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In this paper we present a Bayesian competing risk proportional hazards model to describe mortgage defaults and prepayments. We develop Bayesian inference for the model using Markov chain Monte Carlo methods. Implementation of the model is illustrated using actual default/prepayment data and additional insights that can be obtained from the Bayesian analysis are discussed.

De Facto, signal processing is the interpolation and extrapolation of a sequence of observations viewed as a realization of a stochastic process. Its role in applied statistics ranges from scenarios in forecasting and time series analysis, to image reconstruction, machine learning, and the degradation modeling for reliability assessment. A general solution to the problem of filtering and prediction entails some formidable mathematics. Efforts to circumvent the mathematics has resulted in the need for introducing more explicit descriptions of the underlying process. One such example, and a noteworthy one, is the Kalman Filter Model, which is a special case of state space models or what statisticians refer to as Dynamic Linear Models. Implementing the Kalman Filter Model in the era of "big and high velocity non-Gaussian data" can pose computational challenges with respect to efficiency and timeliness. Particle filtering is a way to ease such computational burdens. The purpose of this paper is to trace the historical evolution of this development from its inception to its current state, with an expository focus on two versions of the particle filter, namely, the propagate first-update next and the update first-propagate next version. By way of going beyond a pure review, this paper also makes transparent the importance and the role of a less recognized principle, namely the principle of conditionalization, in filtering and prediction based on Bayesian methods. Furthermore, the paper also articulates the philosophical underpinnings of the filtering and prediction set-up, a matter that needs to ne made explicit, and Yule's decomposition of a random variable in terms of a sequence of innovations.

We develop a new class of dynamic multivariate Poisson count models that allow for fast online updating and we refer to these models as multivariate Poisson-scaled beta (MPSB). The MPSB model allows for serial dependence in the counts as well as dependence across multiple series with a random common environment. Other notable features include analytic forms for state propagation and predictive likelihood densities. Sequential updating occurs through the updating of the sufficient statistics for static model parameters, leading to a fully adapted particle learning algorithm and a new class of predictive likelihoods and marginal distributions which we refer to as the (dynamic) multivariate confluent hyper-geometric negative binomial distribution (MCHG-NB) and the the dynamic multivariate negative binomial (DMNB) distribution. To illustrate our methodology, we use various simulation studies and count data on weekly non-durable goods consumer demand., Comment: 31 pages, 9 figures

Managing risk at the aggregate level is crucial for banks and financial institutions as required by the Basel III framework. In this paper, we introduce discrete time Bayesian state space models with Poisson measurements to model aggregate mortgage default rate. We discuss parameter updating, filtering, smoothing, forecasting and estimation using Markov chain Monte Carlo methods. In addition, we investigate the dynamic behavior of the default rate and the effects of macroeconomic variables. We illustrate the use of the proposed models using actual U.S. residential mortgage data and discuss insights gained from Bayesian analysis., Comment: Published in at http://dx.doi.org/10.1214/13-AOAS632 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The Bayesian measure of sample information about the parameter, known as Lindley's measure, is widely used in various problems such as developing prior distributions, models for the likelihood functions and optimal designs. The predictive information is defined similarly and used for model selection and optimal designs, though to a lesser extent. The parameter and predictive information measures are proper utility functions and have been also used in combination. Yet the relationship between the two measures and the effects of conditional dependence between the observable quantities on the Bayesian information measures remain unexplored. We address both issues. The relationship between the two information measures is explored through the information provided by the sample about the parameter and prediction jointly. The role of dependence is explored along with the interplay between the information measures, prior and sampling design. For the conditionally independent sequence of observable quantities, decompositions of the joint information characterize Lindley's measure as the sample information about the parameter and prediction jointly and the predictive information as part of it. For the conditionally dependent case, the joint information about parameter and prediction exceeds Lindley's measure by an amount due to the dependence. More specific results are shown for the normal linear models and a broad subfamily of the exponential family. Conditionally independent samples provide relatively little information for prediction, and the gap between the parameter and predictive information measures grows rapidly with the sample size., Comment: Published in at http://dx.doi.org/10.1214/10-STS329 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)