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We introduce a Bayesian approach for analyzing (possibly) high-dimensional dependent data that are distributed according to a member from the natural exponential family of distributions. This problem requires extensive methodological advancements, as jointly modeling high-dimensional dependent data leads to the so-called "big n problem." The computational complexity of the "big n problem" is further exacerbated when allowing for non-Gaussian data models, as is the case here. Thus, we develop new computationally efficient distribution theory for this setting. In particular, we introduce something we call the "conjugate multivariate distribution," which is motivated by the univariate distribution introduced in Diaconis and Ylvisaker (1979). Furthermore, we provide substantial theoretical and methodological development including: results regarding conditional distributions, an asymptotic relationship with the multivariate normal distribution, conjugate prior distributions, and full-conditional distributions for a Gibbs sampler. The results in this manuscript are extremely general, and can be adapted to many different settings. We demonstrate the proposed methodology through simulated examples and analyses based on estimates obtained from the US Census Bureaus' American Community Survey (ACS).

The long term goal of this interdisciplinary research program continues to be the demonstration of Bayesian Hierarchical Model (BHM) utility in several aspects of operational ocean forecasting. The specific goals in the current phase of the research are: 1) publications of results from ensemble ocean forecast experiments driven by the surface vector wind (SVW) BHM; b) consolidating impacts results in MFS reforecast experiments for the time-dependent error-covariance BHM; and c) running the first multi-model and multi-parameter super-ensemble BHM experiments. Research objectives leading to the publication of manuscripts regarding SVW-BHM include: 1) constructing 3 appendices for Milliff et al (2009) to: a) demonstrate a systematic approach to (future) process model development; b) document, in probability model notation, the complete SVW-BHM, as well as expressions for the full conditional distributions; and c) document the SVW-BHM hyperprior specifications; 2) Re-writing the text and updating figures for Bonazzi et al. (2009); and 3) incorporating co-author final edits for Milliff et al. (2009) and Bonazzi et al. (2009). Research objectives leading to consolidation of results for the time-dependent error-covariance BHM include: 1) supplying an anomaly-only data stage version of the error-covariance BHM to MFS for reforecast experiments; and 2) interpreting reforecast results, and iterating with MFS for future reforecast experimental design. Research objectives leading to the first multi-model and multi-parameter super-enesmble BHM runs include: 1) evaluating preliminary experiments based on a Levantine Intermediate Water (LIW) formation rate target process; 2) re-organizing the target process to focus on temperature (T) and salinity (S) profile evolution at 2 locations in the region of LIW formation (Rhodes Gyre); and 3) providing T(z,t) and S(z,t) data files (.mat) to Mark Berliner for preliminary modelling.

Experimental semantics has shed light on the influence of various factors on the meaning of quantifiers (e.g., many, most). In addition to their truth-conditional value, experimental semantics has established that meaning of words is also influenced by individual characteristics, even if contextual clues are accounted for. That is, individuals differ at what threshold they perceive a quantifier to be true and how strict they are with respect to this threshold. The Bayesian framework is well suited to describe meaning representations as computational models, since it is capable of incorporating into the models directly the fields rich prior knowledge, its logic-derived assumptions, but also account for individual differences. But how can we use Bayesian methods to describe and test our theoretical expectations about the meaning of quantifiers? This project serves a dual purpose. First, we expand upon the Bayesian hierarchical logistic model of meaning representation proposed by Ramotowska et al. (2022). To align the model with domain expertise, we assign informative prior distributions that restrict parameters to admissible ranges, align with literature-derived expectations, and generate meaningful predictions. Furthermore, we extend upon its existing model by adding a covariance structure on the threshold and response noise parameters across time points. This addition not only facilitates studying the stability of meaning representations but also opens up the possibility for exploring relationships between model parameters across different paradigms or tasks. Second, we demonstrate the utility of the Bayes factor as a powerful tool for testing competing theories. Researchers working with Bayesian models for meaning representation commonly rely on parameter estimation results (including Bayesian model averaging) and model fit indices to inform their inferences. However, to effectively test specific hypotheses, researchers need to adopt model comparison methods using Bayes factors. The Bayes factor compares the predictive adequacy of one model over the other and indicates the support by the data given the model. In particular, we propose a methodology to test informed hypotheses using the unconditional encompassing approach proposed (Gelfand, Smith, & Lee, 1992; Klugkist, Kato, & Hoijtink, 2005; Sedransk, Monahan, & Chiu, 1985). This methodology is intuitive, and easy-to-implement: all it requires are MCMC samples from the prior and posterior distributions of the parameters of interest.

