When observations are organized into groups where commonalties exist amongst them, the dependent random measures can be an ideal choice for modeling. One of the propositions of the dependent random measures is that the atoms of the posterior distribution are shared amongst groups, and hence groups can borrow information from each other. When normalized dependent random measures prior with independent increments are applied, we can derive appropriate exchangeable probability partition function (EPPF), and subsequently also deduce its inference algorithm given any mixture model likelihood. We provide all necessary derivation and solution to this framework. For demonstration, we used mixture of Gaussians likelihood in combination with a dependent structure constructed by linear combinations of CRMs. Our experiments show superior performance when using this framework, where the inferred values including the mixing weights and the number of clusters both respond appropriately to the number of completely random measure used.

Many representation schemes combining first-order logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable elimination and belief propagation, neither of which take logical structure into account. We propose the first method that has the full power of both graphical model inference and first-order theorem proving (in finite domains with Herbrand interpretations). We first define probabilistic theorem proving (PTP), their generalization, as the problem of computing the probability of a logical formula given the probabilities or weights of a set of formulas.

A goal of research in artificial intelligence and machine learning since the early days of expert systems has been to develop automated reasoning methods that combine logic and probability. Why is there a need to combine logic and probability? Probability theory allows one to quantify uncertainty over a set of propositions--ground facts about the world--and a probabilistic reasoning system allows one to infer the probability of unknown (hidden) propositions conditioned on the knowledge of other propositions. However, probability theory alone has nothing to say about how propositions are constructed from relationships over entities or tuples of entities, and how general knowledge at the level of relationships is to be represented and applied.

We consider a group of Bayesian agents who try to estimate a state of the world $\theta$ through interaction on a social network. Each agent $v$ initially receives a private measurement of $\theta$: a number $S_v$ picked from a Gaussian distribution with mean $\theta$ and standard deviation one. Then, in each discrete time iteration, each reveals its estimate of $\theta$ to its neighbors, and, observing its neighbors' actions, updates its belief using Bayes' Law. This process aggregates information efficiently, in the sense that all the agents converge to the belief that they would have, had they access to all the private measurements. We show that this process is computationally efficient, so that each agent's calculation can be easily carried out. We also show that on any graph the process converges after at most $2N \cdot D$ steps, where $N$ is the number of agents and $D$ is the diameter of the network. Finally, we show that on trees and on distance transitive-graphs the process converges after $D$ steps, and that it preserves privacy, so that agents learn very little about the private signal of most other agents, despite the efficient aggregation of information. Our results extend those in an unpublished manuscript of the first and last authors.

A common analytical problem in neuroscience is the interpretation of neural activity with respect to sensory input or behavioral output. This is typically achieved by regressing measured neural activity against known stimuli or behavioral variables to produce a "tuning function" for each neuron. Unfortunately, because this approach handles neurons individually, it cannot take advantage of simultaneous measurements from spatially adjacent neurons that often have similar tuning properties. On the other hand, sharing information between adjacent neurons can errantly degrade estimates of tuning functions across space if there are sharp discontinuities in tuning between nearby neurons. In this paper, we develop a computationally efficient block Gibbs sampler that effectively pools information between neurons to de-noise tuning function estimates while simultaneously preserving sharp discontinuities that might exist in the organization of tuning across space. This method is fully Bayesian and its computational cost per iteration scales sub-quadratically with total parameter dimensionality. We demonstrate the robustness and scalability of this approach by applying it to both real and synthetic datasets. In particular, an application to data from the spinal cord illustrates that the proposed methods can dramatically decrease the experimental time required to accurately estimate tuning functions.

We propose a probabilistic modeling framework for learning the dynamic patterns in the collective behaviors of social agents and developing profiles for different behavioral groups, using data collected from multiple information sources. The proposed model is based on a hierarchical Bayesian process, in which each observation is a finite mixture of an set of latent groups and the mixture proportions (i.e., group probabilities) are drawn randomly. Each group is associated with some distributions over a finite set of outcomes. Moreover, as time evolves, the structure of these groups also changes; we model the change in the group structure by a hidden Markov model (HMM) with a fixed transition probability. We present an efficient inference method based on tensor decompositions and the expectation-maximization (EM) algorithm for parameter estimation.

