The Value of a Question
This past quarter I was able to take a class at Stanford on Information Theory which had some really interesting material. While the class ended up not covering any significant applications to statistics or machine learning, I was able to glean some some small insights here and there after some contemplation. Below, I present in the form of an approachable gamified series of questions, a relatively interesting and intuitively insightful result at the intersection of statistics and information theory that I came across while musing around a couple days ago.
Gambler’s 20 Questions
I figured the header needed a catchy name so I’ll present the results in a form of gambling scenario. A certain online platform is hosting a prediction contest! The rules are as follows: the contest hosts will randomly sample \(N\) variables \(X_i \sim N(0, \sigma^2)\) and take their sum \(Y = \sum_{i=1}^N X_i\). Each participant submits a prediction \(\hat{Y}\) and receives reward linearly proportional to the negative squared error \((Y\hat{Y})^2\).

Compute the entropy \(H(Y)\).

Which predictor is optimal in this scenario? What is your expected reward?

You have an insider on the contest who’s willing to help you out. You can ask them a single yes or no question to help you decide on your predictor. What is an optimal question to ask? Denote the random variable \(Z\) as the answer to your question. Compute \(H(Y\vert Z)\) and your expected reward conditioning on this information. Hint: The expected value of the halfnormal distribution is given by \(E[A \vert A > 0] = \sigma \sqrt{\frac{2}{\pi}}\) where \(A\sim N(0, \sigma^2)\).

Is there another question you could have asked, \(Z'\), that gives you the same amount of information, but yields lower reward?

Instead of asking a question, the insider now proposes a deal: for each \(c\) units of reward you pay, they will give you the value of one of the variables \(X_i\) (i.e \(kc\) units of reward gives you the first \(k\) variable values). How much should you pay to optimize the expected rewards?

Compute the marginal information gained from the \(m\)th payment. Is it increasing or decreasing in \(m\)? Analyze the ratio of cost to marginal information gained. Is this quantity bounded in \(m\) under the asymptotic limit \(N\to \infty\)?

Given the choice between learning the first \(m\) variable values, or the ability to ask \(k\) yes or no questions in succession, which should you choose? Describe your answers in terms of \(m,k\) and the given variables. Hint: You may want to check out the Truncated Normal and try writing some code as this part can get messy.

Does the optimal series of \(k\) questions satisfy \(H(Z_k) = 1\)?