The problem of reducing SO2 emissions in Europe is considered. The costs of reduction are assumed to be uncertain and are modeled by a set of possible scenarios. A mean-variance model of the problem is formulated and ...
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The problem of reducing SO2 emissions in Europe is considered. The costs of reduction are assumed to be uncertain and are modeled by a set of possible scenarios. A mean-variance model of the problem is formulated and a specialized computational procedure is developed. The approach is applied to the trans-boundary air pollution model with real-world data.
probabilistic programming languages (PPLs) provide language support for expressing flexible probabilistic models and solving Bayesian inference problems. PPLs with programmable inference make it possible for users to ...
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probabilistic programming languages (PPLs) provide language support for expressing flexible probabilistic models and solving Bayesian inference problems. PPLs with programmable inference make it possible for users to obtain improved results by customizing inference engines using guide programs that are tailored to a corresponding model program. However, errors in guide programs can compromise the statistical soundness of the inference. This article introduces a novel coroutine-based framework for verifying the correctness of user-written guide programs for a broad class of Markov chain Monte Carlo (MCMC) inference algorithms. Our approach rests on a novel type system for describing communication protocols between a model program and a sequence of guides that each update only a subset of random variables. We prove that, by translating guide types to context-free processes with finite norms, it is possible to check structural type equality between models and guides in polynomial time. This connection gives rise to an efficient type-inference algorithm for probabilistic programs with flexible constructs such as general recursion and branching. We also contribute a coverage-checking algorithm that verifies the support of sequentially composed guide programs agrees with that of the model program, which is a key soundness condition for MCMC inference with multiple guides. Evaluations on diverse benchmarks show that our type-inference and coverage-checking algorithms efficiently infer types and detect sound and unsound guides for programs that existing static analyses cannot handle.
The methods of activity analysis (Koopmans [6,7]) are re-examined in the presence of technological uncertainty. In particular, such uncertainty arises when new emerging technologies are employed in the production proc...
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The methods of activity analysis (Koopmans [6,7]) are re-examined in the presence of technological uncertainty. In particular, such uncertainty arises when new emerging technologies are employed in the production process and/or when new product designs are being developed. Both input coefficients and output coefficients may be uncertain. If activity levels are to be determined and fixed a priori, one may not be able to require in advance that total output suffice to cover total demand. (Indeed, demand itself may also be uncertain.) Instead, the requirement is written as a chance-constraint, to hold on some predetermined probability level only. The purpose of the present note is to discuss the economic properties of the resulting optimal solution. The certainty equivalent of the chance-constrained program and the corresponding Kuhn-Tucker conditions are written down. At the point of optimum, each producer will hold some inventories of finished goods as a contingency against variation in the output coefficients and in demand. Equilibrium prices will suffice to provide each activity some expected positive profit (an explicit formula for the calculation of the magnitude of this profit is provided). In choosing between several risky activities, each producer may attempt to establish an optimal portfolio of activities, providing a trade-off between expected cost and risk. The nature of an emerging theory of activity portfolios, developed along the lines of standard concepts in financial portfolio analysis, is indicated.
We use probabilistic program synthesis to solve questions in MIT and Harvard Probability and Statistics courses. Traditional approaches using the latest GPT-3 language model without program synthesis achieve a solve r...
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ISBN:
(纸本)9783031116476;9783031116469
We use probabilistic program synthesis to solve questions in MIT and Harvard Probability and Statistics courses. Traditional approaches using the latest GPT-3 language model without program synthesis achieve a solve rate of 0.2 in these classes. In contrast, by turning course questions into probabilistic programs using the latest program synthesis Transformer, OpenAI Codex, and executing the programs, our solve rates are 0.9 and 0.88, which are on par with human performance.
Decision makers faced with uncertain information often experience regret upon learning that an alternative action would have been preferable to the one actually selected. Models that minimize the maximum regret can be...
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Decision makers faced with uncertain information often experience regret upon learning that an alternative action would have been preferable to the one actually selected. Models that minimize the maximum regret can be useful in such situations, especially when decisions are subject to ex post review. Of particular interest are those decision problems that can be modeled as linear programs with interval objective function coefficients. The minimax regret solution for these formulations can be found using an algorithm that, at each iteration, solves first a linear program to obtain a candidate solution and then a mixed integer program (MIP) to maximize the corresponding regret. The exact solution of the MIP is computationally expensive and becomes impractical as the problem size increases. In this paper, we develop a heuristic for the MTP and investigate its performance both alone and in combination with exact procedures. The heuristic is shown to be effective for problems that are significantly larger than those previously reported in the literature. (C) 1999 Elsevier Science B.V. All rights reserved.
