The Long and the Short of It: Summarising Event Sequences with Serial Episodes

An ideal outcome of pattern mining is a small set of informative patterns, containing no redundancy or noise, that identifies the key structure of the data at hand. Standard frequent pattern miners do not achieve this goal, as due to the pattern explosion typically very large numbers of highly redundant patterns are returned. We pursue the ideal for sequential data, by employing a pattern set mining approach-an approach where, instead of ranking patterns individually, we consider results as a whole. Pattern set mining has been successfully applied to transactional data, but has been surprisingly under studied for sequential data. In this paper, we employ the MDL principle to identify the set of sequential patterns that summarises the data best. In particular, we formalise how to encode sequential data using sets of serial episodes, and use the encoded length as a quality score. As search strategy, we propose two approaches: the first algorithm selects a good pattern set from a large candidate set, while the second is a parameter-free any-time algorithm that mines pattern sets directly from the data. Experimentation on synthetic and real data demonstrates we efficiently discover small sets of informative patterns.

We Are Not Your Real Parents: Telling Causal from Confounded using MDL

Given data over variables $(X_1,...,X_m, Y)$ we consider the problem of finding out whether $X$ jointly causes $Y$ or whether they are all confounded by an unobserved latent variable $Z$. To do so, we take an information-theoretic approach based on Kolmogorov complexity. In a nutshell, we follow the postulate that first encoding the true cause, and then the effects given that cause, results in a shorter description than any other encoding of the observed variables. The ideal score is not computable, and hence we have to approximate it. We propose to do so using the Minimum Description Length (MDL) principle. We compare the MDL scores under the models where $X$ causes $Y$ and where there exists a latent variables $Z$ confounding both $X$ and $Y$ and show our scores are consistent. To find potential confounders we propose using latent factor modeling, in particular, probabilistic PCA (PPCA). Empirical evaluation on both synthetic and real-world data shows that our method, CoCa, performs very well -- even when the true generating process of the data is far from the assumptions made by the models we use. Moreover, it is robust as its accuracy goes hand in hand with its confidence.

Discovering Reliable Dependencies from Data: Hardness and Improved Algorithms

The reliable fraction of information is an attractive score for quantifying (functional) dependencies in high-dimensional data. In this paper, we systematically explore the algorithmic implications of using this measure for optimization. We show that the problem is NP-hard, which justifies the usage of worst-case exponential-time as well as heuristic search methods. We then substantially improve the practical performance for both optimization styles by deriving a novel admissible bounding function that has an unbounded potential for additional pruning over the previously proposed one. Finally, we empirically investigate the approximation ratio of the greedy algorithm and show that it produces highly competitive results in a fraction of time needed for complete branch-and-bound style search.

Causal Discovery by Telling Apart Parents and Children

We consider the problem of inferring the directed, causal graph from observational data, assuming no hidden confounders. We take an information theoretic approach, and make three main contributions. First, we show how through algorithmic information theory we can obtain SCI, a highly robust, effective and computationally efficient test for conditional independence---and show it outperforms the state of the art when applied in constraint-based inference methods such as stable PC. Second, building upon on SCI, we show how to tell apart the parents and children of a given node based on the algorithmic Markov condition. We give the Climb algorithm to efficiently discover the directed, causal Markov blanket---and show it is at least as accurate as inferring the global network, while being much more efficient. Last, but not least, we detail how we can use the Climb score to direct those edges that state of the art causal discovery algorithms based on PC or GES leave undirected---and show this improves their precision, recall and F1 scores by up to 20%.

Causal Inference on Multivariate and Mixed-Type Data

Given data over the joint distribution of two random variables $X$ and $Y$, we consider the problem of inferring the most likely causal direction between $X$ and $Y$. In particular, we consider the general case where both $X$ and $Y$ may be univariate or multivariate, and of the same or mixed data types. We take an information theoretic approach, based on Kolmogorov complexity, from which it follows that first describing the data over cause and then that of effect given cause is shorter than the reverse direction. The ideal score is not computable, but can be approximated through the Minimum Description Length (MDL) principle. Based on MDL, we propose two scores, one for when both $X$ and $Y$ are of the same single data type, and one for when they are mixed-type. We model dependencies between $X$ and $Y$ using classification and regression trees. As inferring the optimal model is NP-hard, we propose Crack, a fast greedy algorithm to determine the most likely causal direction directly from the data. Empirical evaluation on a wide range of data shows that Crack reliably, and with high accuracy, infers the correct causal direction on both univariate and multivariate cause-effect pairs over both single and mixed-type data.

