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## Bayesian program induction assignment help

Solution.pdfPart 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.

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