No abstract

The spread of COVID‐19 has been greatly impacted by regulatory policies and behavior patterns that vary across counties, states, and countries. Population‐level dynamics of COVID‐19 can generally be described using a set of ordinary differential equations, but these deterministic equations are insufficient for modeling the observed case rates, which can vary due to local testing and case reporting policies and nonhomogeneous behavior among individuals. To assess the impact of population mobility on the spread of COVID‐19, we have developed a novel Bayesian time‐varying coefficient state‐space model for infectious disease transmission. The foundation of this model is a time‐varying coefficient compartment model to recapitulate the dynamics among susceptible, exposed, undetected infectious, detected infectious, undetected removed, hospitalized, detected recovered, and detected deceased individuals. The infectiousness and detection parameters are modeled to vary by time, and the infectiousness component in the model incorporates information on multiple sources of population mobility. Along with this compartment model, a multiplicative process model is introduced to allow for deviation from the deterministic dynamics. We apply this model to observed COVID‐19 cases and deaths in several U.S. states and Colorado counties. We find that population mobility measures are highly correlated with transmission rates and can explain complicated temporal variation in infectiousness in these regions. Additionally, the inferred connections between mobility and epidemiological parameters, varying across locations, have revealed the heterogeneous effects of different policies on the dynamics of COVID‐19.

A probability model exhibits instability if small changes in a data outcome result in large and, often unanticipated, changes in probability. This instability is a property of the probability model, given by a distributional form and a given configuration of parameters. For correlated data structures found in several application areas, there is increasing interest in identifying such sensitivity in model probability structure. We consider the problem of quantifying instability for general probability models defined on sequences of observations, where each sequence of length $N$ has a finite number of possible values that can be taken at each point. A sequence of probability models, indexed by $N$, and an associated parameter sequence result to accommodate data of expanding dimension. Model instability is formally shown to occur when a certain log probability ratio under such models grows faster than $N$. In this case, a one component change in the data sequence can shift probability by orders of magnitude. Also, as instability becomes more extreme, the resulting probability models are shown to tend to degeneracy, placing all their probability on potentially small portions of the sample space. These results on instability apply to large classes of models commonly used in random graphs, network analysis and machine learning contexts.

For spatial and network data, we consider models formed from a Markov random field (MRF) structure and the specification of a conditional distribution for each observation. At issue, fast simulation from such MRF models is often an important consideration, particularly when repeated generation of large numbers of data sets is required (e.g., for approximating sampling distributions). However, a standard Gibbs strategy for simulating from MRF models involves single-updates, performed with the conditional distribution of each observation in a sequential manner, whereby a Gibbs iteration may become computationally involved even for relatively small samples. As an alternative, we describe a general way to simulate from MRF models using Gibbs sampling with "concliques" (i.e., groups of non-neighboring observations). Compared to standard Gibbs sampling, this simulation scheme can be much faster by reducing Gibbs steps and by independently updating all observations per conclique at once. We detail the simulation method, establish its validity, and assess its computational performance through numerical studies, where speed advantages are shown for several spatial and network examples. arXiv:1808.04739v1 [stat.CO]

A restricted Boltzmann machine (RBM) is an undirected graphical model constructed for discrete or continuous random variables, with two layers, one hidden and one visible, and no conditional dependency within a layer. In recent years, RBMs have risen to prominence due to their connection to deep learning. By treating a hidden layer of one RBM as the visible layer in a second RBM, a deep architecture can be created. RBMs thereby are thought to have the ability to encode very complex and rich structures in data, making them attractive for supervised learning. However, the generative behavior of RBMs largely is unexplored and typical fitting methodology does not easily allow for uncertainty quantification in addition to point estimates. In this paper, we discuss the relationship between RBM parameter specification in the binary case and model properties such as degeneracy, instability and uninterpretability. We also describe the associated difficulties that can arise with likelihood‐based inference and further discuss the potential Bayes fitting of such (highly flexible) models, especially as Gibbs sampling (quasi‐Bayes) methods often are advocated for the RBM model structure.