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Stochastic kinetic models (SKMs) are increasingly used to account for the inherent stochasticity exhibited by interacting populations of species in areas such as epidemiology, population ecology and systems biology. Species numbers are modelled using a continuous-time stochastic process, and, depending on the application area of interest, this will typically take the form of a Markov jump process or an It\^o diffusion process. Widespread use of these models is typically precluded by their computational complexity. In particular, performing exact fully Bayesian inference in either modelling framework is challenging due to the intractability of the observed data likelihood, necessitating the use of computationally intensive techniques such as particle Markov chain Monte Carlo (particle MCMC). We propose to increase the computational and statistical efficiency of this approach by leveraging the tractability of an inexpensive surrogate derived directly from either the jump or diffusion process. The surrogate is used in three ways: in the design of a gradient-based parameter proposal, to construct an appropriate bridge construct and in the first stage of a delayed-acceptance step. We find that the resulting approach offers substantial gains in efficiency over a standard particle MCMC implementation.

We introduce a new type of Krasnoselskii's result. Using a simple differentiability condition, we relax the nonexpansive condition in Krasnoselskii's theorem. More clearly, we analyze the convergence of the sequence $x_{n+1}=\frac{x_n+g(x_n)}{2}$ based on some differentiability condition of $g$ and present some fixed point results. We introduce some iterative sequences that for any real differentiable function $g$ and any starting point $x_0\in \mathbb [a,b]$ converge monotonically to the nearest root of $g$ in $[a,b]$ that lay to the right or left side of $x_0$. Based on this approach, we present an efficient and novel method for finding the real roots of real functions. We prove that no root will be missed in our method. It is worth mentioning that our iterative method is free from the derivative evaluation which can be regarded as an advantage of this method in comparison with many other methods. Finally, we illustrate our results with some numerical examples.

We define the information threshold as the point of maximum curvature in the prior vs. posterior Bayesian curve, both of which are described as a function of the true positive and negative rates of the classification system in question. The nature of the threshold is such that for sufficiently adequate binary classification systems, retrieving excess information beyond the threshold does not significantly alter the reliability of our classification assessment. We hereby introduce the "marital status thought experiment" to illustrate this idea and report a previously undefined mathematical relationship between the Bayesian prior and posterior, which may have significant philosophical and epistemological implications in decision theory. Where the prior probability is a scalar between 0 and 1 given by $\phi$ and the posterior is a scalar between 0 and 1 given by $\rho$, then at the information threshold, $\phi_e$: $\phi_e + \rho_e = 1$ Otherwise stated, given some degree of prior belief, we may assert its persuasiveness when sufficient quality evidence yields a posterior so that their combined sum equals 1. Retrieving further evidence beyond this point does not significantly improve the posterior probability, and may serve as a benchmark for confidence in decision-making.

A growing body of research runs human subject evaluations to study whether providing users with explanations of machine learning models can help them with practical real-world use cases. However, running user studies is challenging and costly, and consequently each study typically only evaluates a limited number of different settings, e.g., studies often only evaluate a few arbitrarily selected explanation methods. To address these challenges and aid user study design, we introduce Use-Case-Grounded Simulated Evaluations (SimEvals). SimEvals involve training algorithmic agents that take as input the information content (such as model explanations) that would be presented to each participant in a human subject study, to predict answers to the use case of interest. The algorithmic agent's test set accuracy provides a measure of the predictiveness of the information content for the downstream use case. We run a comprehensive evaluation on three real-world use cases (forward simulation, model debugging, and counterfactual reasoning) to demonstrate that Simevals can effectively identify which explanation methods will help humans for each use case. These results provide evidence that SimEvals can be used to efficiently screen an important set of user study design decisions, e.g. selecting which explanations should be presented to the user, before running a potentially costly user study.

The problem of constrained Markov decision process is considered. An agent aims to maximize the expected accumulated discounted reward subject to multiple constraints on its costs (the number of constraints is relatively small). A new dual approach is proposed with the integration of two ingredients: entropy regularized policy optimizer and Vaidya's dual optimizer, both of which are critical to achieve faster convergence. The finite-time error bound of the proposed approach is provided. Despite the challenge of the nonconcave objective subject to nonconcave constraints, the proposed approach is shown to converge (with linear rate) to the global optimum. The complexity expressed in terms of the optimality gap and the constraint violation significantly improves upon the existing primal-dual approaches.

