Lifted inference algorithms for probabilistic first-order logic frameworks such as Markov logic networks (MLNs) have received significant attention in recent years. These algorithms use so called lifting rules to identify symmetries in the first-order representation and reduce the inference problem over a large probabilistic model to an inference problem over a much smaller model. In this paper, we present two new lifting rules, which enable fast MAP inference in a large class of MLNs. Our first rule uses the concept of single occurrence equivalence class of logical variables, which we define in the paper. The rule states that the MAP assignment over an MLN can be recovered from a much smaller MLN, in which each logical variable in each single occurrence equivalence class is replaced by a constant (i.e., an object in the domain of the variable).

Variational inference algorithms have proven successful for Bayesian analysis in large data settings, with recent advances using stochastic variational inference (SVI). However, such methods have largely been studied in independent or exchangeable data settings. We develop an SVI algorithm to learn the parameters of hidden Markov models (HMMs) in a time-dependent data setting. The challenge in applying stochastic optimization in this setting arises from dependencies in the chain, which must be broken to consider minibatches of observations. We propose an algorithm that harnesses the memory decay of the chain to adaptively bound errors arising from edge effects. We demonstrate the effectiveness of our algorithm on synthetic experiments and a large genomics dataset where a batch algorithm is computationally infeasible.

We present an algorithm for finding a chordal Markov network that maximizes any given decomposable scoring function. The algorithm is based on a recursive characterization of clique trees, and it runs in O(4 n) time for n vertices. On an eight-vertex benchmark instance, our implementation turns out to be about ten million times faster than a recently proposed, constraint satisfaction based algorithm (Corander et al., NIPS 2013). Within a few hours, it is able to solve instances up to 18 vertices, and beyond if we restrict the maximum clique size. We also study the performance of a recent integer linear programming algorithm (Bartlett and Cussens, UAI 2013).

A core goal in systems neuroscience and neuroethology is to understand how neural circuits generate naturalistic behavior. One foundational idea is that complex naturalistic behavior may be composed of sequences of stereotyped behavioral syllables, which combine to generate rich sequences of actions. To investigate this, a common approach is to use autoregressive hidden Markov models (ARHMMs) to segment video into discrete behavioral syllables. While these approaches have been successful in extracting syllables that are interpretable, they fail to account for other forms of behavioral variability, such as differences in speed, which may be better described as continuous in nature. To overcome these limitations, we introduce a class of warped ARHMMs (WARHMM). As is the case in the ARHMM, behavior is modeled as a mixture of autoregressive dynamics.

We introduce a novel sampling algorithm for Markov chain Monte Carlo-based Bayesian inference for factorial hidden Markov models. This algorithm is based on an auxiliary variable construction that restricts the model space allowing iterative exploration in polynomial time. The sampling approach overcomes limitations with common conditional Gibbs samplers that use asymmetric updates and become easily trapped in local modes. Instead, our method uses symmetric moves that allows joint updating of the latent sequences and improves mixing. We illustrate the application of the approach with simulated and a real data example.

Markov Logic Networks (MLNs) are weighted first-order logic templates for generating large (ground) Markov networks. Lifted inference algorithms for them bring the power of logical inference to probabilistic inference. These algorithms operate as much as possible at the compact first-order level, grounding or propositionalizing the MLN only as necessary. As a result, lifted inference algorithms can be much more scalable than propositional algorithms that operate directly on the much larger ground network. Unfortunately, existing lifted inference algorithms suffer from two interrelated problems, which severely affects their scalability in practice. First, for most real-world MLNs having complex structure, they are unable to exploit symmetries and end up grounding most atoms (the grounding problem).

In this paper, we propose a learning approach for the Mixture of Hidden Markov Models (MHMM) based on the Method of Moments (MoM). Computational advantages of MoM make MHMM learning amenable for large data sets. It is not possible to directly learn an MHMM with existing learning approaches, mainly due to a permutation ambiguity in the estimation process. We show that it is possible to resolve this ambiguity using the spectral properties of a global transition matrix even in the presence of estimation noise. We demonstrate the validity of our approach on synthetic and real data.

In this paper, we present a new approach for lifted MAP inference in Markov logic networks (MLNs). The key idea in our approach is to compactly encode the MAP inference problem as an Integer Polynomial Program (IPP) by schematically applying three lifted inference steps to the MLN: lifted decomposition, lifted conditioning, and partial grounding. Our IPP encoding is lifted in the sense that an integer assignment to a variable in the IPP may represent a truth-assignment to multiple indistinguishable ground atoms in the MLN. We show how to solve the IPP by first converting it to an Integer Linear Program (ILP) and then solving the latter using state-of-the-art ILP techniques. Experiments on several benchmark MLNs show that our new algorithm is substantially superior to ground inference and existing methods in terms of computational efficiency and solution quality.

We show that the usual score function for conditional Markov networks can be written as the expectation over the scores of their spanning trees. We also show that a small random sample of these output trees can attain a significant fraction of the margin obtained by the complete graph and we provide conditions under which we can perform tractable inference. The experimental results confirm that practical learning is scalable to realistic datasets using this approach.

The Markov property serves as a foundational assumption in most existing work on vehicle driving behavior, positing that future states depend solely on the current state, not the series of preceding states. This study validates the Markov properties of vehicle trajectories for both Autonomous Vehicles (AVs) and Human-driven Vehicles (HVs). A statistical method used to test whether time series data exhibits Markov properties is applied to examine whether the trajectory data possesses Markov characteristics. t test and F test are additionally introduced to characterize the differences in Markov properties between AVs and HVs. Based on two public trajectory datasets, we investigate the presence and order of the Markov property of different types of vehicles through rigorous statistical tests. Our findings reveal that AV trajectories generally exhibit stronger Markov properties compared to HV trajectories, with a higher percentage conforming to the Markov property and lower Markov orders. In contrast, HV trajectories display greater variability and heterogeneity in decision-making processes, reflecting the complex perception and information processing involved in human driving. These results have significant implications for the development of driving behavior models, AV controllers, and traffic simulation systems. Our study also demonstrates the feasibility of using statistical methods to test the presence of Markov properties in driving trajectory data.