Quando uma matriz de transições é representada como o produto de duas matrizes estocásticas, ao trocarmos a ordem dos fatores obtemos uma outra matriz de transições que retém características fundamentais da sua precursora (e portanto do processo estocástico subjacente). Como a nova matriz será potencialmente muito menor do que a original, a ideia de substituir a segunda pela primeira surge naturalmente. Tal substituição, apelidada de “O Truque da Fatoração Estocástica,” pode de fato ser feita em algumas situações, levando a ganhos tanto computacionais quanto estatísticos. O seminário terá como foco o truque da fatoração estocástica. Partindo de uma definição formal, discutirei algumas situações em que é possível aplicá-lo e as dificuldades em se fazê-lo na prática, assim como possíveis soluções. Aplicações específicas serão usadas para ilustrar o assunto, como o cálculo do PageRank –operação realizada pelo Google para ordenar as páginas retornadas em uma busca– e problemas de tomada de decisão tão diversos quanto controle viral de HIV, gerenciamento de processos industriais e jogos de carta.

We extend classical pyramidal algorithms of signal processing on the torus to the case of generic weighted graphs. This is done by making a connection with “local equilibria” appearing in Diaconis and Fill intertwining relation. This connection actually provides a way to identify local equilibria of generic Markov chains starting from the adaptation, through random spanning forests, of the classical subsampling procedure of signal processing. This is work in progress with Luca Avena, Fabienne Castell and Clothilde Mélot.

In this talk we assume a multivariate risk model has been developed for a portfolio and its capital derived as a homogeneous risk measure. The Euler (or gradient) principle, then, states that the capital to be allocated to each component of the portfolio has to be calculated as an expectation conditional to a rare event, which can be challenging to evaluate in practice. We exploit the copula-dependence within the portfolio risks to design a Sequential Monte Carlo Samplers based estimate to the marginal conditional expectations involved in the problem, showing its efficiency through a series of computational examples.

Consider a graph whose vertices represent n uniform random points in the unit square. One may form a random geometric graph by connecting two points by an edge if the distance of the points is at most r. Let c be a positive integer. We form an irrigation subgraph of the random geometric graph by selecting, at random, c vertices among the neighbors of any given vertex, and keeping only the edges joining the vertex to the selected neighbors. We present various results about connectivity properties of such graphs. Joint work with Nicolas Broutin and Luc Devroye.

Eleições, especialmente em países grandes como o Brasil, nos fornecem uma grande quantidade de informações a partir de bancos de dados disponíveis na Internet, que podem nos ajudar a entender como indivíduos interagem e influenciam uns aos outros. Neste seminário, analisaremos extensivamente os dados de eleições brasileiras durante o período 1970-2014. Através da distribuição estatística de votos dos candidatos a deputado e senador, faremos uma análise comparativa de diferentes eleições. Iremos discutir também um modelo baseado em um sistema de equações diferenciais não-lineares acopladas com parâmetros estocásticos. Este modelo reproduz bem o comportamento dos dados observados e nos permite relacionar um parâmetro da distribuição de votos com as redes de interações sociais entre candidatos e eleitores, e entre os eleitores entre si.

Considere um modelo de Percolação de sítios, independente, com parâmetro $p in (0,1)$ em $Z^d, dgeq 2$, onde há apenas elos de comprimento unitário (primeiros vizinhos) e elos de longo alcance de comprimento $k$ paralelos aos eixos coordenados. Mostramos que o ponto crítico deste modelo converge para $p_c(Z^{2d})$, o ponto crítico da Percolação ordinária (primeiros vizinhos) em $Z^{2d}$ , quando $k$ vai para o infinito. Nós também generalizamos este resultado para modelos cujos elos de longo alcance têm comprimentos variados. Trabalho conjunto com Rémy Sanchis (MAT-UFMG) e Roger Silva (EST-UFMG).