Experimental semantics has shed light on the influence of various factors on the meaning of quantifiers (e.g., many, most). In addition to their truth-conditional value, experimental semantics has established that meaning of words is also influenced by individual characteristics, even if contextual clues are accounted for. That is, individuals differ at what threshold they perceive a quantifier to be true and how strict they are with respect to this threshold. The Bayesian framework is well suited to describe meaning representations as computational models, since it is capable of incorporating into the models directly the fields rich prior knowledge, its logic-derived assumptions, but also account for individual differences. But how can we use Bayesian methods to describe and test our theoretical expectations about the meaning of quantifiers? This project serves a dual purpose. First, we expand upon the Bayesian hierarchical logistic model of meaning representation proposed by Ramotowska et al. (2022). To align the model with domain expertise, we assign informative prior distributions that restrict parameters to admissible ranges, align with literature-derived expectations, and generate meaningful predictions. Furthermore, we extend upon its existing model by adding a covariance structure on the threshold and response noise parameters across time points. This addition not only facilitates studying the stability of meaning representations but also opens up the possibility for exploring relationships between model parameters across different paradigms or tasks. Second, we demonstrate the utility of the Bayes factor as a powerful tool for testing competing theories. Researchers working with Bayesian models for meaning representation commonly rely on parameter estimation results (including Bayesian model averaging) and model fit indices to inform their inferences. However, to effectively test specific hypotheses, researchers need to adopt model comparison methods using Bayes factors. The Bayes factor compares the predictive adequacy of one model over the other and indicates the support by the data given the model. In particular, we propose a methodology to test informed hypotheses using the unconditional encompassing approach proposed (Gelfand, Smith, & Lee, 1992; Klugkist, Kato, & Hoijtink, 2005; Sedransk, Monahan, & Chiu, 1985). This methodology is intuitive, and easy-to-implement: all it requires are MCMC samples from the prior and posterior distributions of the parameters of interest.

In this note we provide detailed derivations of two versions of small-variance asymptotics for hierarchical Dirichlet process (HDP) mixture models and the HDP hidden Markov model (HDP-HMM, a.k.a. the infinite HMM). We include derivations for the probabilities of certain CRP and CRF partitions, which are of more general interest.

Hierarchical models are versatile tools for joint modeling of data sets arising from different, but related, sources. Fully Bayesian inference may, however, become computationally prohibitive if the source-specific data models are complex, or if the number of sources is very large. To facilitate computation, we propose an approach, where inference is first made independently for the parameters of each data set, whereupon the obtained posterior samples are used as observed data in a substitute hierarchical model, based on a scaled likelihood function. Compared to direct inference in a full hierarchical model, the approach has the advantage of being able to speed up convergence by breaking down the initial large inference problem into smaller individual subproblems with better convergence properties. Moreover it enables parallel processing of the possibly complex inferences of the source-specific parameters, which may otherwise create a computational bottleneck if processed jointly as part of a hierarchical model. The approach is illustrated with both simulated and real data.

Providing direct and indirect contributions of more than $18 billion to the United States` gross output in 2004, the computer and video gaming industry is one of the fastest-growing sectors of entertainment. A significant portion of that market includes team-oriented online games. Players in these games often have a high-level of interest in statistics that help them assess their ability compared to other players. However few models exist that estimate individual player ratings from team competitions. The TrueSkill rating system is an example of one way to rate individuals from group competitions. This presentation presents a different model that also describes team abilities in terms of how well the individual players on the teams contribute to their team`s winning. In addition, the models presented include parameters that estimate other characteristics of the games themselves. The models are posed in a hierarchical Bayesian framework so the priors on the parameter variances can be inferred separately. The models are initially fit and evaluated using Markov-Chain Monte Carlo (MCMC) integration. Unfortunately, the amount of time it takes to fit the models using MCMC is longer than the average length of the competitions used to create the models, and therefore MCMC can not be used to rate the players real-time. The second part of this presentation gives an efficient recursive Newton-Raphson approximate Bayesian inference method to solve this problem. As with the TrueSkill system, the ratings and rankings derived can be used in order to improve gameplay in current matches and for helping players decide which matches to participate in. Companies and servers that apply well-developed statistics for assessing their players` abilities are more likely to attract and retain players, leading to greater success in the industry. ©2006 Microsoft Corporation. All rights reserved.

