Supplement to "Markov Chain Monte Carlo Based on Deterministic Transformations"
This is a supplement to the article "Markov Chain Monte Carlo Based on Deterministic Transformations" available at http://arxiv.org/abs/1106.5850
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arXiv subjects
Sourabh Bhattacharya
Publications and source records attributed to Sourabh Bhattacharya.
Markov Chain Monte Carlo Based on Deterministic Transformations
In this article we propose a novel MCMC method based on deterministic transformations T: X x D --> X where X is the state-space and D is some set which may or may not be a subset of X. We refer to our new methodology as Transformation-based Markov chain Monte Carlo (TMCMC). One of the remarkable advantages of our proposal is that even if the underlying target distribution is very high-dimensional, deterministic transformation of a one-dimensional random variable is sufficient to generate an appropriate Markov chain that is guaranteed to converge to the high-dimensional target distribution. Apart from clearly leading to massive computational savings, this idea of deterministically transforming a single random variable very generally leads to excellent acceptance rates, even though all the random variables associated with the high-dimensional target distribution are updated in a single block. Since it is well-known that joint updating of many random variables using Metropolis-Hastings (MH) algorithm generally leads to poor acceptance rates, TMCMC, in this regard, seems to provide a significant advance. We validate our proposal theoretically, establishing the convergence properties. Furthermore, we show that TMCMC can be very effectively adopted for simulating from doubly intractable distributions. TMCMC is compared with MH using the well-known Challenger data, demonstrating the effectiveness of of the former in the case of highly correlated variables. Moreover, we apply our methodology to a challenging posterior simulation problem associated with the geostatistical model of Diggle et al. (1998), updating 160 unknown parameters jointly, using a deterministic transformation of a one-dimensional random variable. Remarkable computational savings as well as good convergence properties and acceptance rates are the results.
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Perfect Simulation for Mixtures with Known and Unknown Number of components
We propose and develop a novel and effective perfect sampling methodology for simulating from posteriors corresponding to mixtures with either known (fixed) or unknown number of components. For the latter we consider the Dirichlet process-based mixture model developed by these authors, and show that our ideas are applicable to conjugate, and importantly, to non-conjugate cases. As to be expected, and, as we show, perfect sampling for mixtures with known number of components can be achieved with much less effort with a simplified version of our general methodology, whether or not conjugate or non-conjugate priors are used. While no special assumption is necessary in the conjugate set-up for our theory to work, we require the assumption of bounded parameter space in the non-conjugate set-up. However, we argue, with appropriate analytical, simulation, and real data studies as support, that such boundedness assumption is not unrealistic and is not an impediment in practice. Not only do we validate our ideas theoretically and with simulation studies, but we also consider application of our proposal to three real data sets used by several authors in the past in connection with mixture models. The results we achieved in each of our experiments with either simulation study or real data application, are quite encouraging.
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On Bayesian "central clustering": Application to landscape classification of Western Ghats
Landscape classification of the well-known biodiversity hotspot, Western Ghats (mountains), on the west coast of India, is an important part of a world-wide program of monitoring biodiversity. To this end, a massive vegetation data set, consisting of 51,834 4-variate observations has been clustered into different landscapes by Nagendra and Gadgil [Current Sci. 75 (1998) 264--271]. But a study of such importance may be affected by nonuniqueness of cluster analysis and the lack of methods for quantifying uncertainty of the clusterings obtained. Motivated by this applied problem of much scientific importance, we propose a new methodology for obtaining the global, as well as the local modes of the posterior distribution of clustering, along with the desired credible and "highest posterior density" regions in a nonparametric Bayesian framework. To meet the need of an appropriate metric for computing the distance between any two clusterings, we adopt and provide a much simpler, but accurate modification of the metric proposed in [In Felicitation Volume in Honour of Prof. B. K. Kale (2009) MacMillan]. A very fast and efficient Bayesian methodology, based on [Sankhyā Ser. B 70 (2008) 133--155], has been utilized to solve the computational problems associated with the massive data and to obtain samples from the posterior distribution of clustering on which our proposed methods of summarization are illustrated.