Local volatility models are extensively used and well-recognized for hedging and pricing in financial markets. They are frequently used, for instance, in the evaluation of exotic options so as to avoid arbitrage opportunities with respect to other instruments. The PDE (inverse) problem consists in recovering the time and space varying diffusion coefficient in a parabolic equation from limited data. It is known that this corresponds to an ill-posed problem. The ill-posed character of local volatility surface calibration from market prices requires the use of regularization techniques either implicitly or explicitly. Such regularization techniques have been widely studied for a while and are still a topic of intense research. We have employed convex regularization tools and recent inverse problem advances to deal with the local volatility calibration problem. We describe a theoretical approach to calibrate the local volatility surface from quoted derivative prices, by introducing convex regularization techniques and a priori information. We investigate theoretical as well as practical consequences of our methods and illustrate our results both with data from commodity markets.

This work is part of ongoing collaboration with V. Albani (IMPA), A. De Cezaro (FURGS), and O. Scherzer (Vienna).

Jump-diffusions have considerable appeal as flexible families of stochastic models. Making statistical inference based on discrete observations of such processes is a complex and challenging problem. Its infinite-dimensional nature has required from existing inference methodologies the use of discrete approximations that naturally represent a considerable source of error. In this talk, we rely on a novel algorithm to perform exact simulation of jump-diffusions bridges as the basis to develop an MCMC algorithm to make inference for jump-diffusion processes. The resulting infinite-dimensional Markov chain has the exact posterior distribution of the parameters and missing paths as its invariant distribution. More specifically, it is a Gibbs Sampling with Barker’s steps. The methodology is exact in the sense that it is free of discretisation error and Monte Carlo error is the only source of inaccuracy. The exactness feature is related to the simulation of events of unknown probability. A simulated example is presented to illustrate the methodology.

Although more than a decade has passed since the first eukaryotic genome was sequenced, the molecular basis of genome organization and complexity remains a largely unresolved problem. The relationship of genotype to phenotype has proven particularly challenging. I use gametogenesis in Drosophila as a model system to study the evolution and phenotypic expression of genomic features. Gametogenesis is a fascinating biological process; it varies temporally throughout development, and has profound evolutionary impact in that it provides the raw material for the next generation – the gamete. To date, gametogenesis research has primarily focused on single gene studies of fertility. In contrast, I apply a genomic perspective to the overall process of gametogenesis to understand the role sexual selection plays in genome evolution. In my research on genome evolution in Drosophila melanogaster, I have combined bioinformatics and statistics with experimental genomic and molecular genetic methods to obtain large-scale gene expression data on gametogenesis, or spermatogenic- stage-specific transcriptome (SpermPress). The results help to solve two classical problems that have puzzled biologists for decades: evidence for Meiotic Sex Chromosome inactivation and for Post-meiotic transcription. In this talk, I present the results obtained through the application of advanced Bayesian statistics to Gene Chip microarray data. I also introduce another puzzle yet to be solved in the evolutionary biology field related to the role of sperm haploid selection in the evolution of new genes.% The discussion of alternative analyses and models on spermatogenic transcriptome and gene age is pressing and represents part of my current research agenda.

In this seminar we present a multilevel binary model when the outcome is measured with uncertainty. We are interested in obtaining association measures, such as odds ratio, while taking into account specificity and sensitivity of the outcome. The data set come from a sampling scheme called time-location sampling, and therefore the sample design is incorporate into the model via random effects terms. Posterior inference is implemented using Hamiltonian Monte Carlo and also integrated nested Laplace approximation (INLA). A simulation study is provided and the method is applied on Brazilian alcohol and drug abuse data sets.

The entropy and other measures related to mutual information and/or divergence between random vectors, such as the Shannon index and the Kullback-Leibler divergence, have been widely studied in the case of the multivariate normal distribution. We extend these tools to the more flexible families of multivariate skew-elliptical distributions. We study in detail the cases of the multivariate skew-normal and skewt distributions. We illustrate our findings in the context of two real applications.