Maximum A Posteriori Inference Research Articles

Bayesian Lookahead Perturbation Policy for Inference of Regulatory Networks.

The complexity, scale, and uncertainty in regulatory networks (e.g., gene regulatory networks and microbial networks) regularly pose a huge uncertainty in their models. These uncertainties often cannot be entirely reduced using limited and costly data acquired from the normal condition of systems. Meanwhile, regulatory networks often suffer from the non-identifiability issue, which refers to scenarios where the true underlying network model cannot be clearly distinguished from other possible models. Perturbation or excitation is a well-known process in systems biology for acquiring targeted data to reveal the complex underlying mechanisms of regulatory networks and overcome the non-identifiability issue. We consider a general class of Boolean network models for capturing the activation and inactivation of components and their complex interactions. Assuming partial available knowledge about the interactions between components of the networks, this paper formulates the inference process through the maximum aposteriori (MAP) criterion. We develop a Bayesian lookahead policy that systematically perturbs regulatory networks to maximize the performance of MAP inference under the perturbed data. This is achieved by optimally formulating the perturbation process in a reinforcement learning context and deriving a scalable deep reinforcement learning perturbation policy to compute near-optimal Bayesian policy. The proposed method learns the perturbation policy through planning without the need for any real data. The high performance of the proposed approach is demonstrated by comprehensive numerical experiments using the well-known mammalian cell cycle and gut microbial community networks.

MAP inference algorithms without approximation for collective graphical models on path graphs via discrete difference of convex algorithm

Collective graphical model (CGM) is a probabilistic model that provides a framework for analyzing aggregated count data. Maximum a posteriori (MAP) inference of unobserved variables under given observations is one of the essential operations in CGM. Because the MAP inference problem is known to be NP-hard in general, the current mainstream approach is to solve an alternative problem obtained by approximating the objective function and applying continuous relaxation. However, this approach has two significant drawbacks. First, the quality of the solution deteriorates when the values in the count data are negligible due to the inaccuracy of Stirling’s approximation. Second, the application of continuous relaxation causes the violation of integrality constraints. This paper proposes novel algorithms for MAP inference in CGMs on path graphs to overcome these problems. Our method is based on the discrete difference of convex algorithm (DCA); DCA is a general framework to minimize the sum of a convex function and a concave function by repeatedly minimizing surrogate functions. Utilizing the particular structure of path graphs, we efficiently solve the surrogate function minimization by minimum convex cost flow algorithms. Furthermore, our approach also leads to a new method of solving another important task; MAP inference of the sample size in CGM on path graphs. Our method is naturally applicable to this task, allowing us to design very efficient algorithms. Experimental results on synthetic and real-world datasets show the effectiveness of the proposed algorithms.

Bayesian Imaging Using Plug & Play Priors: When Langevin Meets Tweedie

Since the seminal work of Venkatakrishnan et al. in 2013, Plug & Play (PnP) methods have become ubiquitous in Bayesian imaging. These methods derive Minimum Mean Square Error (MMSE) or Maximum A Posteriori (MAP) estimators for inverse problems in imaging by combining an explicit likelihood function with a prior that is implicitly defined by an image denoising algorithm. The PnP algorithms proposed in the literature mainly differ in the iterative schemes they use for optimisation or for sampling. In the case of optimisation schemes, some recent works guarantee the convergence to a fixed point, albeit not necessarily a MAP estimate. In the case of sampling schemes, to the best of our knowledge, there is no known proof of convergence. There also remain important open questions regarding whether the underlying Bayesian models and estimators are well defined, well-posed, and have the basic regularity properties required to support these numerical schemes. To address these limitations, this paper develops theory, methods, and provably convergent algorithms for performing Bayesian inference with PnP priors. We introduce two algorithms: 1) PnP-ULA (Unadjusted Langevin Algorithm) for Monte Carlo sampling and MMSE inference; and 2) PnP-SGD (Stochastic Gradient Descent) for MAP inference. Using recent results on the quantitative convergence of Markov chains, we establish detailed convergence guarantees for these two algorithms under realistic assumptions on the denoising operators used, with special attention to denoisers based on deep neural networks. We also show that these algorithms approximately target a decision-theoretically optimal Bayesian model that is well-posed. The proposed algorithms are demonstrated on several canonical problems such as image deblurring, inpainting, and denoising, where they are used for point estimation as well as for uncertainty visualisation and quantification.

