Mathematical models of physical systems are subject to many uncertainties such as measurement errors and uncertain initial and boundary conditions. After accounting for these uncertainties, it is often revealed that discrepancies between the model output and the observations remain; if so, the model is said to be inadequate. In practice, the inadequate model may be the best that is available or tractable, and so it may be necessary to use the model for prediction despite its inadequacy. In this case, a representation of the inadequacy is necessary, so the impact of the observed discrepancy can be determined. We investigate this problem in the context of chemical kinetics and propose a new technique to account for model inadequacy that is both probabilistic and physically meaningful. A stochastic inadequacy operator S is introduced which is embedded in the ODEs describing the evolution of chemical species concentrations and which respects certain physical constraints such as conservation laws. The parameters of S are governed by probability distributions, which in turn are characterized by a set of hyperparameters. The model parameters and hyperparameters are calibrated using high-dimensional hierarchical Bayesian inference. We apply the method to a typical problem in chemical kinetics-the reaction mechanism of hydrogen combustion.

Mathematical models of epidemiological systems enable investigation of and predictions about potential disease outbreaks. However, commonly used models are often highly simplified representations of incredibly complex systems. Because of these simplifications, the model output, of, say, new cases of a disease over time or when an epidemic will occur, may be inconsistent with the available data. In this case, we must improve the model, especially if we plan to make decisions based on it that could affect human health and safety, but direct improvements are often beyond our reach. In this work, we explore this problem through a case study of the Zika outbreak in Brazil in 2016. We propose an embedded discrepancy operator—a modification to the model equations that requires modest information about the system and is calibrated by all relevant data. We show that the new enriched model demonstrates greatly increased consistency with real data. Moreover, the method is general enough to easily apply to many other mathematical models in epidemiology.

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By treating combinatorial games as dynamical systems, we are able to address a longstanding open question in combinatorial game theory, namely, how the introduction of a "pass" move into a game affects its behavior. We consider two well known combinatorial games, 3-pile Nim and 3-row Chomp. In the case of Nim, we observe that the introduction of the pass dramatically alters the game's underlying structure, rendering it considerably more complex, while for Chomp, the pass move is found to have relatively minimal impact. We show how these results can be understood by recasting these games as dynamical systems describable by dynamical recursion relations. From these recursion relations we are able to identify underlying structural connections between these "games with passes" and a recently introduced class of "generic (perturbed) games." This connection, together with a (non-rigorous) numerical stability analysis, allows one to understand and predict the effect of a pass on a game. arXiv:1204.3222v[math.CO] 14 Apr 2012Combinatorial games like Chess, Checkers, Go, Nim, and Chomp have been the focus of considerable attention in the fields of computer science, mathematics, artificial intelligence, and most recently, chaos and dynamical systems theory. In traditional combinatorial games (under "normal play"), two players alternate moves until one player reaches a terminal position from which no legal move is available, thereupon losing the game1 . An intriguing but surprisingly difficult question in combinatorial game theory centers on what happens when standard game rules are modified so as to allow for a one-time pass -i.e., a pass move which may be used at most once in a game, and not from a terminal position. Once the pass has been used by either player, it is no longer available. Although this question has been raised in various contexts (see, e.g., [1,2]), it touches upon some deep issues relating to the underlying structure and computational complexity of a game, and to date it remains largely unanswered. Indeed, the effect of a pass on even the simplest, most canonical of combinatorial games -Nim -remains an important open question in combinatorial game theory that has defied traditional approaches, and the late mathematician David Gale even offered a monetary prize to the first person to develop a solution for 3-pile Nim with a pass [4]. In this paper we show how tools from dynamical systems theory (wherein we treat "games with passes" as a type of dynamical system) can be used to address such issues.We take up this question of the effects of a pass via two well studied combinatorial games, 3-pile Nim and 3-row Chomp. The first of these games, 3-pile Nim, is a simple combinatorial game which has been fully solved (without the pass); a complete solution was presented byBouton over a century ago [5]. The second, 3-row Chomp (without the pass), is an unsolved, intrinsically more complex combinatorial game [6]. We find that the introduction of a pass has dramatically different effects on these two games. ...

Undirected probabilistic graphical models represent the conditional dependencies, or Markov properties, of a collection of random variables. Knowing the sparsity of such a graphical model is valuable for modeling multivariate distributions and for efficiently performing inference. While the problem of learning graph structure from data has been studied extensively for certain parametric families of distributions, most existing methods fail to consistently recover the graph structure for non-Gaussian data. Here we propose an algorithm for learning the Markov structure of continuous and non-Gaussian distributions. To characterize conditional independence, we introduce a score based on integrated Hessian information from the joint log-density, and we prove that this score upper bounds the conditional mutual information for a general class of distributions. To compute the score, our algorithm sing estimates the density using a deterministic coupling, induced by a triangular transport map, and iteratively exploits sparse structure in the map to reveal sparsity in the graph. For certain non-Gaussian datasets, we show that our algorithm recovers the graph structure even with a biased approximation to the density. Among other examples, we apply sing to learn the dependencies between the states of a chaotic dynamical system with local interactions.