Comment: Bayesian Checking of the Second Levels of Hierarchical Models [arXiv:0802.0743]

Parameter ensembles or sets of point estimates constitute one of the cornerstones of modern statistical practice. This is especially the case in Bayesian hierarchical models, where different decision-theoretic frameworks can be deployed to summarize such parameter ensembles. The estimation of these parameter ensembles may thus substantially vary depending on which inferential goals are prioritised by the modeller. In this note, we consider the problem of classifying the elements of a parameter ensemble above or below a given threshold. Two threshold classification losses (TCLs) --weighted and unweighted-- are formulated. The weighted TCL can be used to emphasize the estimation of false positives over false negatives or the converse. We prove that the weighted and unweighted TCLs are optimized by the ensembles of unit-specific posterior quantiles and posterior medians, respectively. In addition, we relate these classification loss functions on parameter ensembles to the concepts of posterior sensitivity and specificity. Finally, we find some relationships between the unweighted TCL and the absolute value loss, which explain why both functions are minimized by posterior medians.

Parameter ensembles or sets of point estimates constitute one of the cornerstones of modern statistical practice. This is especially the case in Bayesian hierarchical models, where different decision-theoretic frameworks can be deployed to summarize such parameter ensembles. The estimation of these parameter ensembles may thus substantially vary depending on which inferential goals are prioritised by the modeller. In this note, we consider the problem of classifying the elements of a parameter ensemble above or below a given threshold. Two threshold classification losses (TCLs) --weighted and unweighted-- are formulated. The weighted TCL can be used to emphasize the estimation of false positives over false negatives or the converse. We prove that the weighted and unweighted TCLs are optimized by the ensembles of unit-specific posterior quantiles and posterior medians, respectively. In addition, we relate these classification loss functions on parameter ensembles to the concepts of posterior sensitivity and specificity. Finally, we find some relationships between the unweighted TCL and the absolute value loss, which explain why both functions are minimized by posterior medians.

Parameter ensembles or sets of point estimates constitute one of the cornerstones of modern statistical practice. This is especially the case in Bayesian hierarchical models, where different decision-theoretic frameworks can be deployed to summarize such parameter ensembles. The estimation of these parameter ensembles may thus substantially vary depending on which inferential goals are prioritised by the modeller. In this note, we consider the problem of classifying the elements of a parameter ensemble above or below a given threshold. Two threshold classification losses (TCLs) --weighted and unweighted-- are formulated. The weighted TCL can be used to emphasize the estimation of false positives over false negatives or the converse. We prove that the weighted and unweighted TCLs are optimized by the ensembles of unit-specific posterior quantiles and posterior medians, respectively. In addition, we relate these classification loss functions on parameter ensembles to the concepts of posterior sensitivity and specificity. Finally, we find some relationships between the unweighted TCL and the absolute value loss, which explain why both functions are minimized by posterior medians.

Most existing content-based filtering approaches learn user profiles independently without capturing the similarity among users. Bayesian hierarchical models \cite{Zhang:Efficient} learn user profiles jointly and have the advantage of being able to borrow discriminative information from other users through a Bayesian prior. However, the standard Bayesian hierarchical models assume all user profiles are generated from the same prior. Considering the diversity of user interests, this assumption could be improved by introducing more flexibility. Besides, most existing content-based filtering approaches implicitly assume that each user profile corresponds to exactly one user interest and fail to capture a user's multiple interests (information needs). In this paper, we present a flexible Bayesian hierarchical modeling approach to model both commonality and diversity among users as well as individual users' multiple interests. We propose two models each with different assumptions, and the proposed models are called Discriminative Factored Prior Models (DFPM). In our models, each user profile is modeled as a discriminative classifier with a factored model as its prior, and different factors contribute in different levels to each user profile. Compared with existing content-based filtering models, DFPM are interesting because they can 1) borrow discriminative criteria of other users while learning a particular user profile through the factored prior; 2) trade off well between diversity and commonality among users; and 3) handle the challenging classification situation where each class contains multiple concepts. The experimental results on a dataset collected from real users on digg.com show that our models significantly outperform the baseline models of L-2 regularized logistic regression and traditional Bayesian hierarchical model with logistic regression.

