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Dynamic Bayesian networks have been well explored in the literature as discrete-time models; however, their continuous-time extensions have seen comparatively little attention. In this paper, we propose the first constraint-based algorithm for learning the structure of continuous-time Bayesian networks. We discuss the different statistical tests and the underlying hypotheses used by our proposal to establish conditional independence. Finally, we validate its performance using synthetic data, and discuss its strengths and limitations. We find that score-based is more accurate in learning networks with binary variables, while our constraint-based approach is more accurate with variables assuming more than two values. However, more experiments are needed for confirmation.

Purpose Individualised physiotherapy is an effective treatment for low back pain. We sought to determine how this treatment works by using randomised controlled trial data to develop a Bayesian Network model. Methods 300 randomised controlled trial participants (153 male, 147 female, mean age 44.1) with low back pain (of duration 6-26 weeks) received either individualised physiotherapy or advice. Variables with potential to explain how individualised physiotherapy works were included in a multivariate Bayesian Network model. Modelling incorporated the intervention period (0-10 weeks after study commencement-"early"changes) and the follow-up period (10-52 weeks after study commencement-"late"changes). Sequences of variables in the Bayesian Network showed the most common direct and indirect recovery pathways followed by participants with low back pain receiving individualised physiotherapy versus advice. Results Individualised physiotherapy directly reduced early disability in people with low back pain. Individualised physiotherapy exerted indirect effects on pain intensity, recovery expectations, sleep, fear, anxiety, and depression via its ability to facilitate early improvement in disability. Early improvement in disability, led to an early reduction in depression both directly and via more complex pathways involving fear, recovery expectations, anxiety, and pain intensity. Individualised physiotherapy had its greatest influence on early change variables (during the intervention period). Conclusion Individualised physiotherapy for low back pain appears to work predominately by facilitating an early reduction in disability, which in turn leads to improvements in other biopsychosocial outcomes. The current study cannot rule out that unmeasured mechanisms (such as tissue healing or reduced inflammation) may mediate the relationship between individualised physiotherapy treatment and improvement in disability. Further data-driven analyses involving a broad range of plausible biopsychosocial variables are recommended to fully understand how treatments work for people with low back pain.

Learning the structure of Bayesian networks from data is known to be a computationally challenging, NP-hard problem. The literature has long investigated how to perform structure learning from data containing large numbers of variables, following a general interest in high-dimensional applications (“small n, large p”) in systems biology and genetics. More recently, data sets with large numbers of observations (the so-called “big data”) have become increasingly common; and these data sets are not necessarily high-dimensional, sometimes having only a few tens of variables depending on the application. We revisit the computational complexity of Bayesian network structure learning in this setting, showing that the common choice of measuring it with the number of estimated local distributions leads to unrealistic time complexity estimates for the most common class of score based algorithms, greedy search. We then derive more accurate expressions under common distributional assumptions. These expressions suggest that the speed of Bayesian network learning can be improved by taking advantage of the availability of closed form estimators for local distributions with few parents. Furthermore, we find that using predictive instead of in sample goodness-of-fit scores improves speed; and we confirm that is improves the accuracy of network reconstruction as well, as previously observed by Chickering and Heckerman (2000). We demonstrate these results on large real-world environmental and epidemiological data; and on reference data sets available from public repositories.

A classic approach for learning Bayesian networks from data is to select the maximum a posteriori (MAP) network. In the case of discrete Bayesian networks, the MAP network is selected by maximising one of several possible Bayesian Dirichlet (BD) scores; the most famous is the Bayesian Dirichlet equivalent uniform (BDeu) score from Heckerman et al. (1995). The key properties of BDeu arise from its underlying uniform prior, which makes structure learning computationally efficient; does not require the elicitation of prior knowledge from experts; and satisfies score equivalence. In this paper we will discuss the impact of this uniform prior on structure learning from an information theoretic perspective, showing how BDeu may violate the maximum entropy principle when applied to sparse data and how it may also be problematic from a Bayesian model selection perspective. On the other hand, the BDs score proposed in Scutari (2016) arises from a piecewise prior and it does not appear to violate the maximum entropy principle, even though it is asymptotically equivalent to BDeu.

