Conditioning: Exercises

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To return to the top level: Probabilistic Models of Cognition.


1

In the example on Medical Diagnosis from the Generative Models section we suggested understanding the patterns of symptoms for a particular disease by changing the program to make that disease always true.

A) For this example, does this procedure give the same answers as using query to condition on the disease being true?

B) Why would this procedure give different answers than conditioning for more general hypotheticals? Construct an example where these are different. Then translate this into a Church model and show that intervening and observation give different answers. Hint: think about intervening versus observing on a variable that has a causal parent.

2

Use the rules for computing probabilities to compute the marginal distribution on return values from these Church programs:

A)

B)

3

Describe (using ordinary english) what the Church program in exercise 2B means.

How would you change it if you thought people are more likely to smile if they want something? If you think some people are more likely to want something than others? If you think nice people are less likely to want something?

Given your extended model, how would you ask whether someone wants something from you, given that they are smiling and have rarely smiled before? Show the Church program and a histogram of the answers -- in what ways do these answers make intuitive sense or fail to?

4

Suppose that you are playing the following game at a casino. In this game, a machine randomly gives you a letter of the alphabet and the probability of winning depends on which letter you receive. The machine gives the letters a, e, i, o, u, y (the vowels) with probability 0.01 each and the remaining letters (i.e., the consonants) with probability 0.047 each. Let's use the variable <math>h</math> to denote the letter that you receive; the probability of winning for a given <math>h</math> is <math>1/Q(h)^2</math>, where <math>Q(h)</math> denotes the numeric position of the letter (e.g., <math>Q(\textrm{a}) = 1, Q(\textrm{b}) = 2</math>, and so on).

Let's express this in formal terms. The hypothesis space, <math>H</math>, is the set of letters <math>\{a, b, c, d, \dots, y, z\}</math> and the prior probability of a hypothesis <math>h</math> is 0.01 for vowels (a, e, i, o, u, y) and 0.047 for consonants. The likelihood, <math>p(d \mid h) = 1/Q(h)^2</math>, is the probability of winning given that you drew some letter <math>h</math>.

(A) (Here, give your answers in English, not math) What does the <math>d</math> in <math>p(d \mid h)</math> represent? What does the posterior <math>p(h \mid d)</math> represent?

(B) Manually compute <math>p(h \mid d)</math> for each hypothesis (Excel or something like it is helpful here). Remember to normalize - make sure that summing all your <math>p(h \mid d)</math> values gives you 1.

Now, we're going to write this model in Church using the cosh engine. Here is some starter code for you:

(C) What does the my-list-index function do? What would happen if you ran (my-list-index 'mango '(apple banana) 1)?

(D) What does the multinomial function do? Use multinomial to express this distribution:

x P(x)
red 0.5
blue 0.05
green 0.4
black 0.05

(E) Fill in the ...'s in the code to compute <math>p(h \mid d)</math>. Include a screenshot of the resulting graph. What letter has the highest posterior probability? In English, what does it mean that this letter has the highest posterior? Make sure that your Church answers and hand-computed answers agree - note that this demonstrates the equivalence between the program view of conditional probability and the distributional view.

(F) Which is higher, <math>p(\text{vowel} \mid d)</math> or <math>p(\text{consonant} \mid d)</math>? Answer this using the Church code you wrote (hint: use the vowel? function)

(G) What difference do you see between your code and the mathematical notation? What are the advantages and disadvantages of each? Which do you prefer?

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