The Binary Space Partitioning-Tree Process

The Mondrian process represents an elegant and powerful approach for space partition modelling. However, as it restricts the partitions to be axis-aligned, its modelling flexibility is limited. In this work, we propose a self-consistent Binary Space Partitioning (BSP)-Tree process to generalize the Mondrian process. The BSP-Tree process is an almost surely right continuous Markov jump process that allows uniformly distributed oblique cuts in a two-dimensional convex polygon. The BSP-Tree process can also be extended using a non-uniform probability measure to generate direction differentiated cuts. The process is also self-consistent, maintaining distributional invariance under a restricted subdomain. We use Conditional-Sequential Monte Carlo for inference using the tree structure as the high-dimensional variable. The BSP-Tree process's performance on synthetic data partitioning and relational modelling demonstrates clear inferential improvements over the standard Mondrian process and other related methods

Random Tessellation Forests

Space partitioning methods such as random forests and the Mondrian process are powerful machine learning methods for multi-dimensional and relational data, and are based on recursively cutting a domain. The flexibility of these methods is often limited by the requirement that the cuts be axis aligned. The Ostomachion process and the self-consistent binary space partitioning-tree process were recently introduced as generalizations of the Mondrian process for space partitioning with non-axis aligned cuts in the two dimensional plane. Motivated by the need for a multi-dimensional partitioning tree with non-axis aligned cuts, we propose the Random Tessellation Process (RTP), a framework that includes the Mondrian process and the binary space partitioning-tree process as special cases. We derive a sequential Monte Carlo algorithm for inference, and provide random forest methods. Our process is self-consistent and can relax axis-aligned constraints, allowing complex inter-dimensional dependence to be captured. We present a simulation study, and analyse gene expression data of brain tissue, showing improved accuracies over other methods.Comment: 11 pages, 4 figure

Professor Layton and Mental Gymnastics

Whenever we encounter an uncommon problem, we are forced to think of a solution using parts of our brain we do not normally train in our daily life. However, so far, very few papers have been written on how our intellectual abilities are improved by solving puzzles. The goal of this monograph is to analyse a group of puzzles included in the "Professor Layton" series. While strategy guides may provide direct, easy answers, I shall endeavour to show how people process mentally the information required to solve puzzles and how this benefits the brain. The genres of the aforementioned puzzles range from logic and arithmetic to mental rotation and pattern recognition. Ultimately, I submit that puzzle solving is an extremely effective method of improving our intellectual abilities. Thus, its study may yet help neuroscientists progress in their investigation of mental processes.Sempre que confrontamos um problema invulgar, somos forçados a ponderar uma solução, utilizando partes do nosso cérebro que (normalmente) não utilizamos no nosso dia-a-dia. Contudo, muito poucos estudos foram efectuados relativamente a como a resolução de puzzles pode beneficiar as nossas capacidades intelectuais. Por conseguinte, muitos não sabem como aperfeiçoar o seu pensamento crítico. O objectivo da presente monografia é analisar um certo grupo de puzzles que apareceu alguns anos atrás em videojogos destinados para as consolas portáteis: aqueles incluídos na série Professor Layton. Apesar de ser possível encontrar soluções directas em guias de estratégia, tentaremos no presente trabalho mostrar como processamos a informação necessária à resolução de um puzzle, e como isto beneficia o cérebro. Com esta finalidade, foi resolvida uma série de puzzles e registado o processo cerebral, tendo em atenção o que se sabe actualmente da utilização das capacidades intelectuais

Copula mixed-membership stochastic block model

The Mixed-Membership Stochastic Blockmodels (MMSB) is a popular framework for modelling social relationships by fully exploiting each individual node's participation (or membership) in a social network. Despite its powerful representations, MMSB assumes that the membership indicators of each pair of nodes (i.e., people) are distributed independently. However, such an assumption often does not hold in real-life social networks, in which certain known groups of people may correlate with each other in terms of factors such as their membership categories. To expand MMSB's ability to model such dependent relationships, a new framework - a Copula Mixed-Membership Stochastic Blockmodel - is introduced in this paper for modeling intra-group correlations, namely an individual Copula function jointly models the membership pairs of those nodes within the group of interest. This framework enables various Copula functions to be used on demand, while maintaining the membership indicator's marginal distribution needed for modelling membership indicators with other nodes outside of the group of interest. Sampling algorithms for both the finite and infinite number of groups are also detailed. Our experimental results show its superior performance in capturing group interactions when compared with the baseline models on both synthetic and real world datasets

The Wine and the Cup: Syncretism and Subversion in the Late Antique Christian Cento

Senior Project submitted to The Division of Languages and Literature of Bard College