arXiv subjects

Haosui Duanmu

Publications and source records attributed to Haosui Duanmu.

14 recordsLinked to original sources

arXiv2026

Loeb Equivalence for General Internal Probability Spaces

Loeb measure theory stands as one of the most influential concepts in nonstandard analysis, underpinning nearly all applications in probability, stochastic processes, and mathematical economics. The paper resolves a fundamental open problem in Loeb measure theory originally posed by Keisler and Sun: let $(Ω,\mathcal{F},μ)$ and $(Ω,\mathcal{G},ν)$ be two Loeb equivalent internal probability spaces, and $\mathcal H$ be the internal algebra generated from $\mathcal{F}\cup\mathcal{G}$. Does there exist an internal probability measure $P$ on $\mathcal H$ such that $(Ω,\mathcal{H},P)$ is Loeb equivalent to $(Ω,\mathcal{F},μ)$? While arXiv:2112.13955 recently provided a positive answer for hyperfinite probability spaces, the problem remained open for general internal probability spaces. We establish the existence of such an internal probability measure for all internal probability spaces.

math.LO↗

arXiv2026

Existence of Equilibria in Large Competitive Markets with Bads, Production and Comprehensive Externalities

This paper establishes existence of equilibrium in a measure-theoretic general equilibrium (MGE) model with production, bads, and comprehensive externalities. These features are jointly essential for modeling perfect competition in which emissions of production byproducts impose harm on agents. We show that, when bads and externalities are modeled in an economically natural way, equilibrium exists. This yields the first existence theorem with bads for MGE models, the benchmark for perfect competition, overcoming Hara (2005)'s nonexistence example. The proof uses nonstandard analysis, which provides a systematic technique to extend results for finite to infinite models.

econ.TH↗

arXiv2023

General Equilibrium Theory for Climate Change

We propose two general equilibrium models, quota equilibrium and emission tax equilibrium. The government specifies quotas or taxes on emissions, then refrains from further action. Quota equilibrium exists; the allocation of emission property rights strongly impacts the distribution of welfare. If the only externality arises from total net emissions, quota equilibrium is constrained Pareto Optimal. Every quota equilibrium can be realized as an emission tax equilibrium and vice versa. However, for certain tax rates, emission tax equilibrium may not exist, or may exhibit high multiplicity. Full Pareto Optimality of quota equilibrium can often be achieved by setting the right quota.

econ.TH↗

arXiv2023

On Existence of Berk-Nash Equilibria in Misspecified Markov Decision Processes with Infinite Spaces

Model misspecification is a critical issue in many areas of theoretical and empirical economics. In the specific context of misspecified Markov Decision Processes, Esponda and Pouzo (2021) defined the notion of Berk-Nash equilibrium and established its existence in the setting of finite state and action spaces. However, many substantive applications (including two of the three motivating examples presented by Esponda and Pouzo, as well as Gaussian and log-normal distributions, and CARA, CRRA and mean-variance preferences) involve continuous state or action spaces, and are thus not covered by the Esponda-Pouzo existence theorem. We extend the existence of Berk-Nash equilibrium to compact action spaces and sigma-compact state spaces, with possibly unbounded payoff functions. A complication arises because the Berk-Nash equilibrium notion depends critically on Radon-Nikodym derivatives, which are necessarily bounded in the finite case but typically unbounded in misspecified continuous models. The proofs rely on nonstandard analysis and, relative to previous applications of nonstandard analysis in economic theory, draw on novel argumentation traceable to work of the second author on nonstandard representations of Markov processes.

econ.TH↗

arXiv2022

Statistical minimax theorems via nonstandard analysis

For statistical decision problems with finite parameter space, it is well-known that the upper value (minimax value) agrees with the lower value (maximin value). Only under a generalized notion of prior does such an equivalence carry over to the case infinite parameter spaces, provided nature can play a prior distribution and the statistician can play a randomized strategy. Various such extensions of this classical result have been established, but they are subject to technical conditions such as compactness of the parameter space or continuity of the risk functions. Using nonstandard analysis, we prove a minimax theorem for arbitrary statistical decision problems. Informally, we show that for every statistical decision problem, the standard upper value equals the lower value when the $\sup$ is taken over the collection of all internal priors, which may assign infinitesimal probability to (internal) events. Applying our nonstandard minimax theorem, we derive several standard minimax theorems: a minimax theorem on compact parameter space with continuous risk functions, a finitely additive minimax theorem with bounded risk functions and a minimax theorem on totally bounded metric parameter spaces with Lipschitz risk functions.

math.ST↗

arXiv2022

Existence of matching priors on compact spaces

A matching prior at level $1-α$ is a prior such that an associated $1-α$ credible set is also a $1-α$ confidence set. We study the existence of matching priors for general families of credible regions. Our main result gives topological conditions under which matching priors for specific families of credible regions exist. Informally, we prove that, on compact parameter spaces, a matching prior exists if the so-called rejection-probability function is jointly continuous when we adopt the Wasserstein metric on priors. In light of this general result, we observe that typical families of credible regions, such as credible balls, highest-posterior density regions, quantiles, etc., fail to meet this topological condition. We show how to design approximate posterior credible balls and highest-posterior-density regions that meet these topological conditions, yielding matching priors. Finally, we evaluate a numerical scheme for computing approximately matching priors based on discretization and iteration. The proof of our main theorem uses tools from nonstandard analysis and establishes new results about the nonstandard extension of the Wasserstein metric that may be of independent interest.

