Causal inference using the algorithmic Markov condition

Inferring the causal structure that links n observables is usually based upon detecting statistical dependences and choosing simple graphs that make the joint measure Markovian. Here we argue why causal inference is also possible when only single observations are present. We develop a theory how to generate causal graphs explaining similarities between single objects. To this end, we replace the notion of conditional stochastic independence in the causal Markov condition with the vanishing of conditional algorithmic mutual information and describe the corresponding causal inference rules. We explain why a consistent reformulation of causal inference in terms of algorithmic complexity implies a new inference principle that takes into account also the complexity of conditional probability densities, making it possible to select among Markov equivalent causal graphs. This insight provides a theoretical foundation of a heuristic principle proposed in earlier work. We also discuss how to replace Kolmogorov complexity with decidable complexity criteria. This can be seen as an algorithmic analog of replacing the empirically undecidable question of statistical independence with practical independence tests that are based on implicit or explicit assumptions on the underlying distribution.Comment: 16 figure

Kolmogorov Complexity, Cosmic Background Radiation and Irreversibility

We discuss the algorithmic information approach to the analysis of the observational data on the Universe. Kolmogorov complexity is proposed as a descriptor of the Cosmic Microwave Background (CMB) radiation maps. An algorithm of computation of the complexity is described, applied, first, to toy models and then, to the data of the Boomerang experiment. The sky maps obtained via the summing of two independent Boomerang channels reveal threshold independent behavior of the mean ellipticity of the anisotropies, thus indicating correlations present in the sky signal and possibly carrying crucial information on the curvature and the non-Friedmannian, i.e. accelerated expansion of the Universe. Similar effect has been detected for COBE-DMR 4 year maps. Finally, as another application of these concepts, we consider the possible link between the CMB properties, curvature of the Universe and arrows of time.Comment: Talk at XXII Solvay Conference on Physics "The Physics of Communication" (Delphi, November 24-29, 2001); the discussion include