Task
Your goal is to produce a function that computes E[q ∗ j |q˜jt, Njt], the expected quality of a GP given observed rounded rating ˜qjt and number of reviews Njt. The function obviously needs to take ˜qjt, Njt as variable inputs. Some fixed inputs (i.e. inputs that won’t ever change) that you also need are:
• A vector of random draws from the prior P(q ∗ j ). This is so that computing R f(q ∗ j )dP(q ∗ j ) is as simple as taking an average of f(·) over the vector of random draws.
• A vector of 4 cutpoints for the ordered probit.
Some hints:
• Start by building the simplest functions. For example, P(qrt|q ∗ j ) is really easy.
• Think carefully through the combinatorics when computing R(˜qjt, Njt).
– This is one of the hardest (if not the hardest) parts, especially if you’re trying to do so efficiently.
– I suggest using recursion. This shows how to do so for an equivalent problem: https: //www.techiedelight.com/total-ways-sum-with-n-throws-dice-having-k-faces/. Memoization can also really help.
• An excellent “checkpoint” is computing p(˜qjt|Njt, q∗ j ). If you can compute p(˜qjt|Njt, q∗ j ), you’re basically done with the hard part. Let me know when you can compute p(˜qjt|Njt, q∗ j ). I may tell you that the code gives me enough information and you don’t need to finish the exercise.
• Test your functions along the way. For example: do the probabilities sum to 1 when you
integrate your probability over the whole state space?
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