50,160 research outputs found

    Consistency of maximum likelihood estimation for some dynamical systems

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    We consider the asymptotic consistency of maximum likelihood parameter estimation for dynamical systems observed with noise. Under suitable conditions on the dynamical systems and the observations, we show that maximum likelihood parameter estimation is consistent. Our proof involves ideas from both information theory and dynamical systems. Furthermore, we show how some well-studied properties of dynamical systems imply the general statistical properties related to maximum likelihood estimation. Finally, we exhibit classical families of dynamical systems for which maximum likelihood estimation is consistent. Examples include shifts of finite type with Gibbs measures and Axiom A attractors with SRB measures.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1259 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Analyticity of Entropy Rates of Continuous-State Hidden Markov Models

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    The analyticity of the entropy and relative entropy rates of continuous-state hidden Markov models is studied here. Using the analytic continuation principle and the stability properties of the optimal filter, the analyticity of these rates is shown for analytically parameterized models. The obtained results hold under relatively mild conditions and cover several classes of hidden Markov models met in practice. These results are relevant for several (theoretically and practically) important problems arising in statistical inference, system identification and information theory

    Convergence and Convergence Rate of Stochastic Gradient Search in the Case of Multiple and Non-Isolated Extrema

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    The asymptotic behavior of stochastic gradient algorithms is studied. Relying on results from differential geometry (Lojasiewicz gradient inequality), the single limit-point convergence of the algorithm iterates is demonstrated and relatively tight bounds on the convergence rate are derived. In sharp contrast to the existing asymptotic results, the new results presented here allow the objective function to have multiple and non-isolated minima. The new results also offer new insights into the asymptotic properties of several classes of recursive algorithms which are routinely used in engineering, statistics, machine learning and operations research

    Nonlinear Information Bottleneck

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    Information bottleneck (IB) is a technique for extracting information in one random variable XX that is relevant for predicting another random variable YY. IB works by encoding XX in a compressed "bottleneck" random variable MM from which YY can be accurately decoded. However, finding the optimal bottleneck variable involves a difficult optimization problem, which until recently has been considered for only two limited cases: discrete XX and YY with small state spaces, and continuous XX and YY with a Gaussian joint distribution (in which case optimal encoding and decoding maps are linear). We propose a method for performing IB on arbitrarily-distributed discrete and/or continuous XX and YY, while allowing for nonlinear encoding and decoding maps. Our approach relies on a novel non-parametric upper bound for mutual information. We describe how to implement our method using neural networks. We then show that it achieves better performance than the recently-proposed "variational IB" method on several real-world datasets

    Quick and energy-efficient Bayesian computing of binocular disparity using stochastic digital signals

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    Reconstruction of the tridimensional geometry of a visual scene using the binocular disparity information is an important issue in computer vision and mobile robotics, which can be formulated as a Bayesian inference problem. However, computation of the full disparity distribution with an advanced Bayesian model is usually an intractable problem, and proves computationally challenging even with a simple model. In this paper, we show how probabilistic hardware using distributed memory and alternate representation of data as stochastic bitstreams can solve that problem with high performance and energy efficiency. We put forward a way to express discrete probability distributions using stochastic data representations and perform Bayesian fusion using those representations, and show how that approach can be applied to diparity computation. We evaluate the system using a simulated stochastic implementation and discuss possible hardware implementations of such architectures and their potential for sensorimotor processing and robotics.Comment: Preprint of article submitted for publication in International Journal of Approximate Reasoning and accepted pending minor revision

    Improvements to deep convolutional neural networks for LVCSR

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    Deep Convolutional Neural Networks (CNNs) are more powerful than Deep Neural Networks (DNN), as they are able to better reduce spectral variation in the input signal. This has also been confirmed experimentally, with CNNs showing improvements in word error rate (WER) between 4-12% relative compared to DNNs across a variety of LVCSR tasks. In this paper, we describe different methods to further improve CNN performance. First, we conduct a deep analysis comparing limited weight sharing and full weight sharing with state-of-the-art features. Second, we apply various pooling strategies that have shown improvements in computer vision to an LVCSR speech task. Third, we introduce a method to effectively incorporate speaker adaptation, namely fMLLR, into log-mel features. Fourth, we introduce an effective strategy to use dropout during Hessian-free sequence training. We find that with these improvements, particularly with fMLLR and dropout, we are able to achieve an additional 2-3% relative improvement in WER on a 50-hour Broadcast News task over our previous best CNN baseline. On a larger 400-hour BN task, we find an additional 4-5% relative improvement over our previous best CNN baseline.Comment: 6 pages, 1 figur

    Simulated maximum likelihood for general stochastic volatility models: a change of variable approach

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    Maximum likelihood has proved to be a valuable tool for fitting the log-normal stochastic volatility model to financial returns time series. Using a sequential change of variable framework, we are able to cast more general stochastic volatility models into a form appropriate for importance samplers based on the Laplace approximation. We apply the methodology to two example models, showing that efficient importance samplers can be constructed even for highly non-Gaussian latent processes such as square-root diffusions.Change of Variable; Heston Model; Laplace Importance Sampler; Simulated Maximum Likelihood; Stochastic Volatility
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