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Abstract We propose a new approach to deriving quantitative mean field approximations for any probability measure $$P$$ on $$\mathbb {R}^{n}$$ with density proportional to $$e^{f(x)}$$, for $$f$$ strongly concave. We bound the mean field approximation for the log partition function $$\log \int e^{f(x)}dx$$ in terms of $$\sum _{i \neq j}\mathbb {E}_{Q^{*}}|\partial _{ij}f|^{2}$$, for a semi-explicit probability measure $$Q^{*}$$ characterized as the unique mean field optimizer, or equivalently as the minimizer of the relative entropy $$H(\cdot \,|\,P)$$ over product measures. This notably does not involve metric-entropy or gradient-complexity concepts which are common in prior work on nonlinear large deviations. Three implications are discussed, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems. Our arguments are based primarily on functional inequalities and the notion of displacement convexity from optimal transport.
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Hypothetical Frequencies as Approximations
Hájek (Erkenntnis 70(2):211–235, 2009) argues that probabilities cannot be the limits of relative frequencies in counterfactual infinite sequences. I argue for a different understanding of these limits, drawing on Norton’s (Philos Sci 79(2):207–232, 2012) distinction between approximations (inexact descriptions of a target) and idealizations (separate models that bear analogies to the target). Then, I adapt Hájek’s arguments to this new context. These arguments provide excellent reasons not to use hypothetical frequencies as idealizations, but no reason not to use them as approximations.
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- Award ID(s):
- 2043089
- PAR ID:
- 10464375
- Date Published:
- Journal Name:
- Erkenntnis
- ISSN:
- 0165-0106
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10670-022-00584-0
