501 research outputs found
A statistical test for Nested Sampling algorithms
Nested sampling is an iterative integration procedure that shrinks the prior
volume towards higher likelihoods by removing a "live" point at a time. A
replacement point is drawn uniformly from the prior above an ever-increasing
likelihood threshold. Thus, the problem of drawing from a space above a certain
likelihood value arises naturally in nested sampling, making algorithms that
solve this problem a key ingredient to the nested sampling framework. If the
drawn points are distributed uniformly, the removal of a point shrinks the
volume in a well-understood way, and the integration of nested sampling is
unbiased. In this work, I develop a statistical test to check whether this is
the case. This "Shrinkage Test" is useful to verify nested sampling algorithms
in a controlled environment. I apply the shrinkage test to a test-problem, and
show that some existing algorithms fail to pass it due to over-optimisation. I
then demonstrate that a simple algorithm can be constructed which is robust
against this type of problem. This RADFRIENDS algorithm is, however,
inefficient in comparison to MULTINEST.Comment: 11 pages, 7 figures. Published in Statistics and Computing, Springer,
September 201
Abrupt Motion Tracking via Nearest Neighbor Field Driven Stochastic Sampling
Stochastic sampling based trackers have shown good performance for abrupt
motion tracking so that they have gained popularity in recent years. However,
conventional methods tend to use a two-stage sampling paradigm, in which the
search space needs to be uniformly explored with an inefficient preliminary
sampling phase. In this paper, we propose a novel sampling-based method in the
Bayesian filtering framework to address the problem. Within the framework,
nearest neighbor field estimation is utilized to compute the importance
proposal probabilities, which guide the Markov chain search towards promising
regions and thus enhance the sampling efficiency; given the motion priors, a
smoothing stochastic sampling Monte Carlo algorithm is proposed to approximate
the posterior distribution through a smoothing weight-updating scheme.
Moreover, to track the abrupt and the smooth motions simultaneously, we develop
an abrupt-motion detection scheme which can discover the presence of abrupt
motions during online tracking. Extensive experiments on challenging image
sequences demonstrate the effectiveness and the robustness of our algorithm in
handling the abrupt motions.Comment: submitted to Elsevier Neurocomputin
Ensemble Transport Adaptive Importance Sampling
Markov chain Monte Carlo methods are a powerful and commonly used family of
numerical methods for sampling from complex probability distributions. As
applications of these methods increase in size and complexity, the need for
efficient methods increases. In this paper, we present a particle ensemble
algorithm. At each iteration, an importance sampling proposal distribution is
formed using an ensemble of particles. A stratified sample is taken from this
distribution and weighted under the posterior, a state-of-the-art ensemble
transport resampling method is then used to create an evenly weighted sample
ready for the next iteration. We demonstrate that this ensemble transport
adaptive importance sampling (ETAIS) method outperforms MCMC methods with
equivalent proposal distributions for low dimensional problems, and in fact
shows better than linear improvements in convergence rates with respect to the
number of ensemble members. We also introduce a new resampling strategy,
multinomial transformation (MT), which while not as accurate as the ensemble
transport resampler, is substantially less costly for large ensemble sizes, and
can then be used in conjunction with ETAIS for complex problems. We also focus
on how algorithmic parameters regarding the mixture proposal can be quickly
tuned to optimise performance. In particular, we demonstrate this methodology's
superior sampling for multimodal problems, such as those arising from inference
for mixture models, and for problems with expensive likelihoods requiring the
solution of a differential equation, for which speed-ups of orders of magnitude
are demonstrated. Likelihood evaluations of the ensemble could be computed in a
distributed manner, suggesting that this methodology is a good candidate for
parallel Bayesian computations
Reversible Jump Metropolis Light Transport using Inverse Mappings
We study Markov Chain Monte Carlo (MCMC) methods operating in primary sample
space and their interactions with multiple sampling techniques. We observe that
incorporating the sampling technique into the state of the Markov Chain, as
done in Multiplexed Metropolis Light Transport (MMLT), impedes the ability of
the chain to properly explore the path space, as transitions between sampling
techniques lead to disruptive alterations of path samples. To address this
issue, we reformulate Multiplexed MLT in the Reversible Jump MCMC framework