Some Inapproximability Results of MAP Inference and Exponentiated Determinantal Point Processes

We study the computational complexity of two hard problems on determinantal point processes (DPPs). One is maximum a posteriori (MAP) inference, i.e., to find a principal submatrix having the maximum determinant. The other is probabilistic inference on exponentiated DPPs (E-DPPs), which can sharpen or weaken the diversity preference of DPPs with an exponent parameter p. We present several complexity-theoretic hardness results that explain the difficulty in approximating MAP inference and the normalizing constant for E-DPPs. We first prove that unconstrained MAP inference for an n × n matrix is NP-hard to approximate within a factor of 2βn, where β = 10−1013 . This result improves upon the best-known inapproximability factor of (9/8 − ϵ), and rules out the existence of any polynomial-factor approximation algorithm assuming P ≠ NP. We then show that log-determinant maximization is NP-hard to approximate within a factor of 5/4 for the unconstrained case and within a factor of 1 + 10−1013 for the size-constrained monotone case. In particular, log-determinant maximization does not admit a polynomial-time approximation scheme unless P = NP. As a corollary of the first result, we demonstrate that the normalizing constant for E-DPPs of any (fixed) constant exponent p ≥ β-1 = 101013 is NP-hard to approximate within a factor of 2βpn, which is in contrast to the case of p ≤ 1 admitting a fully polynomial-time randomized approximation scheme.

Sparse connectivity for MAP inference in linear models using sister mitral cells.

Sensory processing is hard because the variables of interest are encoded in spike trains in a relatively complex way. A major goal in studies of sensory processing is to understand how the brain extracts those variables. Here we revisit a common encoding model in which variables are encoded linearly. Although there are typically more variables than neurons, this problem is still solvable because only a small number of variables appear at any one time (sparse prior). However, previous solutions require all-to-all connectivity, inconsistent with the sparse connectivity seen in the brain. Here we propose an algorithm that provably reaches the MAP (maximum a posteriori) inference solution, but does so using sparse connectivity. Our algorithm is inspired by the circuit of the mouse olfactory bulb, but our approach is general enough to apply to other modalities. In addition, it should be possible to extend it to nonlinear encoding models.

Strong replica symmetry for high-dimensional disordered log-concave Gibbs measures

Abstract We consider a generic class of log-concave, possibly random, (Gibbs) measures. We prove the concentration of an infinite family of order parameters called multioverlaps. Because they completely parametrize the quenched Gibbs measure of the system, this implies a simple representation of the asymptotic Gibbs measures, as well as the decoupling of the variables in a strong sense. These results may prove themselves useful in several contexts. In particular in machine learning and high-dimensional inference, log-concave measures appear in convex empirical risk minimization, maximum a-posteriori inference or M-estimation. We believe that they may be applicable in establishing some type of ‘replica symmetric formulas’ for the free energy, inference or generalization error in such settings.

Exact Lifted Inference with Distinct Soft Evidence on Every Object

The presence of non-symmetric evidence has been a barrier for the application of lifted inference since the evidence destroys the symmetry of the first-order probabilistic model. In the extreme case, if distinct soft evidence is obtained about each individual object in the domain then, often, all current exact lifted inference methods reduce to traditional inference at the ground level. However, it is of interest to ask whether the symmetry of the model itself before evidence is obtained can be exploited. We present new results showing that this is, in fact, possible. In particular, we show that both exact maximum a posteriori (MAP) and marginal inference can be lifted for the case of distinct soft evidence on a unary Markov Logic predicate. Our methods result in efficient procedures for MAP and marginal inference for a class of graphical models previously thought to be intractable.

