We compare Karl Popper’s ideas concerning the falsifiability of a theory with similar notions from the part of statistical learning theory known as VC-theory . Popper’s notion of the dimension of a theory is contrasted with the apparently very similar VC-dimension. Having located some divergences, we discuss how best to view Popper’s work from the perspective of statistical learning theory, either as a precursor or as aiming to capture a different learning activity.

Compared to constraint-based causal discovery, causal discovery based on functional causal models is able to identify the whole causal model under appropriate assumptions [Shimizu et al. 2006; Hoyer et al. 2009; Zhang and Hyvärinen 2009b]. Functional causal models represent the effect as a function of the direct causes together with an independent noise term. Examples include the linear non-Gaussian acyclic model, nonlinear additive noise model, and post-nonlinear model. Currently, there are two ways to estimate the parameters in the models: dependence (...) minimization and maximum likelihood. In this article, we show that for any acyclic functional causal model, minimizing the mutual information between the hypothetical cause and the noise term is equivalent to maximizing the data likelihood with a flexible model for the distribution of the noise term. We then focus on estimation of the PNL causal model and propose to estimate it with the warped Gaussian process with the noise modeled by the mixture of Gaussians. As a Bayesian nonparametric approach, it outperforms the previous one based on mutual information minimization with nonlinear functions represented by multilayer perceptrons; we also show that unlike the ordinary regression, estimation results of the PNL causal model are sensitive to the assumption on the noise distribution. Experimental results on both synthetic and real data support our theoretical claims. (shrink)

It is commonplace to encounter nonstationary or heterogeneous data, of which the underlying generating process changes over time or across data sets. Such a distribution shift feature presents both challenges and opportunities for causal discovery. In this paper we develop a principled framework for causal discovery from such data, called Constraint-based causal Discovery from Nonstationary/heterogeneous Data, which addresses two important questions. First, we propose an enhanced constraint-based procedure to detect variables whose local mechanisms change and recover the skeleton of the (...) causal structure over observed variables. Second, we present a way to determine causal orientations by making use of independence changes in the data distribution implied by the underlying causal model, benefiting from information carried by changing distributions. Experimental results on various synthetic and real-world data sets are presented to demonstrate the efficacy of our methods. (shrink)

We study the identifiability and estimation of functional causal models under selection bias, with a focus on the situation where the selection depends solely on the effect variable, which is known as outcome-dependent selection. We address two questions of identifiability: the identifiability of the causal direction between two variables in the presence of selection bias, and, given the causal direction, the identifiability of the model with outcome-dependent selection. Regarding the first, we show that in the framework of post-nonlinear causal models, (...) once outcome-dependent selection is properly modeled, the causal direction between two variables is generically identifiable; regarding the second, we identify some mild conditions under which an additive noise causal model with outcome-dependent selection is to a large extent identifiable. We also propose two methods for estimating an additive noise model from data that are generated with outcome-dependent selection. (shrink)