Optimal strategy for classroom behaviour

During the course of attending Information Theory by Prof. Lalitha Sankar, I encountered the following question from Thomas & Clover in one of the homeworks:

Calculate the capacity of the following channel with the probability transistion matrix,

$$\begin{bmatrix}p & 1-p & 0 & 0\\ 1-p & p & 0 & 0\\ 0 & 0 & q & 1-q\\ 0 & 0 & 1-q & q \end{bmatrix}$$

Mind’s muscle memory kicked in as the math derivation lead to the conclusion that optimal strategy is to use the channel according to the ratio: $\frac{2^{-H(p)}}{2^{-H(p)} +2^{-H(q)}}$, where $H(\alpha)$ would denote the entropy of Bernoulli($\alpha$).

A few moment of stagnant staring at the problem got me thinking of a potential use case.

A student is attending a lecture and the BSC channels model the following

1. First BSC channel : Denotes the capability of the student for understanding the material taught in the lecture. Let $X$ denote the event that the student is attentive to what is being taught in the lecture or not and $Y$ model whether the student understood the concept or not. $\mathbb{P}(Y|X)$ denotes the probability matrix of the outcomes. In order for the channel to behave as a Bernoulli($\alpha$), we make the following assumptions:
   • If the student is in tune with the teaching style of the lecturer, then $p\approx 1$ (a perfect learner). $\mathbb{P}(Y=1 | X=1) \approx 1$ i.e. given complete attention, student understands it completely. On the other hand, given no attention student doesnt gain anything hence $P(Y=0 | X=0)\approx 1$
   • If the student is completely oblivious to the lecturer’s teaching patterns and is attending the course because his grad advisor forced his hand, then we model it as a Bernoulli($\frac{1}{2}$). This is because whether the student is attentive or not, the conclusion is that the student is not understanding the course content.

2. Second BSC channel : Denotes the capability of student in alternate pursuits in classroom –
   • Thinking of research project the student is working on (this is me)
   • Thinking of strategies in board games (also me. Terra Mystica !!!)
   • Chatting on phone, replying to mails, twitter ,etc. The parameter $q$ models how clearly one can think of that other task being performed while in classroom.

If we consider this model then the surprising conclusion of this homework problem comes to :

It is information theoritically suboptimal to completely focus on the content being taught in class

There is an ideal ratio in which both needs to be pursued during your classroom time. So, the next time you have are caught distracted in class you have an argument to make.

Though you cant completely zone out as well, since that is also a suboptimal strategy :)

License

Copyright 2017 Parth Thaker .

Released under the MIT license.