In synthetic aperture radar (SAR), images are formed by focusing the response of stationary objects to a single spatial location. On the other hand, moving targets cause phase errors in the standard formation of SAR images that cause displacement and defocusing effects. SAR imagery also contains significant sources of non-stationary spatially-varying noises, including antenna gain discrepancies, angular scintillation (glints) and complex speckle. In order to account for this intricate phenomenology, this work combines the knowledge of the physical, kinematic, and statistical properties of SAR imaging into a single unified Bayesian structure that simultaneously (a) estimates the nuisance parameters such as clutter distributions and antenna miscalibrations and (b) estimates the target signature required for detection/inference of the target state. Moreover, we provide a Monte Carlo estimate of the posterior distribution for the target state and nuisance parameters that infers the parameters of the model directly from the data, largely eliminating tuning of algorithm parameters. We demonstrate that our algorithm competes at least as well on a synthetic dataset as state-of-the-art algorithms for estimating sparse signals. Finally, performance analysis on a measured dataset demonstrates that the proposed algorithm is robust at detecting/estimating targets over a wide area and performs at least as well as popular algorithms for SAR moving target detection.

Comment: Bayesian Checking of the Second Levels of Hierarchical Models [arXiv:0802.0743]

Sampling from hierarchical Bayesian models is often difficult for MCMC methods, because of the strong correlations between the model parameters and the hyperparameters. Recent Riemannian manifold Hamiltonian Monte Carlo (RMHMC) methods have significant potential advantages in this setting, but are computationally expensive. We introduce a new RMHMC method, which we call semi-separable Hamiltonian Monte Carlo, which uses a specially designed mass matrix that allows the joint Hamiltonian over model parameters and hyperparameters to decompose into two simpler Hamiltonians. This structure is exploited by a new integrator which we call the alternating blockwise leapfrog algorithm. The resulting method can mix faster than simpler Gibbs sampling while being simpler and more efficient than previous instances of RMHMC.

Comment: Bayesian Checking of the Second Level of Hierarchical Models [arXiv:0802.0743]

We develop a new method to sample from posterior distributions in hierarchical models without using Markov chain Monte Carlo. This method, which is a variant of importance sampling ideas, is generally applicable to high-dimensional models involving large data sets. Samples are independent, so they can be collected in parallel, and we do not need to be concerned with issues like chain convergence and autocorrelation. Additionally, the method can be used to compute marginal likelihoods.

Hierarchical models are increasingly used in many applications. Along with this increased use comes a desire to investigate whether the model is compatible with the observed data. Bayesian methods are well suited to eliminate the many (nuisance) parameters in these complicated models; in this paper we investigate Bayesian methods for model checking. Since we contemplate model checking as a preliminary, exploratory analysis, we concentrate on objective Bayesian methods in which careful specification of an informative prior distribution is avoided. Numerous examples are given and different proposals are investigated and critically compared.

We discuss the methods of Evans and Moshonov [Bayesian Analysis 1 (2006) 893--914, Bayesian Statistics and Its Applications (2007) 145--159] concerning checking for prior-data conflict and their relevance to the method proposed in this paper. [arXiv:0802.0743]

We characterise the convergence of the Gibbs sampler which samples from the joint posterior distribution of parameters and missing data in hierarchical linear models with arbitrary symmetric error distributions. We show that the convergence can be uniform, geometric or sub-geometric depending on the relative tail behaviour of the error distributions, and on the parametrisation chosen. Our theory is applied to characterise the convergence of the Gibbs sampler on latent Gaussian process models. We indicate how the theoretical framework we introduce will be useful in analyzing more complex models.

