Zooming in on Probability and Causality (aim & preliminary results)

A B D C E  ACB A C D + + CB ED Motivation - ‘Building Causal Models from Multiple Databases’ Results from three experiments … One model. (under suitable assumptions)  But how can we verify the validity of causal claims we make?

Causal structures Problem: Verification implies somehow comparing with the ‘true’ causal structure of a system. For a system that is described in terms of (causal) relations, e.g. SEMs and CBNs, this is fairly straightforward: the system is the model. But most real world systems are ultimately dynamical systems, governed by differential equations that describe the detailed evolution through phase space, and at that ground level it is no longer clear where the probabilities and causal relations actually reside. Main Question: What, if anything, actually is ‘the causal structure’ of a system? Approach: Look in detail at a (simple) dynamical system for which you have complete information and try to see if you can understand probability and causality at this level.

Representing a system Three crucial ingredients to capture: A simple dynamical system (T,S,f) T - a time interval S - the state space of the entire system f - a set of coupled differential equations with initial values A set R of (random) variables V = {v 1, …, v n } v i - set of possible outcomes of V A context ( ,D) for the system  - set of possible/allowed initial conditions D - relative weight distribution over  Then we have complete information about the (deterministic) system. Note: (fixed) system parameters can also be included in S and .

Sample system - Throwing a drawing pin Two (stable) outcomes Questions: What is the probability of ‘pin up’? What causes (i.e. effectively influences) this probability? What is the causal structure between various quantities (i.e. parameters, intermediate states and/or other random variables)? Point is: can we calculate the answer these questions from the description alone? Experimental setup

Preliminary results - Probability Defining a quantity P representing the relative proportion of initial states in a given context ( ,D) that lead to the outcome V = v i, as Then this proportion turns out to correspond very closely to our everyday notion of probability. It behaves according to the Kolmogorov axioms In the Bayesian interpretation the context corresponds to the prior (though not as ‘belief’) In a frequentist setting the context represents the average, infinite limit properties of the normal, ‘default’ throw (only now made explicit). But, It does make probability a fundamentally relative notion … except for degenerate cases where there is an inherent symmetry in the system and context, e.g. fairly tossing a fair coin; in that case the proportion is not dependent on the context.

Preliminary results - Causal relations Now what about causality? Take ‘effective manipulability’ as a prime characteristic of a causal relation. If probability relates to proportions of trajectories through phase space, then causality relates to changes in these proportions with respect to changes in system parameters, initial and/or intermediate states of the system. However: Causal nodes have no clearly defined boundaries Causal nodes change with every random variable Causal nodes depend on the level of detail in the manipulation Making causal relations also a fundamentally relative notion, without a clear prospect of obtaining ‘the’ causal structure of a system … because in reality it is much more complicated, … or is it? Hopeful for obtaining a causal structure: Causal nodes represent areas in phase space where these changes are effectively induced (e.g. parameter pin length on ‘pin up’) These areas in phase space are connected via the system trajectories Random variables can represent intermediate stages of trajectories