An Iterative Nonlinear Filter Based on Posterior Distribution Approximation via Penalized Kullback–Leibler Divergence Minimization
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This letter deals with Gaussian approximation of complicated posterior distribution involved in the Bayesian paradigm for nonlinear dynamic systems. A general formulation for Gaussian approximation is first provided by equivalently representing posterior distribution as a Gaussian one with some constraint via embedding technique. In this work, it is specified as a penalized Kullback–Leibler divergence minimization problem. This minimization is solved for the expected Gaussian approximation by utilizing a pre-selected cubature rule and the conditional gradient method. Then, a novel iterative filter is developed for nonlinear dynamic systems. In addition, it is also proved to be optimal in linear cases and demonstrated to be effective through simulations.Keywords:
Kullback–Leibler divergence
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We introduce a Markov Chain Monte Carlo (MCMC) method that is designed to sample from target distributions with irregular geometry using an adaptive scheme. In cases where targets exhibit non-Gaussian behaviour, we propose that adaption should be regional rather than global. Our algorithm minimizes the information projection component of the Kullback-Leibler (KL) divergence between the proposal and target distributions to encourage proposals that are distributed similarly to the regional geometry of the target. Unlike traditional adaptive MCMC, this procedure rapidly adapts to the geometry of the target's current position as it explores the surrounding space without the need for many preexisting samples. The divergence minimization algorithms are tested on target distributions with irregularly shaped modes and we provide results demonstrating the effectiveness of our methods.
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An approximation of a true, unknown, posterior probability density (pd) representing some real state-space system is presented as Bayesian decision-making among a set of possible descriptions (models). The decision problem is carefully defined on its basic elements and it is shown how it leads to the use of the Kullback-Leibler (KL) divergence [11] for evaluating a loss of information between the unknown posterior pd and its approximation. The resulting algorithm is derived on a general level, allowing specific algorithms to be designed according to a selected class of the probability distributions. A concrete example of the algorithm is proposed for the Gaussian case. An experiment is performed assuming that none of the possible descriptions is precisely identical with the unknown system.
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When using complex Bayesian models to combine information, the checking for consistency of the information being combined is good statistical practice. Here a new method is developed for detecting prior-data conflicts in Bayesian models based on comparing the observed value of a prior to posterior divergence to its distribution under the prior predictive distribution for the data. The divergence measure used in our model check is a measure of how much beliefs have changed from prior to posterior, and can be thought of as a measure of the overall size of a relative belief function. It is shown that the proposed method is intuitive, has desirable properties, can be extended to hierarchical settings, and is related asymptotically to Jeffreys' and reference prior distributions. In the case where calculations are difficult, the use of variational approximations as a way of relieving the computational burden is suggested. The methods are compared in a number of examples with an alternative but closely related approach in the literature based on the prior predictive distribution of a minimal sufficient statistic.
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In Bayesian machine learning, the posterior distribution is typically computationally intractable, hence variational inference is often required. In this approach, an evidence lower bound on the log likelihood of data is maximized during training. Variational Autoencoders (VAE) are one important example where variational inference is utilized. In this tutorial, we derive the variational lower bound loss function of the standard variational autoencoder. We do so in the instance of a gaussian latent prior and gaussian approximate posterior, under which assumptions the Kullback-Leibler term in the variational lower bound has a closed form solution. We derive essentially everything we use along the way; everything from Bayes' theorem to the Kullback-Leibler divergence.
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When using complex Bayesian models to combine information, the checking for consistency of the information being combined is good statistical practice. Here a new method is developed for detecting prior-data conflicts in Bayesian models based on comparing the observed value of a prior to posterior divergence to its distribution under the prior predictive distribution for the data. The divergence measure used in our model check is a measure of how much beliefs have changed from prior to posterior, and can be thought of as a measure of the overall size of a relative belief function. It is shown that the proposed method is intuitive, has desirable properties, can be extended to hierarchical settings, and is related asymptotically to Jeffreys' and reference prior distributions. In the case where calculations are difficult, the use of variational approximations as a way of relieving the computational burden is suggested. The methods are compared in a number of examples with an alternative but closely related approach in the literature based on the prior predictive distribution of a minimal sufficient statistic.