We characterise the convergence of the Gibbs sampler which samples from the joint posterior distribution of parameters and missing data in hierarchical linear models with arbitrary symmetric error distributions. We show that the convergence can be uniform, geometric or sub-geometric depending on the relative tail behaviour of the error distributions, and on the parametrisation chosen. Our theory is applied to characterise the convergence of the Gibbs sampler on latent Gaussian process models. We indicate how the theoretical framework we introduce will be useful in analyzing more complex models.

This paper presents a sparse Bayesian inference approach that applies to sparse signal representation from overcomplete dictionaries in complex as well as real signal models. The approach is based on the two-layer hierarchical Bayesian prior representation of the Bessel K probability density function for the variable of interest. It allows for the Bayesian modeling of the l1-norm constraint for complex and real signals. In addition, the two-layer model leads to novel priors for the variable of interest that encourage more sparse representations than traditional prior models published in the literature do. An extension of the two-layer model to a three-layer model is also presented. Finally, we apply the fast Bayesian inference scheme by M. Tipping to the two- and three-layer hierarchical prior models to design iterative sparse estimators. We exploit the fact that the popular Fast Relevance Vector Machine (RVM) and Fast Laplace algorithms rely on the same inference scheme, yet on different hierarchical prior models, to compare the impact of the utilized prior model on the estimation performance. The numerical results show that the presented hierarchical prior models for sparse estimation effectively lead to sparse estimators with improved performance over Fast RVM and Fast Laplace in terms of convergence speed, sparseness and achieved mean-squared estimation error. In particular, our estimators show superior performance in low and moderate signal-to-noise ratio regimes, where state-of-the-art estimators fail to produce sparse signal representations.

Kossovsky recently conjectured that the distribution of leading digits of a chain of probability distributions converges to Benford's law as the length of the chain grows. We prove his conjecture in many cases, and provide an interpretation in terms of products of independent random variables and a central limit theorem. An interesting consequence is that in hierarchical Bayesian models priors tend to satisfy Benford's Law as the number of levels of the hierarchy increases, which allows us to develop some simple tests (based on Benford's law) to test proposed models. We give explicit formulas for the error terms as sums of Mellin transforms, which converges extremely rapidly as the number of terms in the chain grows. We may interpret our results as showing that certain Markov chain Monte Carlo processes are rapidly mixing to Benford's law.

Rejoinder: Bayesian Checking of the Second Levels of Hierarchical Models [arXiv:0802.0743]

Bayesian hierarchical modeling is a popular approach to capturing unobserved heterogeneity across individual units. However, standard estimation methods such as Markov chain Monte Carlo (MCMC) can be impracticable for modeling outcomes from a large number of units. We develop a new method to sample from posterior distributions of Bayesian models, without using MCMC. Samples are independent, so they can be collected in parallel, and we do not need to be concerned with issues like chain convergence and autocorrelation. The algorithm is scalable under the weak assumption that individual units are conditionally independent, making it applicable for large datasets. It can also be used to compute marginal likelihoods.

We introduce a general hierarchical Bayesian framework that incorporates a flexible nonparametric data model specification through the use of empirical likelihood methodology, which we term semiparametric hierarchical empirical likelihood (SHEL) models. Although general dependence structures can be readily accommodated, we focus on spatial modeling, a relatively underdeveloped area in the empirical likelihood literature. Importantly, the models we develop naturally accommodate spatial association on irregular lattices and irregularly spaced point-referenced data. We illustrate our proposed framework by means of a simulation study and through three real data examples. First, we develop a spatial Fay-Herriot model in the SHEL framework and apply it to the problem of small area estimation in the American Community Survey. Next, we illustrate the SHEL model in the context of areal data (on an irregular lattice) through the North Carolina sudden infant death syndrome (SIDS) dataset. Finally, we analyze a point-referenced dataset from the North American Breeding Bird survey that considers dove counts for the state of Missouri. In all cases, we demonstrate superior performance of our model, in terms of mean squared prediction error, over standard parametric analyses.

