maxima of Gaussian Processes
Upper and lower bounds are derived for expected soft maxima of Gaussian processes that recover the majorizing measure theorem as temperature goes to zero and produce a finite-size Parisi formula for the Sherrington-Kirkpatrick model.