Crossposted from the AI Alignment Forum. May contain more technical jargon than usual.

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How to Throw Away Information in Causal DAGs

8th Jan 2020

4philip_b

3johnswentworth

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Instead of saying " contains all information in relevant to ", it would be better to say that, contains all information in that is relevant to if you don't condition on anything. Because it may be the case that if you condition on some additional random variable , no longer contains all relevant information.

Example:

Let be i.i.d. binary uniform random variables, i.e. each of the variables takes the value 0 with probability 0.5 and the value 1 with probability 0.5. Let be a random variable. Let be another random variable, where is the xor operation. Let be the function .

Then contains all information in that is relevant to . But if we know the value of , then no longer contains all information in that is relevant to .

When constructing a high-level abstract causal DAG from a low-level DAG, one operation which comes up quite often is throwing away information from a node. This post is about how to do that.

First, how do we throw away information from random variables in general? Sparknotes:

For more explanation of this, see Probability as Minimal Map.

For our purposes, starting from a low-level causal DAG, we want to:

Here ¯S denotes all the node indices

outsideS. (Specifying S rather than ¯S directly will usually be easier in practice, since XS is usually a small neighborhood of nodes around Xi.) In English: we want to throw away information from Xi, while retaining all information relevant to nodes outside the set S.Two prototypical examples:

In both examples, we’re throwing out “local” information, while maintaining any information which is relevant “globally”. This will mean that local queries - e.g. the voltage in one wire given the voltage in a neighboring wire at the same time - are not supported; short-range correlations violate the abstraction. However, large-scale queries - e.g. the voltage in a wire now given the voltage in a wire a few seconds ago - are supported.

## Modifying Children

We still have one conceptual question to address: when we replace Xi by f(Xi), how do we modify children nodes of Xi to use f(Xi) instead?

The first and most important answer is: it doesn’t matter, so long as whatever they do is consistent with f(Xi). For instance, suppose Xi ranges over {-1, 0, 1}, and f(Xi)=X2i. When f(Xi)=1, the children can act as though Xi were -1 or 1 - it doesn’t matter which, so long as they don’t act like Xi=0. As long as the childrens’ behavior is consistent with the information in f(Xi), we will be able to support long-range queries.

There is one big catch, however: the children do need to all behave as if Xi had the

samevalue, whatever value they choose. The joint distribution P[Xch(i)|Xsp(i),f(Xi)] (where ch(i) = children of i and sp(i) = spouses of i) must be equal to P[Xch(i)|Xsp(i),X∗i] for some value X∗i consistent with f(Xi). The simplest way to achieve this is to pick a particular “representative” value X∗i(f∗) for each possible value f∗ of f(Xi), so that f(X∗i(f∗))=f∗.Example: in the digital circuit case, we would pick one representative “high” voltage (for instance the

supply voltageVDD) and one representative “low” voltage (for instance the ground voltage VSS). X∗i(f(Xi)) would then map any high voltages to VDD and any low voltages to VSS.Once we have our representative value function X∗i(f(Xi)), we just have the children use X∗i(f(Xi)) in place of Xi.

If we want, we could even simplify one step further: we could just choose f to spit out representative values directly. That convention is cleaner for proofs and algorithms, but a bit more confusing for human usage and examples.