[2 marks] 2. ANSWER EITHER THIS QUESTION OR QUESTION 3. Tenenbaum's concept model was used to predict behavior in a "num

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2 Marks 2 Answer Either This Question Or Question 3 Tenenbaum S Concept Model Was Used To Predict Behavior In A Num 1 (213.68 KiB) Viewed 26 times

[2 marks] 2. ANSWER EITHER THIS QUESTION OR QUESTION 3. Tenenbaum's concept model was used to predict behavior in a "number game" experiment, where one player thinks of a concept and gives random example num- bers that are consistent with the concept. Suppose the integers 1 through 100, inclusive, are allowed. (a) For Bayesian models like Tenenbaum's, it's important to understand how they use and define their likelihoods. i. Suppose the likelihood of a set of examples X under concept cı is li, and the likelihood of X under concept ca is 12. If these are the only possible concepts and they have equal prior probabilities, what's the posterior probability of ci? ii. One kind of number concept that is represented in this model is based on intervals. Suppose c is "numbers falling between 1 and 100”, and ca is “numbers falling between 1 and 95”. Are there any sets of numbers - not limited to the sets (or set sizes) used in Tenenbaum's experiment - where we would expect c2 to be strongly favored by the likelihood? If so, give an example and explain why. Otherwise, explain why not. (b) In his paper about the model, Tenenbaum writes "What I call the MIN algorithm ... replaces the step of hypothesis averaging with maximization: p(y e C|X) = 1 if y E argmax p(X\h), and 0 otherwise.” X is a set of numbers from the concept, y is a new number, and h is a concept. i. What's a term for what the MIN algorithm is doing in picking the concept that y belongs to? Briefly explain. ii. Tenenbaum compares the MIN model to his Bayesian model using R”, a measure based on the correlation between human judgments and model predictions. Suppose we want to compare the probabilities of human judgments (estimates that the probability that a number is in the set defined by the concept) under these two models. What is currently miss- ing? Discuss how you might augment the models and, if necessary, the experiment, giving specifics where you can. QUESTION CONTINUES ON NEXT PAGE [3 marks] [2 marks] [6 marks)

QUESTION CONTINUED FROM PREVIOUS PAGE [3 marks) (c) Tenenbaum's paper mentions two free parameters. One is !, which governs the bias toward rule-based versus interval-based concepts. The second is o, which deals with the probability that an interval-based concept will have a particular size of interval. The original value of o was chosen to favor small intervals, with intervals longer than 10 being unlikely. i. Suppose the example numbers (X) are 2 and 8. If o is changed to assign high probability to larger intervals (e.g., longer than 30), what happens to the model's predictions about the relative probabilities of numbers 4 and 7 being in the concept? That is, how does the ratio P(46C|X={2,8}) P(TEC X=(2,8) change? Briefly explain why. Mention the prior and/or likelihood as appropriate. ii. Suppose we believe there are two kinds of participants in the game: those who favour concepts based on mathematical rules and those who favour interval-based concepts. How can we build on Tenenbaum's model to test this idea, supposing that each participant plays 20 different variations on the number game? Assume that o is fixed, and you have a function that gives you the probability of a particular judgment given the examples in the game as well as I, expressed as Pjudgment(y e C|X, 1). (d) Tenenbaum's model used a uniform prior over concepts defined by mathe- matical rules. If we wanted to expand the hypothesis space to include every rule we can possibly think of, why might the decision to use a uniform prior become problematic? Suggest one approach we could take in defining a non- uniform prior over the mathematical rules in Tenenbaum's model, and explain one strength and one weakness of that approach. [5 marks] (4 marks)