Telling Cause from Effect using MDL-based Local and Global Regression

We consider the fundamental problem of inferring the causal direction between two univariate numeric random variables $X$ and $Y$ from observational data. The two-variable case is especially difficult to solve since it is not possible to use standard conditional independence tests between the variables. To tackle this problem, we follow an information theoretic approach based on Kolmogorov complexity and use the Minimum Description Length (MDL) principle to provide a practical solution. In particular, we propose a compression scheme to encode local and global functional relations using MDL-based regression. We infer $X$ causes $Y$ in case it is shorter to describe $Y$ as a function of $X$ than the inverse direction. In addition, we introduce Slope, an efficient linear-time algorithm that through thorough empirical evaluation on both synthetic and real world data we show outperforms the state of the art by a wide margin.

Efficiently Discovering Locally Exceptional yet Globally Representative Subgroups

Subgroup discovery is a local pattern mining technique to find interpretable descriptions of sub-populations that stand out on a given target variable. That is, these sub-populations are exceptional with regard to the global distribution. In this paper we argue that in many applications, such as scientific discovery, subgroups are only useful if they are additionally representative of the global distribution with regard to a control variable. That is, when the distribution of this control variable is the same, or almost the same, as over the whole data. We formalise this objective function and give an efficient algorithm to compute its tight optimistic estimator for the case of a numeric target and a binary control variable. This enables us to use the branch-and-bound framework to efficiently discover the top-$k$ subgroups that are both exceptional as well as representative. Experimental evaluation on a wide range of datasets shows that with this algorithm we discover meaningful representative patterns and are up to orders of magnitude faster in terms of node evaluations as well as time.

Discovering Reliable Approximate Functional Dependencies

Given a database and a target attribute of interest, how can we tell whether there exists a functional, or approximately functional dependence of the target on any set of other attributes in the data? How can we reliably, without bias to sample size or dimensionality, measure the strength of such a dependence? And, how can we efficiently discover the optimal or $\alpha$-approximate top-$k$ dependencies? These are exactly the questions we answer in this paper. As we want to be agnostic on the form of the dependence, we adopt an information-theoretic approach, and construct a reliable, bias correcting score that can be efficiently computed. Moreover, we give an effective optimistic estimator of this score, by which for the first time we can mine the approximate functional dependencies from data with guarantees of optimality. Empirical evaluation shows that the derived score achieves a good bias for variance trade-off, can be used within an efficient discovery algorithm, and indeed discovers meaningful dependencies. Most important, it remains reliable in the face of data sparsity.

Identifying Consistent Statements about Numerical Data with Dispersion-Corrected Subgroup Discovery

Existing algorithms for subgroup discovery with numerical targets do not optimize the error or target variable dispersion of the groups they find. This often leads to unreliable or inconsistent statements about the data, rendering practical applications, especially in scientific domains, futile. Therefore, we here extend the optimistic estimator framework for optimal subgroup discovery to a new class of objective functions: we show how tight estimators can be computed efficiently for all functions that are determined by subgroup size (non-decreasing dependence), the subgroup median value, and a dispersion measure around the median (non-increasing dependence). In the important special case when dispersion is measured using the average absolute deviation from the median, this novel approach yields a linear time algorithm. Empirical evaluation on a wide range of datasets shows that, when used within branch-and-bound search, this approach is highly efficient and indeed discovers subgroups with much smaller errors.

Causal Inference by Stochastic Complexity

The algorithmic Markov condition states that the most likely causal direction between two random variables X and Y can be identified as that direction with the lowest Kolmogorov complexity. Due to the halting problem, however, this notion is not computable. We hence propose to do causal inference by stochastic complexity. That is, we propose to approximate Kolmogorov complexity via the Minimum Description Length (MDL) principle, using a score that is mini-max optimal with regard to the model class under consideration. This means that even in an adversarial setting, such as when the true distribution is not in this class, we still obtain the optimal encoding for the data relative to the class. We instantiate this framework, which we call CISC, for pairs of univariate discrete variables, using the class of multinomial distributions. Experiments show that CISC is highly accurate on synthetic, benchmark, as well as real-world data, outperforming the state of the art by a margin, and scales extremely well with regard to sample and domain sizes.