arXiv subjects
Publications and source records attributed to Lloyd Allison.
Conway's surreal numbers were aptly named by Knuth. This note examines how far one can get towards implementing surreals and the arithmetic operations on them so that they execute efficiently. Lazy evaluation and recursive data structures yield a considerable speed up.
Compared to functions in mathematics, functions in programming languages seem to be under classified. Functional programming languages based on the lambda calculus famously treat functions as first-class values. Object-oriented languages have adopted ``lambdas'', notably for call-back routines in event-based programming. Typically a programming language has functions, a function has a type, and some functions act on other functions and/or return functions but there is generally a lack of (i) ``class Function'' in the OO sense of the word class and particularly (ii) subclasses of Function for functions having specific properties. Some such classes are presented here and programmed in some popular programming languages as an experimental investigation into OO languages missing this opportunity.
A circular program creates a data structure whose computation depends upon itself or refers to itself. The technique is used to implement the classic data structures circular and doubly-linked lists, threaded trees and queues, in a functional programming language. These structures are normally thought to require updatable variables found in imperative languages. For example, a functional program to perform the breadth-first traversal of a tree is given. Some of the examples result in circular data structures when evaluated. Some examples are particularly space-efficient by avoiding the creation of intermediate temporary structures which would otherwise later become garbage. Lastly, the technique can be applied in an imperative language to give an elegant program.
A complete time-parameterized statistical model quantifying the divergent evolution of protein structures in terms of the patterns of conservation of their secondary structures is inferred from a large collection of protein 3D structure alignments. This provides a better alternative to time-parameterized sequence-based models of protein relatedness, that have clear limitations dealing with twilight and midnight zones of sequence relationships. Since protein structures are far more conserved due to the selection pressure directly placed on their function, divergence time estimates can be more accurate when inferred from structures. We use the Bayesian and information-theoretic framework of Minimum Message Length to infer a time-parameterized stochastic matrix (accounting for perturbed structural states of related residues) and associated Dirichlet models (accounting for insertions and deletions during the evolution of protein domains). These are used in concert to estimate the Markov time of divergence of tertiary structures, a task previously only possible using proxies (like RMSD). By analyzing one million pairs of homologous structures, we yield a relationship between the Markov divergence time of structures and of sequences. Using these inferred models and the relationship between the divergence of sequences and structures, we demonstrate a competitive performance in secondary structure prediction against neural network architectures commonly employed for this task. The source code and supplementary information are downloadable from \url{http://lcb.infotech.monash.edu.au/sstsum}.
A library of software for inductive inference guided by the Minimum Message Length (MML) principle was created previously. It contains various (object-oriented-) classes and subclasses of statistical Model and can be used to infer Models from given data sets in machine learning problems. Here transformations of statistical Models are considered and implemented within the library so as to have desirable properties from the object-oriented programming and mathematical points of view. The subclasses of class Function needed to do such transformations are defined.
A circular program contains a data structure whose definition is self-referential or recursive. The use of such a definition allows efficient functional programs to be written and can avoid repeated evaluations and the creation of intermediate data structures that would have to be garbage collected. This paper uses circular programs in various ways, to implement memo-structures and explicit search-trees to hold solutions to constraint-satisfaction problems.
The two most commonly used criteria for assessing causal model discovery with artificial data are edit-distance and Kullback-Leibler divergence, measured from the true model to the learned model. Both of these metrics maximally reward the true model. However, we argue that they are both insufficiently discriminating in judging the relative merits of false models. Edit distance, for example, fails to distinguish between strong and weak probabilistic dependencies. KL divergence, on the other hand, rewards equally all statistically equivalent models, regardless of their different causal claims. We propose an augmented KL divergence, which we call Causal KL (CKL), which takes into account causal relationships which distinguish between observationally equivalent models. Results are presented for three variants of CKL, showing that Causal KL works well in practice.
Causal discovery automates the learning of causal Bayesian networks from data and has been of active interest from their beginning. With the sourcing of large data sets off the internet, interest in scaling up to very large data sets has grown. One approach to this is to parallelize search using Markov Blanket (MB) discovery as a first step, followed by a process of combining MBs in a global causal model. We develop and explore three new methods of MB discovery using Minimum Message Length (MML) and compare them empirically to the best existing methods, whether developed specifically as MB discovery or as feature selection. Our best MML method is consistently competitive and has some advantageous features.