math.ST↗

arXiv2021

Admissibility is Bayes optimality with infinitesimals

We give an exact characterization of admissibility in statistical decision problems in terms of Bayes optimality in a so-called nonstandard extension of the original decision problem, as introduced by Duanmu and Roy. Unlike the consideration of improper priors or other generalized notions of Bayes optimalitiy, the nonstandard extension is distinguished, in part, by having priors that can assign "infinitesimal" mass in a sense that can be made rigorous using results from nonstandard analysis. With these additional priors, we find that, informally speaking, a decision procedure $δ_0$ is admissible in the original statistical decision problem if and only if, in the nonstandard extension of the problem, the nonstandard extension of $δ_0$ is Bayes optimal among the extensions of standard decision procedures with respect to a nonstandard prior that assigns at least infinitesimal mass to every standard parameter value. We use the above theorem to give further characterizations of admissibility, one related to Blyth's method, one to a condition due to Stein which characterizes admissibility under some regularity assumptions; and finally, a characterization using finitely additive priors in decision problems meeting certain regularity requirements. Our results imply that Blyth's method is a sound and complete method for establishing admissibility. Buoyed by this result, we revisit the univariate two-sample common-mean problem, and show that the Graybill--Deal estimator is admissible among a certain class of unbiased decision procedures.

math.ST↗

arXiv2020

Nonstandard Representation of the Dirichlet Form

The Dirichlet form is a generalization of the Laplacian, heavily used in the study of many diffusion-like processes. In this paper we present a nonstandard representation theorem for the Dirichlet form, showing that the usual Dirichlet form can be well-approximated by a hyperfinite sum. One of the main motivations for such a result is to provide a tool for directly translating results about Dirichlet forms on finite or countable state spaces to results on more general state spaces, without having to translate the details of the proofs. As an application, we prove a generalization of a well-known comparison theorem for Markov chains on finite state spaces, and also relate our results to previous generalization attempts.

math.PR↗

arXiv2020

Finitely-additive, countably-additive and internal probability measures

We discuss two ways to construct standard probability measures, called push-down measures, from internal probability measures. We show that the Wasserstein distance between an internal probability measure and its push-down measure is infinitesimal. As an application to standard probability theory, we show that every finitely-additive Borel probability measure $P$ on a separable metric space is a limit of a sequence of countably-additive Borel probability measures if and only if the space is totally bounded.

math.LO↗

arXiv2020

Loeb Extension and Loeb Equivalence

In Keisler and Sun (2004), the authors raise several open problems on Loeb equivalences between various internal probability spaces. We provide counter-examples for the first two open problems. Moreover, we reduce the third open problem to the following question: Is the internal algebra generated by the union of two Loeb equivalent internal algebras a subset of the Loeb extension of any one of the internal algebra?

math.LO↗

arXiv2020

Drift, Minorization, and Hitting Times

The "drift-and-minorization" method, introduced and popularized in (Rosenthal, 1995; Meyn and Tweedie, 1994; Meyn and Tweedie, 2012), remains the most popular approach for bounding the convergence rates of Markov chains used in statistical computation. This approach requires estimates of two quantities: the rate at which a single copy of the Markov chain "drifts" towards a fixed "small set", and a "minorization condition" which gives the worst-case time for two Markov chains started within the small set to couple with moderately large probability. In this paper, we build on (Oliveira, 2012; Peres and Sousi, 2015) and our work (Anderson, Duanmu, Smith, 2019a; Anderson, Duanmu, Smith, 2019b) to replace the "minorization condition" with an alternative "hitting condition" that is stated in terms of only one Markov chain, and illustrate how this can be used to obtain similar bounds that can be easier to use.

math.PR↗

arXiv2019

Mixing Times and Hitting Times for General Markov Processes

The hitting and mixing times are two fundamental quantities associated with Markov chains. In Peres and Sousi[PS2015] and Oliveira[Oli2012], the authors show that the mixing times and "worst-case" hitting times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on general state spaces that satisfy the strong Feller property. Finally, we show that this asymptotic equivalence can be used to find bounds on the mixing times of a large class of Markov chains used in MCMC, such as typical Gibbs samplers and Metroplis-Hastings chains, even though they usually do not satisfy the strong Feller property.

math.PR↗

arXiv2017

On Extended Admissible Procedures and their Nonstandard Bayes Risk

For finite parameter spaces under finite loss, every Bayes procedure derived from a prior with full support is admissible, and every admissible procedure is Bayes. This relationship already breaks down once we move to finite-dimensional Euclidean parameter spaces. Compactness and strong regularity conditions suffice to repair the relationship, but without these conditions, admissible procedures need not be Bayes. Under strong regularity conditions, admissible procedures can be shown to be the limits of Bayes procedures. Under even stricter conditions, they are generalized Bayes, i.e., they minimize the Bayes risk with respect to an improper prior. In both these cases, one must venture beyond the strict confines of Bayesian analysis. Using methods from mathematical logic and nonstandard analysis, we introduce the class of nonstandard Bayes decision procedures---namely, those whose Bayes risk with respect to some prior is within an infinitesimal of the optimal Bayes risk. Among procedures with finite risk functions, we show that a decision procedure is extended admissible if and only if its nonstandard extension is nonstandard Bayes. For problems with continuous risk functions defined on metric parameter spaces, we derive a nonstandard analogue of Blyth's method that can be used to establish the admissibility of a procedure. We also apply the nonstandard theory to derive a purely standard theorem: when risk functions are continuous on a compact Hausdorff parameter space, a procedure is extended admissible if and only if it is Bayes.

math.ST↗

arXiv2012

Productively Lindelof and Indestructibly Lindelof Spaces

There has recently been considerable interest in productively Lindelof spaces, i.e. spaces such that their product with every Lindelof space is Lindelof. Here we make several related remarks about such spaces. Indestructible Lindelof spaces, i.e. spaces that remain Lindelof in every countably closed forcing extension, were introduced by Tall in 1995. Their connection with topological games and selection principles was explored by Scheepers and Tall in 2010. We find further connections here.

math.GN↗