Reference no: EM132346703

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.

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 (attached).

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).

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 (attached) 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.

References - Payne, J. W., Bettman, J. R., & Johnson, E. J. (1988). Adaptive strategy selection in decision making. Journal of Experimental Psychology: Learning, Memory, and Cognition, 14(3), 534.

Attachment:- Assignment File.rar