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The capability of R to do symbolic mathematics is enhanced by the caracas package. This package uses the Python computer algebra library SymPy as a back-end but caracas is tightly integrated in the R environment. This enables the R user with symbolic mathematics within R at a high abstraction level rather than using text strings and text string manipulation as the case would be if using SymPy from R directly. We demonstrate how mathematics and statistics can benefit from bridging computer algebra and data via R. This is done thought a number of examples and we propose some topics for small student projects. The caracas package integrates well with e.g. Rmarkdown, and as such creation of scientific reports and teaching is supported.

We propose Unity Smoothing (US) for handling inconsistencies between a Bayesian network model and new unseen observations. We show that prediction accuracy, using the junction tree algorithm with US is comparable to that of Laplace smoothing. Moreover, in applications were sparsity of the data structures is utilized, US outperforms Laplace smoothing in terms of memory usage. Furthermore, we detail how to avoid redundant calculations that must otherwise be performed during the message passing scheme in the junction tree algorithm which we refer to as Unity Propagation (UP). Experimental results shows that it is always faster to exploit UP on top of the Lauritzen-Spigelhalter message passing scheme for the junction tree algorithm.

In Gaussian graphical models, the likelihood equations must typically be solved iteratively. We investigate two algorithms: A version of iterative proportional scaling which avoids inversion of large matrices, and an algorithm based on convex duality and operating on the covariance matrix by neighbourhood coordinate descent, corresponding to the graphical lasso with zero penalty. For large, sparse graphs, the iterative proportional scaling algorithm appears feasible and has simple convergence properties. The algorithm based on neighbourhood coordinate descent is extremely fast and less dependent on sparsity, but needs a positive definite starting value to converge. We give an algorithm for finding such a starting value for graphs with low colouring number. As a consequence, we also obtain a simplified proof for existence of the maximum likelihood estimator in such cases.

The capability of R to do symbolic mathematics is enhanced by the caracas package. This package uses the Python computer algebra library SymPy as a back-end but caracas is tightly integrated in the R environment, thereby enabling the R user with symbolic mathematics within R. Key components of the caracas package are illustrated in this paper. Examples are taken from statistics and mathematics. The caracas package integrates well with e.g. Rmarkdown, and as such creation of scientific reports and teaching is supported.

A graphical model is a multivariate (potentially very high dimensional) probabilistic model, which is formed by combining lower dimensional components. Inference (computation of conditional probabilities) is based on message passing algorithms that utilize conditional independence structures. In graphical models for discrete variables with finite state spaces, there is a fundamental problem in high dimensions: A discrete distribution is represented by a table of values, and in high dimensions such tables can become prohibitively large. In inference, such tables must be multiplied which can lead to even larger tables. The sparta package meets this challenge by implementing methods that efficiently handles multiplication and marginalization of sparse tables. The package was written in the R programming language and is freely available from the Comprehensive R Archive Network (CRAN). The companion package jti, also on CRAN, was developed to showcase the potential of sparta in connection to the Junction Tree Algorithm. We show, that jti is able to handle highly complex graphical models which are otherwise infeasible due to lack of computer memory, using sparta as a backend for table operations.

There are two main objectives of this paper. The first is to present a statistical framework for models with context specific independence structures, i.e., conditional independences holding only for sepcific values of the conditioning variables. This framework is constituted by the class of split models. Split models are extension of graphical models for contigency tables and allow for a more sophisticiated modelling than graphical models. The treatment of split models include estimation, representation and a Markov property for reading off those independencies holding in a specific context. The second objective is to present a software package named YGGDRASIL which is designed for statistical inference in split models, i.e., for learning such models on the basis of data., Comment: Appears in Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI2000)