The analysis and diagnosis of Alzheimer's disease (AD) can be based on genetic variations, e.g., single nucleotide polymorphisms (SNPs) and phenotypic traits, e.g., Magnetic Resonance Imaging (MRI) features. We consider two important and related tasks: i) to select genetic and phenotypical markers for AD diagnosis and ii) to identify associations between genetic and phenotypical data. While previous studies treat these two tasks separately, they are tightly coupled because underlying associations between genetic variations and phenotypical features contain the biological basis for a disease. Here we present a new sparse Bayesian approach for joint association study and disease diagnosis. In this approach, common latent features are extracted from different data sources based on sparse projection matrices and used to predict multiple disease severity levels; in return, the disease status can guide the discovery of relationships between data sources. The sparse projection matrices not only reveal interactions between data sources but also select groups of biomarkers related to the disease. Moreover, to take advantage of the linkage disequilibrium (LD) measuring the non-random association of alleles, we incorporate a graph Laplacian type of prior in the model. To learn the model from data, we develop an efficient variational inference algorithm. Analysis on an imaging genetics dataset for the study of Alzheimer's Disease (AD) indicates that our model identifies biologically meaningful associations between genetic variations and MRI features, and achieves significantly higher accuracy for predicting ordinal AD stages than the competing methods.

Attack graphs provide compact representations of the attack paths that an attacker can follow to compromise network resources by analysing network vulnerabilities and topology. These representations are a powerful tool for security risk assessment. Bayesian inference on attack graphs enables the estimation of the risk of compromise to the system's components given their vulnerabilities and interconnections, and accounts for multi-step attacks spreading through the system. Whilst static analysis considers the risk posture at rest, dynamic analysis also accounts for evidence of compromise, e.g. from SIEM software or forensic investigation. However, in this context, exact Bayesian inference techniques do not scale well. In this paper we show how Loopy Belief Propagation - an approximate inference technique - can be applied to attack graphs, and that it scales linearly in the number of nodes for both static and dynamic analysis, making such analyses viable for larger networks. We experiment with different topologies and network clustering on synthetic Bayesian attack graphs with thousands of nodes to show that the algorithm's accuracy is acceptable and converge to a stable solution. We compare sequential and parallel versions of Loopy Belief Propagation with exact inference techniques for both static and dynamic analysis, showing the advantages of approximate inference techniques to scale to larger attack graphs.

The goal of ordinal embedding is to represent items as points in a low-dimensional Euclidean space given a set of constraints in the form of distance comparisons like "item $i$ is closer to item $j$ than item $k$". Ordinal constraints like this often come from human judgments. To account for errors and variation in judgments, we consider the noisy situation in which the given constraints are independently corrupted by reversing the correct constraint with some probability. This paper makes several new contributions to this problem. First, we derive prediction error bounds for ordinal embedding with noise by exploiting the fact that the rank of a distance matrix of points in $\mathbb{R}^d$ is at most $d+2$. These bounds characterize how well a learned embedding predicts new comparative judgments. Second, we investigate the special case of a known noise model and study the Maximum Likelihood estimator. Third, knowledge of the noise model enables us to relate prediction errors to embedding accuracy. This relationship is highly non-trivial since we show that the linear map corresponding to distance comparisons is non-invertible, but there exists a nonlinear map that is invertible. Fourth, two new algorithms for ordinal embedding are proposed and evaluated in experiments.

Bayesian Statistics continues to remain incomprehensible in the ignited minds of many analysts. Being amazed by the incredible power of machine learning, a lot of us have become unfaithful to statistics. Our focus has narrowed down to exploring machine learning. We fail to understand that machine learning is only one way to solve real world problems. In several situations, it does not help us solve business problems, even though there is data involved in these problems. To say the least, knowledge of statistics will allow you to work on complex analytical problems, irrespective of the size of data. In 1770s, Thomas Bayes introduced'Bayes Theorem'.