Second Melbourne-Singapore Probability and Statistics Forum
(4 Jul 2016)

~ Abstracts ~


Coupling, concentration, convergence and cut-off
Andrew Barbour, University of Zurich, Switzerland

Many finite Markov chains -- for example, card shuffling - reach their equilibrium distribution abruptly, when distance is measured using the total variation distance; the time to reach equilibrium is much longer than the width of the window on which the total variation distance between the $n$-step distribution and equilibrium changes from 1 to 0. In this talk, we illustrate how concentration can be used to establish this 'cut-off' phenomenon, using the well-known Bernoulli--Laplace model of gas diffusion.
Joint work with Malawian Luczak, QMUL.

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EM algorithm and Stochastic control
Steven Kou, National University of Singapore

We propose a Monte Carlo simulation based approach, called the dynamic EM algorithm, to solve stochastic control problems. In the special case of just searching for an optimal parameter, the algorithm simply becomes the classical Expectation-Maximization (EM) algorithm in statistics. The new algorithm extends the existing literature as follows: (1) We do not assume any particular dynamics of the stochastic processes such as diffusion or jump diffusions. (2) We show the monotonicity of performance improvement in every iteration, which leads to the convergence results. (3) We focus on finite-time horizon problems, where the optimal policy is not necessarily stationary. Various applications are given, such as real business cycle, stochastic growth, and airline network revenue management. This is a joint work with Paul Glasserman, Xianhua Peng, and Xingbo Xu.

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Polya urns with random immigration
Adrian Röllin, National University of Singapore

We discuss a Polya urn scheme, where besides the classical draw-and-replacement mechanism, balls are added to the urn at times given by a renewal process. The limiting distributions obtained depend uniquely on the immigration distribution with no universality anywhere in sight. This is joint work with Erol Peköz and Nathan Ross.

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Shotgun assembly of labelled graphs
Nathan Ross, University of Melbourne, Australia

We consider the problem of reconstructing graphs or labelled graphs from neighborhoods of given radius. The question of inferring a graph from neighborhoods is a generalization of the DNA shotgun problem where the graph is an interval and the nodes are labelled by A, C, G, T. The graph shotgun problem is motivated in part by applications in neuroscience. We provide some necessary and some sufficient conditions for correct recovery both in combinatorial terms and for some generative models including Ising model on lattices and Erdos-Renyi random graphs. Joint work with Elchanan Mossel.

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Phase transitions in (generalized) exponential random graphs
Mei Yin, University of Denver, USA

The exponential random graph model has been a topic of continued research interest. The past few years especially has witnessed (exponentially) growing attention in exponential models and their variations. Emphasis has been made on the variational principle of the limiting free energy, concentration of the limiting probability distribution, phase transitions, and asymptotic structures. This talk is based on joint work with several collaborators, including Sukhada Fadnavis (Harvard University), Richard Kenyon (Brown University), Charles Radin (University of Texas at Austin), and Alessandro Rinaldo (Carnegie Mellon University), and will focus on the phenomenon of phase transitions in (generalized) exponential random graphs.

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