Neutron stars are a prime laboratory for testing physical processes under conditions of strong gravity, high density, and extreme magnetic fields. Among the zoo of neutron star phenomena, magnetars stand out for their bursting behaviour, ranging from extremely bright, rare giant flares to numerous, less energetic recurrent bursts. The exact trigger and emission mechanisms for these bursts are not known; favoured models involve either a crust fracture and subsequent energy release into the magnetosphere, or explosive reconnection of magnetic field lines. In the absence of a predictive model, understanding the physical processes responsible for magnetar burst variability is difficult. Here, we develop an empirical model that decomposes magnetar bursts into a superposition of small spike-like features with a simple functional form, where the number of model components is itself part of the inference problem. The cascades of spikes that we model might be formed by avalanches of reconnection, or crust rupture aftershocks. Using Markov Chain Monte Carlo (MCMC) sampling augmented with reversible jumps between models with different numbers of parameters, we characterise the posterior distributions of the model parameters and the number of components per burst. We relate these model parameters to physical quantities in the system, and show for the first time that the variability within a burst does not conform to predictions from ideas of self-organised criticality. We also examine how well the properties of the spikes fit the predictions of simplified cascade models for the different trigger mechanisms.

Long-term goals: The overall project goal has been to test the feasibility and practicality of Bayesian Hierarchical Model (BHM) methods in aspects of the Mediterranean Forecast System (MFS); an operational ocean data assimilation and forecast system. Three main objectives have been pursued in support of the project goal. They are: 1. a surface wind BHM (MFS-Wind-BHM) to drive ensemble ocean data assimilation and forecasts in MFS; 2. a time- and depth-dependent background error covariance BHM (MFS-Error-BHM) to evolve the background error covariance matrix in 13 sub-regions of the MFS forecast domain; and 3. a BHM to demonstrate super-ensemble forecast capabilities (MFS-SuperEnsemble-BHM) for ocean applications.

Year 2 of Phase 1 Bayesian Hierarchical Models (BHM) to Augment the Mediterranean Forecast System ( MFS) ended in May 2007. Long-term goals for Phase I included: a) development of an ensemble ocean forecast methodology based on a surface wind BHM (MFS-Wind-BHM) in data assimilation and forecast steps of the MFS; and b) development of a BHM for time-dependent background error covariance evolution (MFS-Error-BHM) in the MFS data assimilation system . Phase II of the project was initiated in June 2007. Long term goals for the second phase include the development of a BHM to guide ocean model super-ensemble experiments, in both multi-model and the multi-parameter experimental designs. The MFS ocean forecast model will be modified for multiparameter super-ensemble experiments, and MFS will be joined by a Mediterranean Sea implementation of the Regional Ocean Modeling System (MedROMS: http://www.med-roms.org) in multi-model super-ensemble experiments.

The long term goal, spanning both phases of the research program, is to demonstrate Bayesian Hierarchical Model (BHM) utility in several aspects of operational ocean forecasting. The specific goal in Phase II (beginning June 2007) is to complete a proof-of-concept demonstration of BHM methods in SuperEnsemble (SE) ocean forecasts. Multi-model and multi-parameter demonstrations are being developed in MFS-SuperEnsemble-BHM.

The long term goal, spanning both phases of the research program, is to demonstrate Bayesian Hierarchical Model (BHM) utility in several aspects of operational ocean forecasting. The specific goal in Phase II (beginning June 2007) is to complete a proof-of-concept demonstration of BHM methods in SuperEnsemble (SE) ocean forecasts. Multi-model and multi-parameter demonstrations are being developed in MFS-SuperEnsemble-BHM.

The long term goal of this interdisciplinary research program continues to be the demonstration of Bayesian Hierarchical Model (BHM) utility in several aspects of operational ocean forecasting. The specific goals in the current phase of the research are: 1) publications of results from ensemble ocean forecast experiments driven by the surface vector wind (SVW) BHM; b) consolidating impacts results in MFS reforecast experiments for the time-dependent error-covariance BHM; and c) running the first multi-model and multi-parameter super-ensemble BHM experiments.

Hierarchical Bayesian models can be especially useful in precision medicine settings, where clinicians are interested in estimating the patient-level latent variables associated with an individual's current health state and its trajectory. Such models are often fit using batch Markov Chain Monte Carlo (MCMC). However, the slow speed of batch MCMC computation makes it difficult to implement in clinical settings, where immediate latent variable estimates are often desired in response to new patient data. In this report, we discuss how importance sampling (IS) can instead be used to obtain fast, in-clinic estimates of patient-level latent variables. We apply IS to the hierarchical model proposed in Coley et al (2015) for predicting an individual's underlying prostate cancer state. We find that latent variable estimates via IS can typically be obtained in 1-10 seconds per person and have high agreement with estimates coming from longer-running batch MCMC methods. Alternative options for out-of-sample fitting and online updating are also discussed.