MAP Inference for Probabilistic Logic Programming

AbstractIn Probabilistic Logic Programming (PLP) the most commonly studied inference task is to compute the marginal probability of a query given a program. In this paper, we consider two other important tasks in the PLP setting: the Maximum-A-Posteriori (MAP) inference task, which determines the most likely values for a subset of the random variables given evidence on other variables, and the Most Probable Explanation (MPE) task, the instance of MAP where the query variables are the complement of the evidence variables. We present a novel algorithm, included in the PITA reasoner, which tackles these tasks by representing each problem as a Binary Decision Diagram and applying a dynamic programming procedure on it. We compare our algorithm with the version of ProbLog that admits annotated disjunctions and can perform MAP and MPE inference. Experiments on several synthetic datasets show that PITA outperforms ProbLog in many cases.

Leveraging cluster backbones for improving MAP inference in statistical relational models

A wide range of important problems in machine learning, expert system, social network analysis, bioinformatics and information theory can be formulated as a maximum a-posteriori (MAP) inference problem on statistical relational models. While off-the-shelf inference algorithms that are based on local search and message-passing may provide adequate solutions in some situations, they frequently give poor results when faced with models that possess high-density networks. Unfortunately, these situations always occur in models of real-world applications. As such, accurate and scalable maximum a-posteriori (MAP) inference on such models often remains a key challenge. In this paper, we first introduce a novel family of extended factor graphs that are parameterized by a smoothing parameter χ ∈ [0,1]. Applying belief propagation (BP) message-passing to this family formulates a new family of W eighted S urvey P ropagation algorithms (WSP-χ) applicable to relational domains. Unlike off-the-shelf inference algorithms, WSP-χ detects the “backbone” ground atoms in a solution cluster that involve potentially optimal MAP solutions: the cluster backbone atoms are not only portions of the optimal solutions, but they also can be exploited for scaling MAP inference by iteratively fixing them to reduce the complex parts until the network is simplified into one that can be solved accurately using any conventional MAP inference method. We also propose a lazy variant of this WSP-χ family of algorithms. Our experiments on several real-world problems show the efficiency of WSP-χ and its lazy variants over existing prominent MAP inference solvers such as MaxWalkSAT, RockIt, IPP, SP-Y and WCSP.

MAP Inference Via $$\ell _2$$-Sphere Linear Program Reformulation

Maximum a posteriori (MAP) inference is an important task for graphical models. Due to complex dependencies among variables in realistic models, finding an exact solution for MAP inference is often intractable. Thus, many approximation methods have been developed, among which the linear programming (LP) relaxation based methods show promising performance. However, one major drawback of LP relaxation is that it is possible to give fractional solutions. Instead of presenting a tighter relaxation, in this work we propose a continuous but equivalent reformulation of the original MAP inference problem, called LS–LP. We add the $$\ell _2$$-sphere constraint onto the original LP relaxation, leading to an intersected space with the local marginal polytope that is equivalent to the space of all valid integer label configurations. Thus, LS–LP is equivalent to the original MAP inference problem. We propose a perturbed alternating direction method of multipliers (ADMM) algorithm to optimize the LS–LP problem, by adding a sufficiently small perturbation $$\epsilon $$ onto the objective function and constraints. We prove that the perturbed ADMM algorithm globally converges to the $$\epsilon $$-Karush–Kuhn–Tucker ($$\epsilon $$-KKT) point of the LS–LP problem. The convergence rate will also be analyzed. Experiments on several benchmark datasets from Probabilistic Inference Challenge (PIC 2011) and OpenGM 2 show competitive performance of our proposed method against state-of-the-art MAP inference methods.

