Scalable Bayesian Inference for the Inverse Temperature of a Hidden Potts Model

Moores, Matthew T.; Nicholls, Geoff K.; Pettitt, Anthony N.; Mengersen, Kerrie

doi:10.1214/18-ba1130

Cited by 22 publications

(18 citation statements)

References 69 publications

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“…in the absence of informative likelihood terms and with moderately strong interaction term λ), while they perform poorly in cases where the likelihood terms {α i } i∈Vn dominate, like the image analysis context considered here (see see e.g. Hurn [1997] and Moores et al [2015b] for more discussion).…”

Section: Sampling Permutationsmentioning

confidence: 99%

Informed Proposals for Local MCMC in Discrete Spaces

Zanella

2019

Journal of the American Statistical Association

109

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There is a lack of methodological results to design efficient Markov chain Monte Carlo (MCMC) algorithms for statistical models with discrete-valued high-dimensional parameters. Motivated by this consideration, we propose a simple framework for the design of informed MCMC proposals (i.e. Metropolis-Hastings proposal distributions that appropriately incorporate local information about the target) which is naturally applicable to both discrete and continuous spaces. We explicitly characterize the class of optimal proposal distributions under this framework, which we refer to as locallybalanced proposals, and prove their Peskun-optimality in high-dimensional regimes. The resulting algorithms are straightforward to implement in discrete spaces and provide orders of magnitude improvements in efficiency compared to alternative MCMC schemes, including discrete versions of Hamiltonian Monte Carlo. Simulations are performed with both simulated and real datasets, including a detailed application to Bayesian record linkage. A direct connection with gradient-based MCMC suggests that locally-balanced proposals may be seen as a natural way to extend the latter to discrete spaces. arXiv:1711.07424v1 [stat.CO]

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Section: Sampling Permutationsmentioning

confidence: 99%

Informed Proposals for Local MCMC in Discrete Spaces

Zanella

2019

Journal of the American Statistical Association

109

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“…We used the Brier score as a tool of comparison between our method and the finite mixture model of Gaussian distributions (a brief overview of this Gaussian mixture model can be found in Appendix B), and the PFAB algorithm illustrated in Moores et al (2020). The Brier score measures the accuracy of probabilistic prediction for a set of mutually exclusive discrete outcomes.…”

Section: Comparison With Other Modelsmentioning

confidence: 99%

“…Potts models have been used in a variety of areas beyond statistical physics, including neuroscience (Roudi et al, 2009) and quantum computing (King et al, 2018). The hidden Potts model, in which the model for the data depends on an unobserved Potts model configuration, has also seen multiple applications, including in medical image processing (Li et al, 2017) and satellite image analysis (Moores et al, 2020).…”

Section: Introductionmentioning

confidence: 99%

“…Many works have been dedicated to developing approximate Bayesian computation (ABC) algorithms for speeding up the computation by assuming an approximate structure of the likelihood of a hidden Potts model that follows certain properties. Recently, Moores et al (2020) have introduced a parametric functional approximate Bayesian (PFAB) algorithm for inference on the temperature parameter in the hidden Potts model, which surpasses the computational efficiency of previous methods by a long margin while preserving accurate inference.…”

Section: Introductionmentioning

confidence: 99%

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Ordered conditional approximation of Potts models

Chakraborty¹,

Katzfuß²,

Guinness³

2021

Preprint

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Potts models, which can be used to analyze dependent observations on a lattice, have seen widespread application in a variety of areas, including statistical mechanics, neuroscience, and quantum computing. To address the intractability of Potts likelihoods for large spatial fields, we propose fast ordered conditional approximations that enable rapid inference for observed and hidden Potts models. Our methods can be used to directly obtain samples from the approximate joint distribution of an entire Potts field. The computational complexity of our approximation methods is linear in the number of spatial locations; in addition, some of the necessary computations are naturally parallel. We illustrate the advantages of our approach using simulated data and a satellite image.

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“…We assume that the mean temporal dynamics follow a first‐order vector autoregressive process:

μ^{A} false(t false) = A μ^{A} false(t - 1 false) + bolda false(t false), bolda false(t false) false| σ_{a}^{2} \tilde{overset} M V N false(0, σ_{a}^{2} boldI false)

t = 2,…, T ,

μ^{A} false(1 false) \sim M V N false(0, σ_{μ_{1}}^{2} I false)

, with

σ_{μ_{1}}^{2}

known, but

σ_{a}^{2}

unknown. The hyper‐parameter for the Potts model is assigned a uniform prior β ∼Unif[0, β crit ], where β crit is an approximation of the phase transition point of the K ‐state Potts model on a 3D regular lattice (Moores, Pettitt & Mengersen, ),

β_{c r i t} = \frac{2}{3} normallog false{\frac{1}{2} false[\sqrt{2} + \sqrt{4 K - 2} false] false} .

Additional priors completing the model specification are as follows:

α_{l} \tilde{overset} Inverse‐Gamma false(a_{α}, b_{α} false), l = 1, 2, … K

A_{i j} \tilde{overset} N false(0, σ_{A}^{2} false), i = 1, …, K - 1, j = 1, …, K - 1

σ_{q}^{2} \sim Inverse‐Gamma false(a_{q ...}

…”

Section: Spatiotemporal Mixture Modelmentioning

confidence: 99%

A Potts‐mixture spatiotemporal joint model for combined magnetoencephalography and electroencephalography data

2019

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We develop a new methodology for determining the location and dynamics of brain activity from combined magnetoencephalography (MEG) and electroencephalography (EEG) data. The resulting inverse problem is ill-posed and is one of the most difficult problems in neuroimaging data analysis.In our development we propose a solution that combines the data from three different modalities, MRI, MEG, and EEG, together. We propose a new Bayesian spatial finite mixture model that builds on the mesostate-space model developed by Daunizeau and Friston (2007). Our new model incorporates two major extensions: (i) We combine EEG and MEG data together and formulate a joint model for dealing with the two modalities simultaneously; (ii) we incorporate the Potts model to represent the spatial dependence in an allocation process that partitions the cortical surface into a small number of latent states termed mesostates. The cortical surface is obtained from MRI. We formulate the new spatiotemporal model and derive an efficient procedure for simultaneous point estimation and model selection based on the iterated conditional modes algorithm combined with * local polynomial smoothing. The proposed method results in a novel estimator for the number of mixture components and is able to select active brain regions which correspond to active variables in a high-dimensional dynamic linear model. The methodology is investigated using synthetic data and simulation studies and then demonstrated on an application examining the neural response to the perception of scrambled faces. R software implementing the methodology along with several sample datasets are available at the following GitHub repository https://github.com/v2south/PottsMix.

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Scalable Bayesian Inference for the Inverse Temperature of a Hidden Potts Model

Cited by 22 publications

References 69 publications

Informed Proposals for Local MCMC in Discrete Spaces

Informed Proposals for Local MCMC in Discrete Spaces

Ordered conditional approximation of Potts models

A Potts‐mixture spatiotemporal joint model for combined magnetoencephalography and electroencephalography data

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