Chapter 4: Causal and Statistical Dependence

Causal Dependence

expression A depends on expression B if it is ever necessary to evaluate B in order to evaluate A

What about an expression like:

A = C ? B + 2 : 5

Does A depend on B? Answer is only in certain contexts.

Note that A, B and C are evaluations of a function. This incorporates another level of subtlety:

a specific evaluation of A might depend on a specific evaluation of B

However, note that if a specific evaluation of A depends on a specific evaluation of B, then any other specific evaluation of A will depend on some specific evaluation of B. Why?

My interpretation is that if A=5 depends on B=3, A != 5 will depend on some value x in B=x. This implies causation as a mapping from domain of B (the cause) to the domain of A (the effect).

Detecting Dependence Through Intervention

The idea is pretty straight-forward:

If we manipulate A, does B tend to change?

Note how var A is given a value directly.

If setting A to different values in this way changes the distribution of values of B, then B causally depends on A.

var BdoA = function(Aval) {
  return Infer({method: 'enumerate'}, function() {
    var C = flip()
    var A = Aval //we directly set A to the target value
    var B = A ? flip(.1) : flip(.4)
    return {B: B}
  })
}

viz(BdoA(true))
viz(BdoA(false))

Another example:

var cold = flip(0.02)

var cough = (cold && flip(0.5)) || (lungDisease && flip(0.5)) || flip(0.001)

You can set cold = true (or false) manually and see if it changes the distribution of cough (it does). But if you set cough = true (or false), it does not change the distribution of cold.

treating the symptoms of a disease directly doesn’t cure the disease (taking cough medicine doesn’t make your cold go away), but treating the disease does relieve the symptoms.

Statistical Dependence

It simply means

learning information about A tells us something about B, and vice versa.

causal dependencies give rise to statistical dependencies

A simple example:

var BcondA = function(Aval) {
  return Infer({method: 'enumerate'}, function() {
    var C = flip()
    var A = flip()
    var B = A ? flip(.1) : flip(.4)
    condition(A == Aval) //condition on new information about A
    return {B: B}
  })
}

viz(BcondA(true))
viz(BcondA(false))

Because the two distributions on B (when we have different information about A) are different, we can conclude that B statistically depends on A.

Two variables can be statistically dependent even though they is no causal dependence between them. For example, if A and B are leaves of an inverted V graphical model (Ʌ), where the root is C then A and B are not causally related but statistically related.

tags: probabilistic-models-of-cognition