1,548 research outputs found
Nested sampling for Potts models
Nested sampling is a new Monte Carlo method by Skilling [1] intended for general Bayesian computation. Nested sampling provides a robust alternative to annealing-based methods for computing normalizing constants. It can also generate estimates of other quantities such as posterior expectations. The key technical requirement is an ability to draw samples uniformly from the prior subject to a constraint on the likelihood. We provide a demonstration with the Potts model, an undirected graphical model
Mutual information in classical spin models
The total many-body correlations present in finite temperature classical spin
systems are studied using the concept of mutual information. As opposed to
zero-temperature quantum phase transitions, the total correlations are not
maximal at the phase transition, but reach a maximum in the high temperature
paramagnetic phase. The Shannon and Renyi mutual information in both Ising and
Potts models in 2 dimensions are calculated numerically by combining matrix
product states algorithms and Monte Carlo sampling techniques
Joint Image Reconstruction and Segmentation Using the Potts Model
We propose a new algorithmic approach to the non-smooth and non-convex Potts
problem (also called piecewise-constant Mumford-Shah problem) for inverse
imaging problems. We derive a suitable splitting into specific subproblems that
can all be solved efficiently. Our method does not require a priori knowledge
on the gray levels nor on the number of segments of the reconstruction.
Further, it avoids anisotropic artifacts such as geometric staircasing. We
demonstrate the suitability of our method for joint image reconstruction and
segmentation. We focus on Radon data, where we in particular consider limited
data situations. For instance, our method is able to recover all segments of
the Shepp-Logan phantom from angular views only. We illustrate the
practical applicability on a real PET dataset. As further applications, we
consider spherical Radon data as well as blurred data
Bayesian stochastic blockmodeling
This chapter provides a self-contained introduction to the use of Bayesian
inference to extract large-scale modular structures from network data, based on
the stochastic blockmodel (SBM), as well as its degree-corrected and
overlapping generalizations. We focus on nonparametric formulations that allow
their inference in a manner that prevents overfitting, and enables model
selection. We discuss aspects of the choice of priors, in particular how to
avoid underfitting via increased Bayesian hierarchies, and we contrast the task
of sampling network partitions from the posterior distribution with finding the
single point estimate that maximizes it, while describing efficient algorithms
to perform either one. We also show how inferring the SBM can be used to
predict missing and spurious links, and shed light on the fundamental
limitations of the detectability of modular structures in networks.Comment: 44 pages, 16 figures. Code is freely available as part of graph-tool
at https://graph-tool.skewed.de . See also the HOWTO at
https://graph-tool.skewed.de/static/doc/demos/inference/inference.htm
Nesting statistics in the loop model on random planar maps
In the loop model on random planar maps, we study the depth -- in
terms of the number of levels of nesting -- of the loop configuration, by means
of analytic combinatorics. We focus on the `refined' generating series of
pointed disks or cylinders, which keep track of the number of loops separating
the marked point from the boundary (for disks), or the two boundaries (for
cylinders). For the general loop model, we show that these generating
series satisfy functional relations obtained by a modification of those
satisfied by the unrefined generating series. In a more specific model
where loops cross only triangles and have a bending energy, we explicitly
compute the refined generating series. We analyze their non generic critical
behavior in the dense and dilute phases, and obtain the large deviations
function of the nesting distribution, which is expected to be universal. Using
the framework of Liouville quantum gravity (LQG), we show that a rigorous
functional KPZ relation can be applied to the multifractal spectrum of extreme
nesting in the conformal loop ensemble () in the Euclidean
unit disk, as obtained by Miller, Watson and Wilson, or to its natural
generalization to the Riemann sphere. It allows us to recover the large
deviations results obtained for the critical random planar map models.
This offers, at the refined level of large deviations theory, a rigorous check
of the fundamental fact that the universal scaling limits of random planar map
models as weighted by partition functions of critical statistical models are
given by LQG random surfaces decorated by independent CLEs.Comment: 71 pages, 11 figures. v2: minor text and abstract edits, references
adde
Unbiased and Consistent Nested Sampling via Sequential Monte Carlo
We introduce a new class of sequential Monte Carlo methods called Nested
Sampling via Sequential Monte Carlo (NS-SMC), which reframes the Nested
Sampling method of Skilling (2006) in terms of sequential Monte Carlo
techniques. This new framework allows convergence results to be obtained in the
setting when Markov chain Monte Carlo (MCMC) is used to produce new samples. An
additional benefit is that marginal likelihood estimates are unbiased. In
contrast to NS, the analysis of NS-SMC does not require the (unrealistic)
assumption that the simulated samples be independent. As the original NS
algorithm is a special case of NS-SMC, this provides insights as to why NS
seems to produce accurate estimates despite a typical violation of its
assumptions. For applications of NS-SMC, we give advice on tuning MCMC kernels
in an automated manner via a preliminary pilot run, and present a new method
for appropriately choosing the number of MCMC repeats at each iteration.
Finally, a numerical study is conducted where the performance of NS-SMC and
temperature-annealed SMC is compared on several challenging and realistic
problems. MATLAB code for our experiments is made available at
https://github.com/LeahPrice/SMC-NS .Comment: 45 pages, some minor typographical errors fixed since last versio
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