### Learning and using relational theories

Much of human knowledge is organized into sophisticated systems that are often called intuitive theories. We propose that intuitive theories are mentally represented ina logical language, and that the subjective complexity of a theory is determined by the length of its representation in this language. This complexity measure helps to explain how theories are learned from relational data, and how they support inductive inferences about unobserved relations. We describe two experiments that test our approach, and show that it provides a better account of human learning and reasoning than an approach developed by Goodman . What is a theory, and what makes one theory better than another?

### A tutorial on MDL hypothesis testing for graph analysis

When analysing graph structure, it can be difficult to determine whether patterns found are due to chance, or due to structural aspects of the process that generated the data. Hypothesis tests are often used to support such analyses. These allow us to make statistical inferences about which null models are responsible for the data, and they can be used as a heuristic in searching for meaningful patterns. The minimum description length (MDL) principle [6, 4] allows us to build such hypothesis tests, based on efficient descriptions of the data. Broadly: we translate the regularity we are interested in into a code for the data, and if this code describes the data more efficiently than a code corresponding to the null model, by a sufficient margin, we may reject the null model. This is a frequentist approach to MDL, based on hypothesis testing. Bayesian approaches to MDL for model selection rather than model rejection are more common, but for the purposes of pattern analysis, a hypothesis testing approach provides a more natural fit with existing literature. 1 We provide a brief illustration of this principle based on the running example of analysing the size of the largest clique in a graph. We illustrate how a code can be constructed to efficiently represent graphs with large cliques, and how the description length of the data under this code can be compared to the description length under a code corresponding to a null model to show that the null model is highly unlikely to have generated the data.

### Inferring Latent dimension of Linear Dynamical System with Minimum Description Length

Time-invariant linear dynamical system arises in many real-world applications,and its usefulness is widely acknowledged. A practical limitation with this model is that its latent dimension that has a large impact on the model capability needs to be manually specified. It can be demonstrated that a lower-order model class could be totally nested into a higher-order class, and the corresponding likelihood is nondecreasing. Hence, criterion built on the likelihood is not appropriate for model selection. This paper addresses the issue and proposes a criterion for linear dynamical system based on the principle of minimum description length. The latent structure, which is omitted in previous work, is explicitly considered in this newly proposed criterion. Our work extends the principle of minimum description length and demonstrates its effectiveness in the tasks of model training. The experiments on both univariate and multivariate sequences confirm the good performance of our newly proposed method.

### Categorization Under Complexity: A Unified MDL Account of Human Learning of Regular and Irregular Categories

We present an account of human concept learning-that is, learning of categories from examples-based on the principle of minimum description length(MDL). In support of this theory, we tested a wide range of two-dimensional concept types, including both regular (simple) and highly irregular (complex) structures, and found the MDL theory to give a good account of subjects' performance. This suggests that the intrinsic complexityofa concept (that is, its description -length) systematically influences its leamability.

### Discrete MDL Predicts in Total Variation

The Minimum Description Length (MDL) principle selects the model that has the shortest code for data plus model. We show that for a countable class of models, MDL predictions are close to the true distribution in a strong sense. The result is completely general. No independence, ergodicity, stationarity, identifiability, or other assumption on the model class need to be made. More formally, we show that for any countable class of models, the distributions selected by MDL (or MAP) asymptotically predict (merge with) the true measure in the class in total variation distance.