Year 2 of Phase 1 Bayesian Hierarchical Models (BHM) to Augment the Mediterranean Forecast System ( MFS) ended in May 2007. Long-term goals for Phase I included: a) development of an ensemble ocean forecast methodology based on a surface wind BHM (MFS-Wind-BHM) in data assimilation and forecast steps of the MFS; and b) development of a BHM for time-dependent background error covariance evolution (MFS-Error-BHM) in the MFS data assimilation system . Phase II of the project was initiated in June 2007. Long term goals for the second phase include the development of a BHM to guide ocean model super-ensemble experiments, in both multi-model and the multi-parameter experimental designs. The MFS ocean forecast model will be modified for multiparameter super-ensemble experiments, and MFS will be joined by a Mediterranean Sea implementation of the Regional Ocean Modeling System (MedROMS: http://www.med-roms.org) in multi-model super-ensemble experiments.

Eighteen months into the project, the long-term goals and objectives remain as stated in the progress report last year. Our goal is to develop an ensemble ocean forecast methodology, using Bayesian Hierarchical Modelling (BHM) tools. The ocean ensemble forecast methods to be developed should be practical enough to benefit the Mediterranean Forecast System (MFS) in its operational mode, and they should demonstrate forecast uncertainties during difficult to predict regime transitions in the Mediterranean Sea (e.g. the Fall transition, deep water formation). Two main objectives comprise the research plan. First, an ensemble of ocean initial conditions will be derived from realizations of the surface wind forcing as drawn from a posterior distribution of a BHM for the surface wind process. The surface wind field realizations have been used in separate data assimilation steps to produce unique, but realizable ocean initial conditions. The surface wind likelihood distributions are based on QuikSCAT data and ECMWF analyses. The prior distributions are based on a time-dependent augmentation of the stochastic geostrophy model introduced by Royle et al. (1998). The second objective involves the accurate representation of forecast error covariance evolution in MFS. The operational implementation of a reduced order optimal interpolation (ROOI) data assimilation method for MFS involves an ad-hoc truncation in the representation of the background error covariance. BHM can be used to remove this arbitrary truncation. Moreover, as ensemble forecasts are run in MFS for abrupt seasonal transition events, the ensemble spread will be used to refine priors in the BHM for error covariance evolution.

Eighteen months into the project, the long-term goals and objectives remain as stated in the progress report last year. Our goal is to develop an ensemble ocean forecast methodology, using Bayesian Hierarchical Modelling (BHM) tools. The ocean ensemble forecast methods to be developed should be practical enough to benefit the Mediterranean Forecast System (MFS) in its operational mode, and they should demonstrate forecast uncertainties during difficult to predict regime transitions in the Mediterranean Sea (e.g. the Fall transition, deep water formation).

The overall project goal has been to test the feasibility and practicality of Bayesian Hierarchical Model (BHM) methods in aspects of the Mediterranean Forecast System (MFS); an operational ocean data assimilation and forecast system. Three main objectives have been pursued in support of the project goal. They are: 1. a surface wind BHM (MFS-Wind-BHM) to drive ensemble ocean data assimilation and forecasts in MFS;LONG-TERM GOALS The overall project goal has been to test the feasibility and practicality of Bayesian Hierarchical Model (BHM) methods in aspects of the Mediterranean Forecast System (MFS); an operational ocean data assimilation and forecast system. Three main objectives have been pursued in support of the project goal. They are: 1. a surface wind BHM (MFS-Wind-BHM) to drive ensemble ocean data assimilation and forecasts in MFS; 2. a time- and depth-dependent background error covariance BHM (MFS-Error-BHM) to evolve the background error covariance matrix in 13 sub-regions of the MFS forecast domain; and 3. a BHM to demonstrate super-ensemble forecast capabilities (MFS-SuperEnsemble-BHM) for ocean applications.