This work demonstrates how a complete statistical model quantifying the evolution of pairs of aligned proteins can be constructed from a time-parameterised substitution matrix and a time-parameterised 3-state alignment machine. All parameters of such a model can be inferred from any benchmark data-set of aligned protein sequences. This allows us to examine nine well-known substitution matrices on six benchmarks curated using various structural alignment methods; any matrix that does not explicitly model a "time"-dependent Markov process is converted to a corresponding base-matrix that does. In addition, a new optimal matrix is inferred for each of the six benchmarks. Using Minimum Message Length (MML) inference, all 15 matrices are compared in terms of measuring the Shannon information content of each benchmark. This has resulted in a new and clear overall best performed time-dependent Markov matrix, MMLSUM, and its associated 3-state machine, whose properties we have analysed in this work. For standard use, the MMLSUM series of (log-odds) \textit{scoring} matrices derived from the above Markov matrix, are available at https://lcb.infotech.monash.edu.au/mmlsum.
This report concerns the information content of a graph, optionally conditional on one or more background, "common knowledge" graphs. It describes an algorithm to estimate this information content, and includes some examples based on chemical compounds.
A universal code for the (positive) integers can be used to store or compress a sequence of integers. Every universal code implies a probability distribution on integers. This implied distribution may be a reasonable choice when the true distribution of a source of integers is unknown. Wallace Tree Code (WTC) is a universal code for integers based on binary trees. We give the encoding and decoding routines for WTC and analyse the properties of the code in comparison to two well-known codes, the Fibonacci and Elias omega codes. Some improvements on the Elias omega code are also described and examined.
Moral graphs were introduced in the 1980s as an intermediate step when transforming a Bayesian network to a junction tree, on which exact belief propagation can be efficiently done. The moral graph of a Bayesian network can be trivially obtained by connecting non-adjacent parents for each node in the Bayesian network and dropping the direction of each edge. Perhaps because the moralization process looks simple, there has been little attention on the properties of moral graphs and their impact in belief propagation on Bayesian networks. This paper addresses the mistaken claim that it has been previously proved that optimal moral graph triangulation with the constraints of minimum fill-in, treewidth or total states is NP-complete. The problems are in fact NP-complete, but they have not previously been proved. We now prove these.
A family of Markov blankets in a faithful Bayesian network satisfies the symmetry and consistency properties. In this paper, we draw a bijection between families of consistent Markov blankets and moral graphs. We define the new concepts of weak recursive simpliciality and perfect elimination kits. We prove that they are equivalent to graph morality. In addition, we prove that morality can be decided in polynomial time for graphs with maximum degree less than $5$, but the problem is NP-complete for graphs with higher maximum degrees.
Mixture modelling involves explaining some observed evidence using a combination of probability distributions. The crux of the problem is the inference of an optimal number of mixture components and their corresponding parameters. This paper discusses unsupervised learning of mixture models using the Bayesian Minimum Message Length (MML) criterion. To demonstrate the effectiveness of search and inference of mixture parameters using the proposed approach, we select two key probability distributions, each handling fundamentally different types of data: the multivariate Gaussian distribution to address mixture modelling of data distributed in Euclidean space, and the multivariate von Mises-Fisher (vMF) distribution to address mixture modelling of directional data distributed on a unit hypersphere. The key contributions of this paper, in addition to the general search and inference methodology, include the derivation of MML expressions for encoding the data using multivariate Gaussian and von Mises-Fisher distributions, and the analytical derivation of the MML estimates of the parameters of the two distributions. Our approach is tested on simulated and real world data sets. For instance, we infer vMF mixtures that concisely explain experimentally determined three-dimensional protein conformations, providing an effective null model description of protein structures that is central to many inference problems in structural bioinformatics. The experimental results demonstrate that the performance of our proposed search and inference method along with the encoding schemes improve on the state of the art mixture modelling techniques.
Proteins are biomolecules of life. They fold into a great variety of three-dimensional (3D) shapes. Underlying these folding patterns are many recurrent structural fragments or building blocks (analogous to `LEGO bricks'). This paper reports an innovative statistical inference approach to discover a comprehensive dictionary of protein structural building blocks from a large corpus of experimentally determined protein structures. Our approach is built on the Bayesian and information-theoretic criterion of minimum message length. To the best of our knowledge, this work is the first systematic and rigorous treatment of a very important data mining problem that arises in the cross-disciplinary area of structural bioinformatics. The quality of the dictionary we find is demonstrated by its explanatory power -- any protein within the corpus of known 3D structures can be dissected into successive regions assigned to fragments from this dictionary. This induces a novel one-dimensional representation of three-dimensional protein folding patterns, suitable for application of the rich repertoire of character-string processing algorithms, for rapid identification of folding patterns of newly-determined structures. This paper presents the details of the methodology used to infer the dictionary of building blocks, and is supported by illustrative examples to demonstrate its effectiveness and utility.