The Noninvariance Of Deterministic Causal Models

in its title and also the supporting argument for the thesis. Back of this informal and conceptually transparent argument is a substantial technical set of ideas that are explained in some detail but still not at a fully technical level in the remaining sections. The reader who is interested in under standing the general idea of the noninvariance claim I am making about deterministic causal models should be able to get the gist of the argument entirely from what is said in this introductory section. The first point is that the results come out of ergodic theory. What ergodic theory is is explained in more detail in Sections 2 and 3, but it is easy to give a simple and familiar example of an ergodic process. Consider coin flipping. For simplicity's sake, let the coin be a fair one. Then that process is ergodic. It is ergodic for two reasons. First, it is asymptotic ally stationary, which means that the probability distribution of what is happening is, in the long run, unchanged over time and, secondly, it has a unique asymptotic distribution that is independent of the initial starting distribution. For example, we might have agreed always to start with a head, or always to start with a tail, but this does not affect the asymptotic distribution. A slightly more complicated example of an ergodic process is a first-order Markov process, that is, a process in which what happens on trial n + 1 depends on what happened on trial n, where the dependence is a probabilistic one. The same two conditions obtain, namely, asymptotic stationarity and unique asymptotic distribution of states, independent of the initial distribution. (For general processes it is standard to require, for ergodicity, not just asymptotic stationarity but stationarity for all time. This slightly stronger condition will be used in the general definition given later.) The second concept is that of two ergodic processes being measure theoretic isomorphic. This means that their structure of uncertainty is the