Introduction

This seminar was minister by the renowned professor Hélio S. Migon, who is one of the pioneers of statistics in Brazil, especially in the Bayesian statistics field. The first three weeks of lectures were face to face, however due to the COVID-19 pandemic the remain lectures were online.

The seminar was an introduction to Bayesian Dynamic Linear Model (DLM) ideas. We have four lessons discussing the basic concepts and definitions of Bayesian DLM and after that the students had to choose articles related to Bayesian dynamic models to present as a seminar format.

Beyond any doubt the seminar and the possibility to meet professor Migon were a turning point in my academic and professional life. I have been really grateful for that opportunity.

Bayesian Dynamic Linear Model

I shall give a brief introduction to DLM focus on the model specification. I plan to do a specific post about Bayesian DLM with more details and illustrative examples.

The class of DLM is a natural extension of the linear regression model with the parameters varying along time. This class of model was introduced by Harrison and Stevens (1976) in their seminal work entitled “Bayesian Forecasting” published in the Journal of the Royal Statistical Society. Series B (Methodological).

Based on many years of industrial experience working as “operational” forecaster the authors proposed a method which has the following main features:

• parametric form: the model has meaningful dynamic parameters, which can be easily interpretable by practitioners.

• sequential update: model parameters are sequentially update as new data becomes available using Bayes’ theorem.

• probabilistic information about parameters and forecasts are available, which can help decision making under uncertainty.

Let \(Y_t\) be a scalar series at time \(t\). The DLM is defined by the observation and evolution (state) equations, respectively:

\[\begin{eqnarray*} Y_t &=& \mathbf{F}_t^\top \boldsymbol{\theta}_t + \nu_t, \qquad \quad \nu_t \sim N(0, V) \\ \boldsymbol{\theta}_t &=& \mathbf{G}_t \boldsymbol{\theta}_{t-1} + \boldsymbol{\omega}_t, \qquad \boldsymbol{\omega}_t \sim N(0, \mathbf{W}_t) \end{eqnarray*}\] where the error sequences \(\nu_t\) and \(\boldsymbol{\omega}_t\) are internally and mutually independent.

For some prior moments \(\mathbf{m}_0\) and \(\mathbf{C}_0\), the initial prior information is denote by \((\boldsymbol{\theta}_0 \mid D_0) \sim N(\mathbf{m}_0, \mathbf{C}_0)\). By using Bayes’ theorem the posterior and one-step ahead predictive distributions can be obtained analytically.

Course assignments

During the seminar we should have read articles about dynamic models and organized a presentations about them. In the sequel, I list the papers that I have read and the presentations organized (in Portuguese) summarizing the main ideas.