(RJMCMC) and introduce inverse sampling techniques that turn light paths into
the random numbers that would produce them. This allows us to formulate a novel
perturbation that can locally transition between sampling techniques without
changing the geometry of the path, and we derive the correct acceptance
probability using RJMCMC. We investigate how to generalize this concept to
non-invertible sampling techniques commonly found in practice, and introduce
probabilistic inverses that extend our perturbation to cover most sampling
methods found in light transport simulations. Our theory reconciles the
inverses with RJMCMC yielding an unbiased algorithm, which we call Reversible
Jump MLT (RJMLT). We verify the correctness of our implementation in canonical
and practical scenarios and demonstrate improved temporal coherence, decrease
in structured artifacts, and faster convergence on a wide variety of scenes
Learning distributions of shape trajectories from longitudinal datasets: a hierarchical model on a manifold of diffeomorphisms
We propose a method to learn a distribution of shape trajectories from
longitudinal data, i.e. the collection of individual objects repeatedly
observed at multiple time-points. The method allows to compute an average
spatiotemporal trajectory of shape changes at the group level, and the
individual variations of this trajectory both in terms of geometry and time
dynamics. First, we formulate a non-linear mixed-effects statistical model as
the combination of a generic statistical model for manifold-valued longitudinal
data, a deformation model defining shape trajectories via the action of a
finite-dimensional set of diffeomorphisms with a manifold structure, and an
efficient numerical scheme to compute parallel transport on this manifold.
Second, we introduce a MCMC-SAEM algorithm with a specific approach to shape
sampling, an adaptive scheme for proposal variances, and a log-likelihood
tempering strategy to estimate our model. Third, we validate our algorithm on
2D simulated data, and then estimate a scenario of alteration of the shape of
the hippocampus 3D brain structure during the course of Alzheimer's disease.
The method shows for instance that hippocampal atrophy progresses more quickly
in female subjects, and occurs earlier in APOE4 mutation carriers. We finally
illustrate the potential of our method for classifying pathological
trajectories versus normal ageing
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Inference Algorithms and Sensorimotor Representations in Brains and Machines
Animals function in a 3D world in which survival depends on robust, well-controlled actions. Historically, researchers in Artificial Intelligence (AI) and neuroscience have explored sensory and motor systems independently. There is a growing body of literature in AI and neuroscience to suggest that they actually work in tandem. While there has been a great deal of work on vision and audition as sensory modalities in these fields, one could argue that a more fundamental modality in biology is haptics, or the sense of touch. In this thesis, we will look at building computational models that integrate tactile sensing with other sensory modalities to perform manipulation-like tasks in robots and discrimination tasks in mice. We will also explore the problem of inference through the lens of Markov Chain Monte Carlo methods (MCMC). We elaborate on the ideas discussed in this thesis in the introduction presented in Chapter 1. A challenging problem one often faces when applying probabilistic mathematical models to the study of sensory-motor systems and other problems involving learning of inference is sampling. Hamiltonian Markov Chain Monte Carlo (HMC) algorithms can efficiently draw representative samples from complex probabilistic models. Most MCMC methods rely on detailed balance to ensure that we can sample from the correct distribution. This constraint can be relaxed in discrete state spaces such as those employed by HMC type methods. In Chapter 2, we study HMC methods without detailed balance to explore faster convergence. Markov jump processes are stochastic processes on discrete state space but continuous in time. In Chapter 3, we use Markov Jump Processes to simulate waiting times along with generalized detailed balance. This waiting time ,we show, helps generate samples faster. Most MCMC methods are plagued by slow simulation times on discrete computing systems. In Chapter 4, we explore HMC in analog circuits where the problem of generating samples from a distribution is mapped to the problem of sampling charge in a capacitor.The second half of this dissertation focuses on the role of haptics in perception and action. Manipulation is a fundamental problem for artificial and biological agents. High dimensional actuators (say, fingers, trunks,etc) are really hard to control. In Chapter 5, we present an approach to learn to actuate dexterous manipulators to grasp objects