In this paper, we study the problem of fair sequential decision making with biased linear bandit feedback. At each round, a player selects an action described by a covariate and by a sensitive attribute. The perceived reward is a linear combination of the covariates of the chosen action, but the player only observes a biased evaluation of this reward, depending on the sensitive attribute. To characterize the difficulty of this problem, we design a phased elimination algorithm that corrects the unfair evaluations, and establish upper bounds on its regret. We show that the worst-case regret is smaller than $\mathcal{O}(\kappa_*^{1/3}\log(T)^{1/3}T^{2/3})$, where $\kappa_*$ is an explicit geometrical constant characterizing the difficulty of bias estimation. We prove lower bounds on the worst-case regret for some sets of actions showing that this rate is tight up to a possible sub-logarithmic factor. We also derive gap-dependent upper bounds on the regret, and matching lower bounds for some problem instance.Interestingly, these results reveal a transition between a regime where the problem is as difficult as its unbiased counterpart, and a regime where it can be much harder.

Covariance matrix estimation is a fundamental statistical task in many applications, but the sample covariance matrix is sub-optimal when the sample size is comparable to or less than the number of features. Such high-dimensional settings are common in modern genomics, where covariance matrix estimation is frequently employed as a method for inferring gene networks. To achieve estimation accuracy in these settings, existing methods typically either assume that the population covariance matrix has some particular structure, for example sparsity, or apply shrinkage to better estimate the population eigenvalues. In this paper, we study a new approach to estimating high-dimensional covariance matrices. We first frame covariance matrix estimation as a compound decision problem. This motivates defining a class of decision rules and using a nonparametric empirical Bayes g-modeling approach to estimate the optimal rule in the class. Simulation results and gene network inference in an RNA-seq experiment in mouse show that our approach is comparable to or can outperform a number of state-of-the-art proposals.

Recent studies have shown that heavy tails can emerge in stochastic optimization and that the heaviness of the tails has links to the generalization error. While these studies have shed light on interesting aspects of the generalization behavior in modern settings, they relied on strong topological and statistical regularity assumptions, which are hard to verify in practice. Furthermore, it has been empirically illustrated that the relation between heavy tails and generalization might not always be monotonic in practice, contrary to the conclusions of existing theory. In this study, we establish novel links between the tail behavior and generalization properties of stochastic gradient descent (SGD), through the lens of algorithmic stability. We consider a quadratic optimization problem and use a heavy-tailed stochastic differential equation as a proxy for modeling the heavy-tailed behavior emerging in SGD. We then prove uniform stability bounds, which reveal the following outcomes: (i) Without making any exotic assumptions, we show that SGD will not be stable if the stability is measured with the squared-loss $x\mapsto x^2$, whereas it in turn becomes stable if the stability is instead measured with a surrogate loss $x\mapsto |x|^p$ with some $p<2$. (ii) Depending on the variance of the data, there exists a \emph{`threshold of heavy-tailedness'} such that the generalization error decreases as the tails become heavier, as long as the tails are lighter than this threshold. This suggests that the relation between heavy tails and generalization is not globally monotonic. (iii) We prove matching lower-bounds on uniform stability, implying that our bounds are tight in terms of the heaviness of the tails. We support our theory with synthetic and real neural network experiments.

Despite the plethora of post hoc model explanation methods, the basic properties and behavior of these methods and the conditions under which each one is effective are not well understood. In this work, we bridge these gaps and address a fundamental question: Which explanation method should one use in a given situation? To this end, we adopt a function approximation perspective and formalize the local function approximation (LFA) framework. We show that popular explanation methods are instances of this framework, performing function approximations of the underlying model in different neighborhoods using different loss functions. We introduce a no free lunch theorem for explanation methods which demonstrates that no single method can perform optimally across all neighbourhoods and calls for choosing among methods. To choose among methods, we set forth a guiding principle based on the function approximation perspective, considering a method to be effective if it recovers the underlying model when the model is a member of the explanation function class. Then, we analyze the conditions under which popular explanation methods are effective and provide recommendations for choosing among explanation methods and creating new ones. Lastly, we empirically validate our theoretical results using various real world datasets, model classes, and prediction tasks. By providing a principled mathematical framework which unifies diverse explanation methods, our work characterizes the behaviour of these methods and their relation to one another, guides the choice of explanation methods, and paves the way for the creation of new ones.