Decision Theory in Classification
Problem 1
Your so-called friend has convinced you to play the following (“fun!”) game.
- Your friend will flip a biased coin with probability of heads equal to \(q\). As the coin spins in the air, you need to predict whether the coin lands heads or tails.
- Your friend tells you the value of \(q\) before the coin flip.
- You receive a payoff depending on your prediction and the outcome of the coin flip, according to the following table:
| Prediction | Outcome | Payoff |
|---|---|---|
| Heads | Heads | +1 |
| Tails | Tails | +1 |
| Heads | Tails | \(-a\) |
| Tails | Heads | \(-b\) |
Here, \(a\) and \(b\) are positive numbers that represent the cost of calling the flip wrong.
Part A
Suppose you predict heads. Write down a formula for your expected payoff in terms of \(q\), \(a\), and \(b\). Write down a similar formula for the expected payoff if you predict tails.
Part B
Suppose that you decide to follow a policy: you pick \(t \in [0,1]\) in advance. Then, when your friend tells you the value of \(q\), you predict heads if \(q \geq t\) and tails otherwise. What is the value of \(t\) that maximizes your expected payoff?