Blind image deblurring via hybrid deep priors modeling

Blind image deblurring is a challenging low-level vision problem which aims to restore a sharp image only from the blurry observation. Few known information makes this problem fundamentally ill-posed. Most recent works focus on designing various priors on both latent image and blur kernel based on the maximum a posteriori (MAP) model to restrict the solution space. However, their performance is highly related to these hand-crafted explicit priors. In fact, the pre-designed explicit priors may have less flexibility to fit different image structures in real-world scenarios. To overcome these difficulties, we propose a novel framework, named Hybrid Deep Priors Model (HDPM), to simulate the propagation of sharp latent image used in kernel estimation and final deconvolution. Specifically, we introduce the learnable implicit deep prior and hand-crafted explicit prior as regularizations into the MAP inference process to extract the detailed texture and sharp structures of latent image, respectively. In HDPM, we can successfully take the advantages of explicit cues based on task information and implicit deep priors from training data to facilitate the propagation of sharp latent image which is beneficial for the kernel estimation. Extensive experiments demonstrate that the proposed method performs favorably against the state-of-the-art deblurring methods on benchmarks, challenging scenarios and non-uniform images.

Geometric Matrix Completion With Deep Conditional Random Fields.

The problem of completing high-dimensional matrices from a limited set of observations arises in many big data applications, especially recommender systems. The existing matrix completion models generally follow either a memory- or a model-based approach, whereas geometric matrix completion (GMC) models combine the best from both approaches. Existing deep-learning-based geometric models yield good performance, but, in order to operate, they require a fixed structure graph capturing the relationships among the users and items. This graph is typically constructed by evaluating a pre-defined similarity metric on the available observations or by using side information, e.g., user profiles. In contrast, Markov-random-fields-based models do not require a fixed structure graph but rely on handcrafted features to make predictions. When no side information is available and the number of available observations becomes very low, existing solutions are pushed to their limits. In this article, we propose a GMC approach that addresses these challenges. We consider matrix completion as a structured prediction problem in a conditional random field (CRF), which is characterized by a maximum a posteriori (MAP) inference, and we propose a deep model that predicts the missing entries by solving the MAP inference problem. The proposed model simultaneously learns the similarities among matrix entries, computes the CRF potentials, and solves the inference problem. Its training is performed in an end-to-end manner, with a method to supervise the learning of entry similarities. Comprehensive experiments demonstrate the superior performance of the proposed model compared to various state-of-the-art models on popular benchmark data sets and underline its superior capacity to deal with highly incomplete matrices.

Lifted Hinge-Loss Markov Random Fields

Statistical relational learning models are powerful tools that combine ideas from first-order logic with probabilistic graphical models to represent complex dependencies. Despite their success in encoding large problems with a compact set of weighted rules, performing inference over these models is often challenging. In this paper, we show how to effectively combine two powerful ideas for scaling inference for large graphical models. The first idea, lifted inference, is a wellstudied approach to speeding up inference in graphical models by exploiting symmetries in the underlying problem. The second idea is to frame Maximum a posteriori (MAP) inference as a convex optimization problem and use alternating direction method of multipliers (ADMM) to solve the problem in parallel. A well-studied relaxation to the combinatorial optimization problem defined for logical Markov random fields gives rise to a hinge-loss Markov random field (HLMRF) for which MAP inference is a convex optimization problem. We show how the formalism introduced for coloring weighted bipartite graphs using a color refinement algorithm can be integrated with the ADMM optimization technique to take advantage of the sparse dependency structures of HLMRFs. Our proposed approach, lifted hinge-loss Markov random fields (LHL-MRFs), preserves the structure of the original problem after lifting and solves lifted inference as distributed convex optimization with ADMM. In our empirical evaluation on real-world problems, we observe up to a three times speed up in inference over HL-MRFs.

