Stefano Peluso

Bayesian Cluster Analysis: Point Estimation and Credible Balls. Contributed discussion

I begin my discussion by giving an overview of the main results. Then I proceed to touch upon issues about whether the credible ball constructed can be interpreted as a confidence ball, suggestions on reducing computational costs, and posterior consistency or contraction rates.Clustering is widely studied in statistics and machine learning, with applications in a variety of fields. As opposed to popular algorithms such as agglomerative hierarchical clustering or k-means which return a single clustering solution, Bayesian nonparametric models provide a posterior over the entire space of partitions, allowing one to assess statistical properties, such as uncertainty on the number of clusters. However, an important problem is how to summarize the posterior; the huge dimension of partition space and difficulties in visualizing it add to this problem. In a Bayesian analysis, the posterior of a real-valued parameter of interest is often summarized by reporting a point estimate such as the posterior mean along with 95% credible intervals to characterize uncertainty. In this paper, we extend these ideas to develop appropriate point estimates and credible sets to summarize the posterior of the clustering structure based on decision and information theoretic techniques.Clustering is widely studied in statistics and machine learning, with applications in a variety of fields. As opposed to popular algorithms such as agglomerative hierarchical clustering or k-means which return a single clustering solution, Bayesian nonparametric models provide a posterior over the entire space of partitions, allowing one to assess statistical properties, such as uncertainty on the number of clusters. However, an important problem is how to summarize the posterior; the huge dimension of partition space and difficulties in visualizing it add to this problem. In a Bayesian analysis, the posterior of a real-valued parameter of interest is often summarized by reporting a point estimate such as the posterior mean along with 95% credible intervals to characterize uncertainty. In this paper, we extend these ideas to develop appropriate point estimates and credible sets to summarize the posterior of the clustering structure based on decision and information theoretic techniques.

Learning Markov Equivalence Classes of Directed Acyclic Graphs: An Objective Bayes Approach

A Markov equivalence class contains all the Directed Acyclic Graphs (DAGs) encoding the same conditional independencies, and is represented by a Completed Partially Directed DAG (CPDAG), also named Essential Graph (EG). We approach the problem of model selection among noncausal sparse Gaussian DAGs by directly scoring EGs, using an objective Bayes method. Specifically, we construct objective priors for model selection based on the Fractional Bayes Factor, leading to a closed form expression for the marginal likelihood of an EG. Next we propose an MCMC strategy to explore the space of EGs, possibly accounting for sparsity constraints, and illustrate the performance of our method on simulation studies, as well as on a real dataset. Our method is fully Bayesian and thus provides a coherent quantification of inferential uncertainty, requires minimal prior specification, and shows to be competitive in learning the structure of the data-generating EG when compared to alternative state-of-the-art algorithms.

Hidden Leaders: Identifying High-Frequency Lead-Lag Structures in a Multivariate Price Formation Framework

Motivated by the empirical evidence of high-frequency lead-lag effects and cross-asset linkages, we introduce a multi-asset price formation model which generalizes standard univariate microstructure models of lagged price adjustment. Econometric inference on such model provides: (i) a unified statistical test for the presence of lead-lag correlations in the latent price process and for the existence of a multi-asset price formation mechanism; (ii) separate estimation of contemporaneous and lagged dependencies; (iii) an unbiased estimator of the integrated covariance of the efficient martingale price process that is robust to microstructure noise, asynchronous trading and lead-lag dependencies. Through an extensive simulation study, we compare the proposed estimator to alternative approaches and show its advantages in recovering the true lead-lag structure of the latent price process. Our application to a set of NYSE stocks provides empirical evidence for the existence of a multi-asset price formation mechanism and sheds light on its market microstructure determinants.

Robust identification of highly persistent interest rate regimes

Parametric specifications in State Space Models (SSMs) are a source of bias in case of mismatch between modeling assumptions and reality. We propose a Bayesian semiparametric SSM that is robust to misspecified emission distributions. The Markovian nature of the latent stochastic process creates a temporal dependence and links the random probability distributions of the observations in a mixture of products of Dirichlet processes (MPDP). The model is shown to be adequate and it is applied to simulated data and to the motivating empirical problem of regime shifts in interest rates with latent state persistence. Bayesian semiparametric State Space model robust to imprecise emission distribution.Application: robustly identify highly persistent regime shifts in interest rates.Adequacy is shown to theoretically justify the model.