Approximate computations for binary Markov random fields and their use in Bayesian models

Austad, Haakon Michael; Tjelmeland, Håkon

doi:10.1007/s11222-016-9685-7

Cited by 9 publications

(8 citation statements)

References 27 publications

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“…The first training image, shown in Figure 3(a), is a mortality map for liver and gallbladder cancers for white males between 1950 and 1959 in the eastern United States, analyzed by Riggan et al (1987). This data set is previously considered by Sherman et al (2006), Liang (2010) and Austad and Tjelmeland (2016) using Markov random field models, see also Liang et al (2011). In Figure 3(a) the black (y v = 1) and white (y v = 0) pixels represent counties with high and low cancer mortality rates, respectively.…”

Section: Methodsmentioning

confidence: 99%

A multiple-try Metropolis–Hastings algorithm with tailored proposals

Luo

Tjelmeland

2019

Comput Stat

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We present a new multiple-try Metropolis-Hastings algorithm designed to be especially beneficial when a tailored proposal distribution is available. The algorithm is based on a given acyclic graph G, where one of the nodes in G, k say, contains the current state of the Markov chain and the remaining nodes contain proposed states generated by applying the tailored proposal distribution. The Metropolis-Hastings algorithm alternates between two types of updates. The first update type is using the tailored proposal distribution to generate new states in all nodes in G except in node k. The second update type is generating a new value for k, thereby changing the value of the current state. We evaluate the effectiveness of the proposed scheme in an example with previously defined target and proposal distributions.

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Section: Methodsmentioning

confidence: 99%

A multiple-try Metropolis–Hastings algorithm with tailored proposals

Luo

Tjelmeland

2019

Comput Stat

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“…Still for HMRF, Friel et al (2009) proposed an algorithm for computing the likelihood which relies on the merging of exact computations on small sub-lattices of the original lattice. More recently, Austad & Tjelmeland (2017) applied variable elimination on an approximation of the pseudo Boolean expression of a MRF distribution.…”

Section: The Treewidth To Characterise Variable Elimination Complexitymentioning

confidence: 99%

Exact or approximate inference in graphical models: why the choice is dictated by the treewidth, and how variable elimination can be exploited

Peyrard¹,

Cros²,

Givry³

et al. 2019

Aus NZ J of Statistics

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Probabilistic graphical models offer a powerful framework to account for the dependence structure between variables, which is represented as a graph. However, the dependence between variables may render inference tasks intractable. In this paper, we review techniques exploiting the graph structure for exact inference, borrowed from optimisation and computer science. They are built on the principle of variable elimination whose complexity is dictated in an intricate way by the order in which variables are eliminated. The so-called treewidth of the graph characterises this algorithmic complexity: low-treewidth graphs can be processed efficiently. The first point that we illustrate is therefore the idea that for inference in graphical models, the number of variables is not the limiting factor, and it is worth checking the width of several tree decompositions of the graph before resorting to the approximate method. We show how algorithms providing an upper bound of the treewidth can be exploited to derive a 'good' elimination order enabling to realise exact inference. The second point is that when the treewidth is too large, algorithms for approximate inference linked to the principle of variable elimination, such as loopy belief propagation and variational approaches, can lead to accurate results while being much less time consuming than Monte-Carlo approaches. We illustrate the techniques reviewed in this article on benchmarks of inference problems in genetic linkage analysis and computer vision, as well as on hidden variables restoration in coupled Hidden Markov Models.

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“…In our algorithm for simulating from p(x 1:t |y 1:t ) we combine single site Gibbs updates of each element in x 1:t with a one-block Metropolis-Hastings update of all elements in x 1:t . To get a reasonable acceptance rate for the one-block proposals we adopt the approximation procedure introduced in Austad and Tjelmeland (2017) to obtain a partially ordered Markov model (Cressie and Davidson, 1998) approximation to p(x 1:t |y 1:t ), propose potential new values for x 1:t from this approximate posterior, and accept or reject the proposed values according to the usual Metropolis-Hastings acceptance probability. For each value of t we run the Metropolis-Hastings algorithm for a large number of iterations and discard a burn-in period.…”

Section: Specification Of Simulation Examplementioning

confidence: 99%

Ensemble Updating of Binary State Vectors by Maximising the Expected Number of Unchanged Components

Loe¹,

Tjelmeland²

2019

Petroleum Geostatistics 2019

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In recent years, several ensemble-based filtering methods have been proposed and studied. The main challenge in such procedures is the updating of a prior ensemble to a posterior ensemble at every step of the filtering recursions. In the famous ensemble Kalman filter, the assumption of a linear-Gaussian state space model is introduced in order to overcome this issue, and the prior ensemble is updated with a linear shift closely related to the traditional Kalman filter equations. In the current article, we consider how the ideas underlying the ensemble Kalman filter can be applied when the components of the state vectors are binary variables. While the ensemble Kalman filter relies on Gaussian approximations of the forecast and filtering distributions, we instead use first order Markov chains. To update the prior ensemble, we simulate samples from a distribution constructed such that the expected number of equal components in a prior and posterior state vector is maximised. We demonstrate the performance of our approach in a simulation example inspired by the movement of oil and water in a petroleum reservoir, where also a more naïve updating approach is applied for comparison. Here, we observe that the Frobenius norm of the difference between the estimated and the true marginal filtering probabilities is reduced to the half with our method compared to the naïve approach, indicating that our method is superior. Finally, we discuss how our methodology can be generalised from the binary setting to more complicated situations. arXiv:1904.05107v1 [stat.ME]

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Approximate computations for binary Markov random fields and their use in Bayesian models

Cited by 9 publications

References 27 publications

A multiple-try Metropolis–Hastings algorithm with tailored proposals

A multiple-try Metropolis–Hastings algorithm with tailored proposals

Exact or approximate inference in graphical models: why the choice is dictated by the treewidth, and how variable elimination can be exploited

Ensemble Updating of Binary State Vectors by Maximising the Expected Number of Unchanged Components

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