The prediction of phenotypic traits using high-density genomic data has many applications such as the selection of plants and animals of commercial interest; and it is expected to play an increasing role in medical diagnostics. Statistical models used for this task are usually tested using cross-validation, which implicitly assumes that new individuals (whose phenotypes we would like to predict) originate from the same population the genomic prediction model is trained on. In this paper we propose an approach based on clustering and resampling to investigate the effect of increasing genetic distance between training and target populations when predicting quantitative traits. This is important for plant and animal genetics, where genomic selection programs rely on the precision of predictions in future rounds of breeding. Therefore, estimating how quickly predictive accuracy decays is important in deciding which training population to use and how often the model has to be recalibrated. We find that the correlation between true and predicted values decays approximately linearly with respect to either $\F$ or mean kinship between the training and the target populations. We illustrate this relationship using simulations and a collection of data sets from mice, wheat and human genetics., 36 pages, 9 figures

The structure of a Bayesian network includes a great deal of information about the probability distribution of the data, which is uniquely identified given some general distributional assumptions. Therefore it's important to study its variability, which can be used to compare the performance of different learning algorithms and to measure the strength of any arbitrary subset of arcs. In this paper we will introduce some descriptive statistics and the corresponding parametric and Monte Carlo tests on the undirected graph underlying the structure of a Bayesian network, modeled as a multivariate Bernoulli random variable. A simple numeric example and the comparison of the performance of some structure learning algorithm on small samples will then illustrate their use., 19 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:0909.1685

Causal inference for testing clinical hypotheses from observational data presents many difficulties because the underlying data-generating model and the associated causal graph are not usually available. Furthermore, observational data may contain missing values, which impact the recovery of the causal graph by causal discovery algorithms: a crucial issue often ignored in clinical studies. In this work, we use data from a multi-centric study on endometrial cancer to analyze the impact of different missingness mechanisms on the recovered causal graph. This is achieved by extending state-of-the-art causal discovery algorithms to exploit expert knowledge without sacrificing theoretical soundness. We validate the recovered graph with expert physicians, showing that our approach finds clinically-relevant solutions. Finally, we discuss the goodness of fit of our graph and its consistency from a clinical decision-making perspective using graphical separation to validate causal pathways.

Dynamic Bayesian networks have been well explored in the literature as discrete-time models: however, their continuous-time extensions have seen comparatively little attention. In this paper, we propose the first implementation of a constraint-based algorithm for learning the structure of continuous-time Bayesian networks. We discuss the different statistical tests and the underlying hypotheses used by our proposal to establish conditional independence. Furthermore, we analyze and discuss the computational complexity of the best and worst cases for the proposed algorithm. Finally, we validate its performance using synthetic data, and we discuss its strengths and limitations comparing it with the score-based structure learning algorithm from Nodelman et al. [23] . We find the latter to be more accurate in learning networks with binary variables, while our constraint-based approach is more accurate with variables assuming more than two values. Numerical experiments confirm that score-based and constraint-based algorithms are comparable in terms of computation time.

Incomplete data are a common feature in many domains, from clinical trials to industrial applications. Bayesian networks (BNs) are often used in these domains because of their graphical and causal interpretations. BN parameter learning from incomplete data is usually implemented with the Expectation-Maximisation algorithm (EM), which computes the relevant sufficient statistics (&ldquo, soft EM&rdquo, ) using belief propagation. Similarly, the Structural Expectation-Maximisation algorithm (Structural EM) learns the network structure of the BN from those sufficient statistics using algorithms designed for complete data. However, practical implementations of parameter and structure learning often impute missing data (&ldquo, hard EM&rdquo, ) to compute sufficient statistics instead of using belief propagation, for both ease of implementation and computational speed. In this paper, we investigate the question: what is the impact of using imputation instead of belief propagation on the quality of the resulting BNs? From a simulation study using synthetic data and reference BNs, we find that it is possible to recommend one approach over the other in several scenarios based on the characteristics of the data. We then use this information to build a simple decision tree to guide practitioners in choosing the EM algorithm best suited to their problem.