Blind Image Deblurring via Adaptive Optimization with Flexible Sparse Structure Control

Blind image deblurring is a long-standing ill-posed inverse problem which aims to recover a latent sharp image given only a blurry observation. So far, existing studies have designed many effective priors w.r.t. the latent image within the maximum a posteriori (MAP) framework in order to narrow down the solution space. These non-convex priors are always integrated into the final deblurring model, which makes the optimization challenging. However, due to unknown image distribution, complex kernel structure and non-uniform noises in real-world scenarios, it is indeed challenging to explicitly design a fixed prior for all cases. Thus we adopt the idea of adaptive optimization and propose the sparse structure control (SSC) for the latent image during the optimization process. In this paper, we only formulate the necessary optimization constraints in a lightweight MAP model with no priors. Then we develop an inexact projected gradient scheme to incorporate flexible SSC in MAP inference. Besides lp-norm based SSC in our previous work, we also train a group of denoising convolutional neural networks (CNNs) to learn the sparse image structure automatically from the training data under different noise levels, and we show that CNNs-based SSC can achieve similar results compared with lp-norm but are more robust to noise. Extensive experiments demonstrate that the proposed adaptive optimization scheme with two types of SSC achieves the state-of-the-art results on both synthetic data and real-world images.

Maximum A Posteriori Inference in Sum-Product Networks

Sum-product networks (SPNs) are a class of probabilistic graphical models that allow tractable marginal inference. However, the maximum a posteriori (MAP) inference in SPNs is NP-hard. We investigate MAP inference in SPNs from both theoretical and algorithmic perspectives. For the theoretical part, we reduce general MAP inference to its special case without evidence and hidden variables; we also show that it is NP-hard to approximate the MAP problem to 2nε for fixed 0 ≤ ε < 1, where n is the input size. For the algorithmic part, we first present an exact MAP solver that runs reasonably fast and could handle SPNs with up to 1k variables and 150k arcs in our experiments. We then present a new approximate MAP solver with a good balance between speed and accuracy, and our comprehensive experiments on real-world datasets show that it has better overall performance than existing approximate solvers.

MAP inference in dynamic hybrid Bayesian networks

In this paper, we study the maximum a posteriori (MAP) problem in dynamic hybrid Bayesian networks. We are interested in finding the sequence of values of a class variable that maximizes the posterior probability given evidence. We propose an approximate solution based on transforming the MAP problem into a simpler belief update problem. The proposed solution constructs a set of auxiliary networks by grouping consecutive instantiations of the variable of interest, thus capturing some of the potential temporal dependences between these variables while ignoring others. Belief update is carried out independently in the auxiliary models, after which the results are combined, producing a configuration of values for the class variable along the entire time sequence. Experiments have been carried out to analyze the behavior of the approach. The algorithm has been implemented using Java 8 streams, and its scalability has been evaluated.

Dirichlet Priors for MAP Inference of Protein Conformation Abundances from SAXS

Estimation of mixture coefficients of protein conformations in solution find applications in understanding protein behavior. We describe a method for maximum a posteriori (MAP) estimation of the mixture coefficients of ensemble of conformations in a protein mixture solution using measured small angle X-ray scattering (SAXS) intensities. The proposed method builds upon a model for the measurements of crystallographically determined conformations. Assuming that a priori information on the protein mixture is available, and that priori information follows a Dirichlet distribution, we develop a method to estimate the relative abundances with MAP estimator. The Dirichlet distribution depends on concentration parameters which may not be known in practice and thus need to be estimated. To estimate these unknown concentration parameters we developed an expectation-maximization (EM) method. Adenylate kinase (ADK) protein was selected as the test bed due to its known conformations Beckstein et al. (Journal of Molecular Biology, 394(1), 160 1). Known conformations are assumed to form the full vector bases that span the measurement space. In Monte Carlo simulations, mixture coefficient estimation performances of MAP and maximum likelihood (ML) (which assumes a uniform prior on the mixture coefficients) estimators are compared. MAP estimators using known and unknown concentration parameters are also compared in terms of estimation performances. The results show that prior knowledge improves estimation accuracy, but performance is sensitive to perturbations in the Dirichlet distribution's concentration parameters. Moreover, the estimation method based on EM algorithm shows comparable results to approximately known prior parameters.