The problem of probabilistic modeling and inference, at a high-level, can be viewed as constructing a (model, query, inference) tuple, where an inference algorithm implements a query on a model. Notably, the derivatio...
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ISBN:
(纸本)9781450349888
The problem of probabilistic modeling and inference, at a high-level, can be viewed as constructing a (model, query, inference) tuple, where an inference algorithm implements a query on a model. Notably, the derivation of inference algorithms can be a difficult and error-prone task. Hence, researchers have explored how ideas from probabilistic prog ramming can be applied. In the context of constructing these tuples, probabilistic programming can be seen as taking a language-based approach to probabilistic modeling and inference. For instance, by using (1) appropriate languages for expressing models and queries and (2) devising inference techniques that operate on encodings of models (and queries) as program expressions, the task of inference can be automated. In this paper, we describe a compiler that transforms a probabilistic model written in a restricted modeling language and a query for posterior samples given observed data into a Markov Chain Monte Carlo (MCMC) inference algorithm that implements the query. The compiler uses a sequence of intermediate languages (ILs) that guide it in gradually and successively refining a declarative specification of a probabilistic model and the query into an executable MCMC inference algorithm. The compilation strategy produces composable MCMC algorithms for execution on a CPU or GPU.
In this paper a probability maximization model of a stochastic linear knapsack problem is considered where the random variables consist of several groups with mutually correlated ones. We propose a solution algorithm ...
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In this paper a probability maximization model of a stochastic linear knapsack problem is considered where the random variables consist of several groups with mutually correlated ones. We propose a solution algorithm to the equivalent nonlinear fractional programming problem with a simple ranking method. This approach will be effectively applied to one of the portfolio selection problems.
We present Zar: a formally verified compiler pipeline from discrete probabilistic programs with unbounded loops in the conditional probabilistic guarded command language (cpGCL) to proved-correct executable samplers i...
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We present Zar: a formally verified compiler pipeline from discrete probabilistic programs with unbounded loops in the conditional probabilistic guarded command language (cpGCL) to proved-correct executable samplers in the random bit model. We exploit the key idea that all discrete probability distributions can be reduced to unbiased coin-flipping schemes. The compiler pipeline first translates cpGCL programs into choice-fix trees, an intermediate representation suitable for reduction of biased probabilistic choices. Choice-fix trees are then translated to coinductive interaction trees for execution within the random bit model. The correctness of the composed translations establishes the sampling equidistribution theorem: compiled samplers are correct wrt. the conditional weakest pre-expectation semantics of cpGCL source programs. Zar is implemented and fully verified in the Coq proof assistant. We extract verified samplers to OCaml and Python and empirically validate them on a number of illustrative examples.
Computing the posterior distribution of a probabilistic program is a hard task for which no one-fit-for-all solution exists. We propose Gaussian Semantics, which approximates the exact probabilistic semantics of a bou...
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Computing the posterior distribution of a probabilistic program is a hard task for which no one-fit-for-all solution exists. We propose Gaussian Semantics, which approximates the exact probabilistic semantics of a bounded program by means of Gaussian mixtures. It is parametrized by a map that associates each program location with the moment order to be matched in the approximation. We provide two main contributions. The first is a universal approximation theorem stating that, under mild conditions, Gaussian Semantics can approximate the exact semantics arbitrarily closely. The second is an approximation that matches up to second-order moments analytically in face of the generally difficult problem of matching moments of Gaussian mixtures with arbitrary moment order. We test our second-order Gaussian approximation (SOGA) on a number of case studies from the literature. We show that it can provide accurate estimates in models not supported by other approximation methods or when exact symbolic techniques fail because of complex expressions or non-simplified integrals. On two notable classes of problems, namely collaborative filtering and programs involving mixtures of continuous and discrete distributions, we show that SOGA significantly outperforms alternative techniques in terms of accuracy and computational time.
An approach to linear programs with random requirements is suggested. The procedure involves choosing actions which minimize the expected value of a certain loss function. These actions are then taken as goals, and op...
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An approach to linear programs with random requirements is suggested. The procedure involves choosing actions which minimize the expected value of a certain loss function. These actions are then taken as goals, and optimal values of the decision variables are found by solving a simple linear goal programming problem. [ABSTRACT FROM AUTHOR]
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