Bayesian program induction assignment help

Part I: Bayesian program induction
Write a program which computes the posterior probability that any given set of strings (e.g., aab,
bbaa, aaaaab) was generated by Program 1 versus Program 2. Here each candidate program is
represented by a probabilistic context-free grammar. Write your code in such a way that it can
be easily extended to handle a larger space of potential programs. Test whether your code is
working and be prepared demonstrate and explain the result.
1. P(Program 1)=0.7
2. P(Program 2)=0.3
3. Program 1:
4. P(S-->aSb)=0.5
5. P(S-->bSa)=0.2
6. P(S-->a)=0.1
7. P(S-->b)=0.2
8. Program 2:
9. theta ~ Beta(0.1,0.1)
10. P(S-->Xa)=theta
11. P(S-->bY)=1-theta
12. P(Y-->aY)=0.4
13. P(Y-->a)=0.6
14. P(X-->Xb)=0.2
15. P(X-->b)=0.8
Hint: If you do not already have a favorite probabilistic program induction framework, you can
use WebPPL (http://webppl.org).
Part II: Inducing decision rules from demonstrations
In the Mouselab paradigm (Payne, Bettman, & Johnson, 1988) for studying multi-alternative
risky choice participants repeatedly choose between n gambles based on a payoff matrix that
specifies how much each gamble will pay depend on which of the k possible outcomes will
occur. The probabilities of the k possible outcomes are shown in the first column of the payoff
matrix and the payoffs of the n gambles are shown in columns 2 to n+1. Critically, the entries of
the payoff matrix are initially concealed from the participant; to uncover the payoff of Gamble g
in the event of outcome the participant has to click on the corresponding cell of the payoff matrix
as illustrated in Figure 1.
Previous research has found that people use different decision strategies in this paradigm. One
of these decision rules is the Take-The-Best heuristic (TTB). TTB inspects only the payoffs for
the most probably outcome (e.g., BROWN in the example show in Figure 1) and then chooses
the gamble whose payoff in the event of the most probable outcome is highest. Another strategy
that people have been found to use is called “Satisficing” (SAT). Satisficing starts by inspecting
all payoffs of the first gamble. If all of them are larger than its aspiration level (e.g., $0.15) then it
selects that gamble. Otherwise, it inspects all payoffs of the second gamble, compares their

minimum to its aspiration level and so on. A third strategy that people often use is Satisficing-
Take-The-Best (SAT-TTB). SAT-TTB is like TTB except that it stops inspecting payoffs as soon

as it finds a single payoff that is above its aspiration level (e.g., $0.15).

Figure 1: Illustration of the Mouselab paradigm. The number of balls conveys the probability of the corresponding event in
percent.
The Mouselab paradigm records the sequence of clicks that the participant makes, and this
sequence of clicks, in turn, reveals the participants decision-making strategy. Figure 2 illustrates
the sequence of clicks that the SAT-TTB strategy generated in one example problem. In a
decision-making experiment participants make a series of decisions and the payoffs and
probabilities are different every time.
Tasks:
a) Express the TTB, SAT, and SAT-TTB heuristics as rules that can be used to generate
the corresponding click sequences.
b) Formulate a grammar for decision rules from which the decision rules you formulated
can be derived as sentences.
c) Use the decision rules you formulated in part a) to generate click sequences for a
Mouselab paradigm with 5 gambles and 4 outcomes.
d) Develop a method that takes in the grammar you formulated and the click sequences
you generated with one of the rules and returns a simple decision rule that is consistent
with those click sequences.