Simple approximate MAP inference for Dirichlet processes mixtures

The Dirichlet process mixture model (DPMM) is a ubiquitous, flexible Bayesian nonparametric statistical model. However, full probabilistic inference in this model is analytically intractable, so that computationally intensive techniques such as Gibbs sampling are required. As a result, DPMM-based methods, which have considerable potential, are restricted to applications in which computational resources and time for inference is plentiful. For example, they would not be practical for digital signal processing on embedded hardware, where computational resources are at a serious premium. Here, we develop a simplified yet statistically rigorous approximate maximum a-posteriori (MAP) inference algorithm for DPMMs. This algorithm is as simple as DP-means clustering, solves the MAP problem as well as Gibbs sampling, while requiring only a fraction of the computational effort. (For freely available code that implements the MAP-DP algorithm for Gaussian mixtures see http://www.maxlittle.net/.) Unlike related small variance asymptotics (SVA), our method is non-degenerate and so inherits the “rich get richer” property of the Dirichlet process. It also retains a non-degenerate closed-form likelihood which enables out-of-sample calculations and the use of standard tools such as cross-validation. We illustrate the benefits of our algorithm on a range of examples and contrast it to variational, SVA and sampling approaches from both a computational complexity perspective as well as in terms of clustering performance. We demonstrate the wide applicabiity of our approach by presenting an approximate MAP inference method for the infinite hidden Markov model whose performance contrasts favorably with a recently proposed hybrid SVA approach. Similarly, we show how our algorithm can applied to a semiparametric mixed-effects regression model where the random effects distribution is modelled using an infinite mixture model, as used in longitudinal progression modelling in population health science. Finally, we propose directions for future research on approximate MAP inference in Bayesian nonparametrics.

Simultaneous segmentation and multiresolution nonrigid atlas registration.

In this paper, a novel Markov random field (MRF)-based approach is presented for segmenting medical images while simultaneously registering an atlas nonrigidly. In the literature, both segmentation and registration have been studied extensively. For applications that involve both, such as segmentation via atlas-based registration, earlier studies proposed addressing these problems iteratively by feeding the output of each to initialize the other. This scheme, however, cannot guarantee an optimal solution for the combined task at hand, since these two individual problems are then treated separately. In this paper, we formulate simultaneous registration and segmentation (SRS) as a maximum a-posteriori (MAP) problem. We decompose the resulting probabilities such that the MAP inference can be done using MRFs. An efficient hierarchical implementation is employed, allowing coarse-to-fine registration while estimating segmentation at pixel level. The method is evaluated on two clinical data sets: 1) mandibular bone segmentation in 3D CT and 2) corpus callosum segmentation in 2D midsaggital slices of brain MRI. A video tracking example is also given. Our implementation allows us to directly compare the proposed method with the individual segmentation/registration and the iterative approach using the exact same potential functions. In a leave-one-out evaluation, SRS demonstrated more accurate results in terms of dice overlap and surface distance metrics for both data sets. We also show quantitatively that the SRS method is less sensitive to the errors in the registration as opposed to the iterative approach.

RockIt: Exploiting Parallelism and Symmetry for MAP Inference in Statistical Relational Models

RockIt is a maximum a-posteriori (MAP) query engine for statistical relational models. MAP inference in graphical models is an optimization problem which can be compiled to integer linear programs (ILPs).We describe several advances in translating MAP queries to ILP instances and present the novel meta-algorithm cutting plane aggregation (CPA). CPA exploits local context-specific symmetries and bundles up sets of linear constraints. The resulting counting constraints lead to more compact ILPs and make the symmetry of the ground model more explicit to state-of-the-art ILP solvers. Moreover, RockIt parallelizes most parts of the MAP inference pipeline taking advantage of ubiquitous shared-memory multi-core architectures. We report on extensive experiments with Markov logic network (MLN) benchmarks showing that RockIt outperforms the state-of-the-art systems Alchemy, Markov TheBeast, and Tuffy both in terms of efficiency and quality of results.