in simulation. Haptics as a sensory modality is critical to many manipulation tasks. Employing haptics in high dimensional dextrous actuators is challenging. In Chapter 6, we explore how intrinsic curiosity and haptics can be used to learn exploration strategies for discrimination of objects with dextrous hands. A key component to make tactile sensing a possibility is the availability of cheap, efficient, scalable hardware. Chapter 7 presents results for tactile servoing using a physical gelsight sensor. Traditional neuroscience texts delineate sensory and motor systems as two independent systems yet recent results suggest that this may not be entirely complete. That is, there is evidence to suggest that the representations in the cortex is more distributed than is accepted. Finally in Chapter 8, we explore building a computational model of spiking neural data collected from both the barrel and motor cortices during free and active whisking. These works help towards understanding sensorimotor representations in the context of haptics and high dimensional controls. We conclude with a discussion on future directions in Chapter 9
Scalable and flexible inference framework for stochastic dynamic single-cell models
Understanding the inherited nature of how biological processes dynamically change over time and exhibit intra- and inter-individual variability, due to the different responses to environmental stimuli and when interacting with other processes, has been a major focus of systems biology. The rise of single-cell fluorescent microscopy has enabled the study of those phenomena. The analysis of single-cell data with mechanistic models offers an invaluable tool to describe dynamic cellular processes and to rationalise cell-to-cell variability within the population. However, extracting mechanistic information from single-cell data has proven difficult. This requires statistical methods to infer unknown model parameters from dynamic, multi-individual data accounting for heterogeneity caused by both intrinsic (e.g. variations in chemical reactions) and extrinsic (e.g. variability in protein concentrations) noise. Although several inference methods exist, the availability of efficient, general and accessible methods that facilitate modelling of single-cell data, remains lacking. Here we present a scalable and flexible framework for Bayesian inference in state-space mixed-effects single-cell models with stochastic dynamic. Our approach infers model parameters when intrinsic noise is modelled by either exact or approximate stochastic simulators, and when extrinsic noise is modelled by either time-varying, or time-constant parameters that vary between cells. We demonstrate the relevance of our approach by studying how cell-to-cell variation in carbon source utilisation affects heterogeneity in the budding yeast Saccharomyces cerevisiae SNF1 nutrient sensing pathway. We identify hexokinase activity as a source of extrinsic noise and deduce that sugar availability dictates cell-to-cell variability
Proximal Markov chain Monte Carlo algorithms
This paper presents a new Metropolis-adjusted Langevin algorithm (MALA) that uses convex analysis to simulate efficiently from high-dimensional densities that are log-concave, a class of probability distributions that is widely used in modern high-dimensional statistics and data analysis. The method is based on a new first-order approximation for Langevin diffusions that exploits log-concavity to construct Markov chains with favourable convergence properties. This approximation is closely related to Moreau--Yoshida regularisations for convex functions and uses proximity mappings instead of gradient mappings to approximate the continuous-time process. The proposed method complements existing MALA methods in two ways. First, the method is shown to have very robust stability properties and to converge geometrically for many target densities for which other MALA are not geometric, or only if the step size is sufficiently small. Second, the method can be applied to high-dimensional target densities that are not continuously differentiable, a class of distributions that is increasingly used in image processing and machine learning and that is beyond the scope of existing MALA and HMC algorithms. To use this method it is necessary to compute or to approximate efficiently the proximity mappings of the logarithm of the target density. For several popular models, including many Bayesian models used in modern signal and image processing and machine learning, this can be achieved with convex optimisation algorithms and with approximations based on proximal splitting techniques, which can be implemented in parallel. The proposed method is demonstrated on two challenging high-dimensional and non-differentiable models related to image resolution enhancement and low-rank matrix estimation that are not well addressed by existing MCMC methodology
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