Year 2 of Phase 1 Bayesian Hierarchical Models (BHM) to Augment the Mediterranean Forecast System ( MFS) ended in May 2007. Long-term goals for Phase I included: a) development of an ensemble ocean forecast methodology based on a surface wind BHM (MFS-Wind-BHM) in data assimilation and forecast steps of the MFS; and b) development of a BHM for time-dependent background error covariance evolution (MFS-Error-BHM) in the MFS data assimilation system . Phase II of the project was initiated in June 2007. Long term goals for the second phase include the development of a BHM to guide ocean model super-ensemble experiments, in both multi-model and the multi-parameter experimental designs. The MFS ocean forecast model will be modified for multiparameter super-ensemble experiments, and MFS will be joined by a Mediterranean Sea implementation of the Regional Ocean Modeling System (MedROMS: http://www.med-roms.org) in multi-model super-ensemble experiments.

The long term goal of this interdisciplinary research program continues to be the demonstration of Bayesian Hierarchical Model (BHM) utility in several aspects of operational ocean forecasting.

The long term goal of this interdisciplinary research program continues to be the demonstration of Bayesian Hierarchical Model (BHM) utility in several aspects of operational ocean forecasting. The specific goals in the current phase of the research are: 1) publications of results from ensemble ocean forecast experiments driven by the surface vector wind (SVW) BHM; b) consolidating impacts results in MFS reforecast experiments for the time-dependent error-covariance BHM; and c) running the first multi-model and multi-parameter super-ensemble BHM experiments. Research objectives leading to the publication of manuscripts regarding SVW-BHM include: 1) constructing 3 appendices for Milliff et al (2009) to: a) demonstrate a systematic approach to (future) process model development; b) document, in probability model notation, the complete SVW-BHM, as well as expressions for the full conditional distributions; and c) document the SVW-BHM hyperprior specifications; 2) Re-writing the text and updating figures for Bonazzi et al. (2009); and 3) incorporating co-author final edits for Milliff et al. (2009) and Bonazzi et al. (2009). Research objectives leading to consolidation of results for the time-dependent error-covariance BHM include: 1) supplying an anomaly-only data stage version of the error-covariance BHM to MFS for reforecast experiments; and 2) interpreting reforecast results, and iterating with MFS for future reforecast experimental design.

The long term goal, spanning both phases of the research program, is to demonstrate Bayesian Hierarchical Model (BHM) utility in several aspects of operational ocean forecasting. The specific goal in Phase II (beginning June 2007) is to complete a proof-of-concept demonstration of BHM methods in SuperEnsemble (SE) ocean forecasts. Multi-model and multi-parameter demonstrations are being developed in MFS-SuperEnsemble-BHM.

LONG-TERM GOALS. Eighteen months into the project, the long-term goals and objectives remain as stated in the progress report last year. Our goal is to develop an ensemble ocean forecast methodology, using Bayesian Hierarchical Modelling (BHM) tools. The ocean ensemble forecast methods to be developed should be practical enough to benefit the Mediterranean Forecast System (MFS) in its operational mode, and they should demonstrate forecast uncertainties during difficult to predict regime transitions in the Mediterranean Sea (e.g. the Fall transition, deep water formation).

The research undertaken under this effort has two major components. First the application of Bayesian inference theory is applied to problems ranging from Distributed Detection with multiple sensors clutter scene characterization/identification for airborne radar systems to adaptive CFAR detection with heterogeneous clutter. Secondly, multichannel radar detection algorithms are developed that are particularly suitable for airborne radar surveillance systems operating in a complex clutter/interference/noise environments.

The first full year of research for the project entitled Bayesian Hierarchical Models (BHM) to Augment the Mediterranean Ocean Forecast System (MFS) completed at the end of May 2006. Project achievements have met or exceeded plans put forth in the proposal. Prof. Nadia Pinardi (Univ. Bologna INGV) and Dr. Ralph F. Milliff (NWRA/CoRA) presented early results to Physical Oceanography Program Managers in a seminar at ONR Headquarters in Arlington in early May. This annual report reviews highlights from that presentation.

The overall project goal has been to test the feasibility and practicality of Bayesian Hierarchical Model (BHM) methods in aspects of the Mediterranean Forecast System (MFS); an operational ocean data assimilation and forecast system.