Chapter 3: Conditioning

Much of cognition can be understood in terms of conditional inference. In its most basic form, causal attribution is conditional inference: given some observed effects, what were the likely causes? Predictions are conditional inferences in the opposite direction: given that I have observed some cause, what are its likely effects?

Inference can be done in various ways. The most basic way is rejection sampling:

var model = function () {
    var A = flip()
    var B = flip()
    var C = flip()
    var D = A + B + C
    condition(D >= 2)
    return A
}
var dist = Infer({method: 'rejection', samples: 100}, model)
viz(dist)

Another way is to enumerate all possibilities and use Bayes theorem.

In the case of a WebPPL Infer statement with a condition, A = a will be the “event” that the return value is a while B = b will be the event that the value passed to condition is true. Because each of these is a regular (unconditional) probability, they and their ratio can often be computed exactly using the rules of probability. In WebPPL the inference method 'enumerate' attempts to do this calculation (by first enumerating all the possible executions of the model):

var model = function () {
    var A = flip()
    var B = flip()
    var C = flip()
    var D = A + B + C
    condition(D >= 2)
    return A
}
var dist = Infer({method: 'enumerate'}, model)
viz(dist)

Other ways to implement Infer

Much of the difficulty of implementing the WebPPL language (or probabilistic models in general) is in finding useful ways to do conditional inference—to implement Infer.

Conditions and observations

You can add complex propositions to condition without assigning a variable to them:

var dist = Infer(
  function () {
    var A = flip()
    var B = flip()
    var C = flip()
    condition(A + B + C >= 2)
    return A
});
viz(dist)

Using condition allows the flexibility to build complex random expressions like this as needed, making assumptions that are phrased as complex propositions, rather than simple observations. Hence the effective number of queries we can construct for most programs will not merely be a large number but countably infinite, much like the sentences in a natural language.

This will run forever:

var model = function(){
  var trueX = sample(Gaussian({mu: 0, sigma: 1}))
  var obsX = sample(Gaussian({mu: trueX, sigma: 0.1}))
  condition(obsX == 0.2)
  return trueX
}
viz(Infer({method: 'rejection', samples:1000}, model))

Instead of conditioning, it is better to observe:

var model = function(){
  var trueX = sample(Gaussian({mu: 0, sigma: 1}))
  observe(Gaussian({mu: trueX, sigma: 0.1}), 0.2)
  return trueX
}
viz(Infer({method: 'rejection', samples:1000, maxScore: 2}, model))

observe does the same thing as shown below but in a much more efficient manner:

var x = sample(distribution);
condition(x === value);
return x;

In particular, it’s essential to use observe to condition on the value drawn from a continuous distribution.

Factors

In WebPPL, condition and observe are actually special cases of a more general operator: factor. Whereas condition is like making an assumption that must be true, then factor is like making a soft assumption that is merely preferred to be true.

For example if we use observe, A will always be true:

var dist = Infer(
  function () {
    var A = flip(0.001)
    condition(A)
    return A
});
viz(dist)

But if we use factor, we can tweak how often we want A to be true:

var dist = Infer(
  function () {
    var A = flip(0.01)
    factor(A?10:0)
    return A
});
viz(dist)

factor(A?10:0) gives a much higher preference to true than factor(A?5:0) for example.

factor(A?x:y) technically means:

P(A=true) = e^x / (e^x + e^y)

Reasoning about Tug of War

An interesting question posed here is:

For instance, how likely is it that Bob is strong, given that he’s been on a series of winning teams?

Some points to reflect on:

• If team [Bob, randomly chosen player] almost always defeats [Tom, Y], is Bob strong or is [Tom, Y] weak? Do these beliefs change as the number of matches between them goes up?
• Does the likelihood of Bob being strong go up if the team he is in defeats different teams?

tags: probabilistic-models-of-cognition