Palestra: Probabilistic model for integer partitions
Palestrante: Stella Brassesco (Instituto Venezolano de Investigaciones Científicas)

Data: 9 de março de 2020 (segunda-feira)
Hora: 15:30 h

Resumo: A family of independent random variables can be associated to the sequence p(n), which counts the number of partitions of a natural number n. The sum of those variables, suitably normalized, can be seen to converge to a Gaussian random variable, which suggests a method to obtain detailed asymptotics for p(n) an n goes to infinity. Moreover, the representation is useful to deduce asymptotic properties when the uniform distribution is considered on the set of partitions of n.

The problem is related with questions arising in several contexts.

Título: SRW on the plane conditioned on not hitting the origin
Palestrante: Daniel Ungaretti Borges (Unicamp)

Data: 11 de novembro de 2019 (segunda-feira)
Hora: 15:30 h
Local: B106-b – Bloco B - CT – Instituto de Matemática - UFRJ

Resumo: We will discuss the two-dimensional simple random walk conditioned on never hitting the origin, which is, formally speaking, the Doob’s h-transform of the simple random walk with respect to the potential kernel. This random walk is the main building-block of the construction of random interlacements on the plane introduced by Comets, Popov and Vachkovskaia. However, this walk has become an interesting object on its own. To justify this claim we present a few of its properties, citing some of the current literature and presenting the results of a recent joint work with Serguei Popov (UNICAMP) and Leonardo Rolla (UBA/NYU Shanghai).

Título: Dependent Mixtures: Modelling cell lineages
Palestrante: Carlos Tadeu Pagani Zanini (UFRJ)

Data: 25/11/2019 (segunda-feira)
Hora: 15:30
Local: B106-b – Bloco B - CT – IM/UFRJ

Palestrante: Carlos Tadeu Pagani Zanini (UFRJ)

Resumo: Cell lineage data comes from single-cell transcriptomics and it is used to recover the evolutionary path of cells in a given environment. The different evolutionary stages of the cells can be probabilistically described by distinct components in a mixture model. This work proposes a Bayesian dependent mixture model where the dependence on the components of the mixture explicitly incorporates the biological structure that characterizes cell lineage applications. We use a random tree structure (Minimum Spanning Tree) not only to explain the snapshot in the latent space of the continuous development of cells from its initial stage into mature differentiated cells, but also to model the dependence structure between the clusters of cells. Regularization is incorporated in the form of a prior penalization on trees with too many nodes or with redundant edges. Consequently, the model assumes the partition of cells to depend on the lineage structure, which is more biologically reasonable then the usual multistep approach in which partitions are estimated disregarding the underlying tree structure that characterizes cell lineage data. We are able to provide full inference (with uncertainty captured by the posterior samples obtained through MCMC) on the clusters of cells (including number of clusters), on the underlying tree structure and also on pseudotimes.

Authors: Zanini, C. T. P, Paulon, G., Mueller, P.

Título: The Luce model with replicas
Palestrante: José Heleno Faro (Insper)

Data: 7 de outubro de 2019 (segunda-feira)
Hora: 15:30 
Local: B106-b – Bloco B - CT – Instituto de Matemática - UFRJ

Resumo: We propose the notion of replicas in the context of discrete choices and introduce axioms that support which we call the Luce model with replicas. Unlike other relations proposed in the literature that can deal with the duplicates problem, ours entails replicas as a combination of duplicates and stochastically perfect substitutes, which induces a partition of the entire set of alternatives into endogeneous nests of replicas. Our model is less restrictive than Luce.s model and more parsimonious than the available models that may deal with the violation of the constant-ratio rule anticipated by Debreu (1960).

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