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Switching Strategies for Linear Feedback Stabilization with Sparsified State Measurements
In this paper, we address the problem of stabilization in continuous time linear dynamical systems using state feedback when compressive sampling techniques are used for state measurement and reconstruction. In [5], we had introduced the concept of using l1 reconstruction technique, commonly used in sparse data reconstruction, for state measurement and estimation in a discrete time linear system. In this work, we extend the previous scenario to analyse continuous time linear systems. We investigate the effect of switching within a set of sparsifiers, introduced in [5], on the stability of a linear plant in continuous time settings. Initially, we analyze the problem of stabilization in low dimensional systems, following which we generalize the results to address the problem of stabilization in systems of arbitrary dimensions.
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A nonstationary nonparametric Bayesian approach to dynamically modeling effective connectivity in functional magnetic resonance imaging experiments
Effective connectivity analysis provides an understanding of the functional organization of the brain by studying how activated regions influence one other. We propose a nonparametric Bayesian approach to model effective connectivity assuming a dynamic nonstationary neuronal system. Our approach uses the Dirichlet process to specify an appropriate (most plausible according to our prior beliefs) dynamic model as the "expectation" of a set of plausible models upon which we assign a probability distribution. This addresses model uncertainty associated with dynamic effective connectivity. We derive a Gibbs sampling approach to sample from the joint (and marginal) posterior distributions of the unknowns. Results on simulation experiments demonstrate our model to be flexible and a better candidate in many situations. We also used our approach to analyzing functional Magnetic Resonance Imaging (fMRI) data on a Stroop task: our analysis provided new insight into the mechanism by which an individual brain distinguishes and learns about shapes of objects.
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Adaptive Resource Allocation in Jamming Teams Using Game Theory
In this work, we study the problem of power allocation and adaptive modulation in teams of decision makers. We consider the special case of two teams with each team consisting of two mobile agents. Agents belonging to the same team communicate over wireless ad hoc networks, and they try to split their available power between the tasks of communication and jamming the nodes of the other team. The agents have constraints on their total energy and instantaneous power usage. The cost function adopted is the difference between the rates of erroneously transmitted bits of each team. We model the adaptive modulation problem as a zero-sum matrix game which in turn gives rise to a a continuous kernel game to handle power control. Based on the communications model, we present sufficient conditions on the physical parameters of the agents for the existence of a pure strategy saddle-point equilibrium (PSSPE).
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Power Allocation in Team Jamming Games in Wireless Ad Hoc Networks
In this work, we study the problem of power allocation in teams. Each team consists of two agents who try to split their available power between the tasks of communication and jamming the nodes of the other team. The agents have constraints on their total energy and instantaneous power usage. The cost function is the difference between the rates of erroneously transmitted bits of each team. We model the problem as a zero-sum differential game between the two teams and use {\it{Isaacs'}} approach to obtain the necessary conditions for the optimal trajectories. This leads to a continuous-kernel power allocation game among the players. Based on the communications model, we present sufficient conditions on the physical parameters of the agents for the existence of a pure strategy Nash equilibrium (PSNE). Finally, we present simulation results for the case when the agents are holonomic.
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Fast Bayesian Semiparametric Curve-Fitting and Clustering in Massive Data With Application to Cosmology
Recent technological advances have led to a flood of new data on cosmology rich in information about the formation and evolution of the universe, e.g., the data collected in Sloan Digital Sky Survey (SDSS) for more than 200 million objects. The analyses of such data demand cutting edge statistical technologies. Here, we have used the concept of mixture model within Bayesian semiparametric methodology to fit the regression curve with the bivariate data for the apparent magnitude and redshift for Quasars in SDSS (2007) catalogue. Associated with the mixture modeling is a highly efficient curve-fitting procedure, which is central to the application considered in this paper. Moreover, we adopt a new method for analysing the posterior distribution of clusterings, also generated as a by-product of our methodology. The results of our analysis of the cosmological data clearly indicate the existence of four change points on the regression curve andposssibiltiy of clustering of quasars specially at high redshift. This sheds new light not only on the issue of evolution, existence of acceleration or decceleration and environment around quasars at high redshift but also help us to estimate thecosmological parameters related to acceleration or decceleration.
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