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In the classical form, the Poisson Multi-Bernoulli Mixture (PMBM) filter uses a PMBM density to describe target birth, surviving, and death, which does not model the appearance of spawned targets. Although such a model can handle target birth, surviving, and death well, its performance may degrade when target spawning arises. The reason for this is that the original PMBM filter treats the spawned targets as birth targets, ignoring the surviving targets' information. In this paper, we propose a Kullback–Leibler Divergence (KLD) minimization based derivation for the PMBM prediction step, including target spawning, in which the spawned targets are modeled using a Poisson Point Process (PPP). Furthermore, to improve the computational efficiency, three approximations are used to implement the proposed algorithm, such as the Variational Multi-Bernoulli (VMB) filter, the Measurement-Oriented marginal MeMBer/Poisson (MOMB/P) filter, and the Track-Oriented marginal MeMBer/Poisson (TOMB/P) filter. Finally, simulation results demonstrate the validity of the proposed filter by using the spawning model in these three approximations.
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Abstract Bayesian inference produces a posterior distribution for the parameters and predictions from a mathematical model that can be used to guide the formation of hypotheses; specifically, the posterior may be searched for evidence of alternative model hypotheses, which serves as a starting point for hypothesis formation and model refinement. Previous approaches to search for this evidence are largely qualitative and unsystematic; further, demonstrations of these approaches typically stop at hypothesis formation, leaving the questions they raise unanswered. Here, we introduce a Kullback-Leibler (KL) divergence-based ranking to expedite Bayesian hypothesis formation and investigate the hypotheses it generates, ultimately generating novel, biologically significant insights. Our approach uses KL divergence to rank parameters by how much information they gain from experimental data. Subsequently, rather than searching all model parameters at random, we use this ranking to prioritize examining the posteriors of the parameters that gained the most information from the data for evidence of alternative model hypotheses. We test our approach with two examples, which showcase the ability of our approach to systematically uncover different types of alternative hypothesis evidence. First, we test our KL divergence ranking on an established example of Bayesian hypothesis formation. Our top-ranked parameter matches the one previously identified to produce alternative hypotheses. In the second example, we apply our ranking in a novel study of a computational model of prolactin-induced JAK2-STAT5 signaling, a pathway that mediates beta cell proliferation. Here, we cluster our KL divergence rankings to select only a subset of parameters to examine for qualitative evidence of alternative hypotheses, thereby expediting hypothesis formation. Within this subset, we find a bimodal posterior revealing two possible ranges for the prolactin receptor degradation rate. We go on to refine the model, incorporating new data and determining which degradation rate is most plausible. Overall, we demonstrate that our approach offers a novel quantitative framework for Bayesian hypothesis formation and use it to produce a novel, biologically-significant insight.
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Bayesian inference produces a posterior distribution for the parameters of a mathematical model that can be used to guide the formation of hypotheses; specifically, the posterior may be searched for evidence of alternative model hypotheses, which serves as a starting point for hypothesis formation and model refinement. Previous approaches to search for this evidence are largely qualitative and unsystematic; further, demonstrations of these approaches typically stop at hypothesis formation, leaving the questions they raise unanswered. Here, we introduce a Kullback-Leibler (KL) divergence-based ranking to expedite Bayesian hypothesis formation and investigate the hypotheses it generates, ultimately generating novel, biologically significant insights. Our approach uses KL divergence to rank parameters by how much information they gain from experimental data. Subsequently, rather than searching all model parameters at random, we use this ranking to prioritize examining the posteriors of the parameters that gained the most information from the data for evidence of alternative model hypotheses. We test our approach with two examples, which showcase the ability of our approach to systematically uncover different types of alternative hypothesis evidence. First, we test our KL divergence ranking on an established example of Bayesian hypothesis formation. Our top-ranked parameter matches the one previously identified to produce alternative hypotheses. In the second example, we apply our ranking in a novel study of a computational model of prolactin-induced JAK2-STAT5 signaling, a pathway that mediates beta cell proliferation. Within the top 3 ranked parameters (out of 33), we find a bimodal posterior revealing two possible ranges for the prolactin receptor degradation rate. We go on to refine the model, incorporating new data and determining which degradation rate is most plausible. Overall, while the effectiveness of our approach depends on having a properly formulated prior and on the form of the posterior distribution, we demonstrate that our approach offers a novel and generalizable quantitative framework for Bayesian hypothesis formation and use it to produce a novel, biologically-significant insight into beta cell signaling.
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Alternative hypothesis
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