For each of the following programs:
Draw the dependency diagram (Bayes net). If you donâ€™t have software on your computer for doing this, Google Docs has a decent interface for creating drawings.
Use informal evaluation order reasoning and the intervention method to determine causal dependency between A and B.
Use conditioning to determine whether A and B are statistically dependent.
var a = flip();
var b = flip();
var c = flip(a && b ? .8 : .5);
var a = flip();
var b = flip(a ? .9 : .2);
var c = flip(b ? .7 : .1);
var a = flip();
var b = flip(a ? .9 : .2);
var c = flip(a ? .7 : .1);
var a = flip(.6);
var c = flip(.1);
var z = flip() ? a : c;
var b = z ? 'foo' : 'bar';
var examFairPrior = Bernoulli({p: .8});
var doesHomeworkPrior = Bernoulli({p: .8});
var examFair = mem(function(exam) { return sample(examFairPrior) });
var doesHomework = mem(function(student) { return sample(doesHomeworkPrior) });
var pass = function(student, exam) {
return flip(examFair(exam) ?
(doesHomework(student) ? .9 : .5) :
(doesHomework(student) ? .2 : .1));
}
var a = pass('alice', 'historyExam');
var b = pass('bob', 'historyExam');
As the end of the chapter mentions, some probabilistic programs do not have corresponding graphical models. This includes programs with recursion. The example below is an example of a recursive probabilistic program:
// compute the total rainfall over the course of a storm
var totalRain = function() {
// amount of rainfall today
var dailyRainfall = sample(Gaussian({mu: 5, sigma: 1}))
// if it rained a lot today, the storm is likely to continue
if (dailyRainfall > 6 && flip(0.8) || flip(0.3)) {
return dailyRainfall + totalRain()
} else {
return dailyRainfall
}
}
viz(Infer({"model": totalRain, "method": "forward"}))
Write another recursive probabilistic program to model some real-world phenomenon below: