Bayes-Optimal Decision Rule for a Noisy Signal
You observe a noisy signal $S \in \{-1, +1\}$ about an ETF's true value move $V \in \{-1, +1\}$. The prior on $V$ is uniform: $P(V = +1) = P(V = -1) = 0.5$. The signal is informative but imperfect: $P(S = V) = 0.65$.
After observing $S$, you must choose an action $a \in \{-1, 0, +1\}$, corresponding to sell, do nothing, or buy. The payoffs are:
- If $a \neq 0$ and $a = V$: you earn $+1$
- If $a \neq 0$ and $a \neq V$: you earn $-1$
- If $a = 0$: you earn $0$
- Derive the Bayes-optimal decision rule -- that is, the action $a(S)$ that maximizes expected payoff given the observed signal.
- Compute the overall expected payoff of this optimal strategy.
Hints
- Start by computing the posterior probability $P(V \mid S)$ using Bayes' theorem. The uniform prior and symmetric signal structure make this straightforward.
- For each possible signal realization, compare the expected payoff of all three actions. The action $a = 0$ always yields $0$ -- so you just need to check whether the expected payoff of trading in either direction beats zero.
- The expected payoff of matching the signal is $0.65(+1) + 0.35(-1)$. Compare this to zero to determine when it is optimal to trade versus sit out.
Worked Solution
How to Think About It: This is a classic signal extraction problem. You have a noisy binary signal about which way an ETF will move, and you need to decide whether to trade on it or sit out. The key tension: the signal is right 65% of the time, so it is informative but far from certain. The question boils down to whether the edge from following the signal (which pays off 65% of the time) outweighs the cost of being wrong (35% of the time). Since doing nothing always pays 0, you just need the expected payoff from acting to be positive. Intuitively, 65% accuracy on a symmetric $\pm 1$ payoff gives you an edge of $0.65 - 0.35 = 0.30$ per trade, which is solidly positive -- so you should always trade in the direction of your signal.
Quick Estimate: If you always follow the signal, you are right 65% of the time and wrong 35% of the time. Expected payoff per round: $0.65 \times (+1) + 0.35 \times (-1) = 0.65 - 0.35 = 0.30$. Since the signal is symmetric and the prior is symmetric, this is the same regardless of which signal you observe. So the overall expected value is $0.30$. The alternative -- doing nothing -- gives $0$. So following the signal is strictly better.
Approach: Compute the posterior on $V$ given each signal realization, then evaluate the expected payoff of each action under that posterior.
Formal Solution:
*Step 1: Posterior probabilities.*
By Bayes' theorem, when $S = +1$:
$$P(V = +1 \mid S = +1) = \frac{P(S = +1 \mid V = +1) \, P(V = +1)}{P(S = +1)}$$
Since $P(S = V) = 0.65$, we have $P(S = +1 \mid V = +1) = 0.65$ and $P(S = +1 \mid V = -1) = 0.35$. With a uniform prior:
$$P(S = +1) = 0.65 \times 0.5 + 0.35 \times 0.5 = 0.5$$
So:
$$P(V = +1 \mid S = +1) = \frac{0.65 \times 0.5}{0.5} = 0.65$$
$$P(V = -1 \mid S = +1) = 0.35$$
By symmetry, $P(V = -1 \mid S = -1) = 0.65$ and $P(V = +1 \mid S = -1) = 0.35$.
*Step 2: Expected payoff of each action.*
Given $S = +1$:
- $a = +1$ (buy): $E[\text{payoff}] = 0.65 \times (+1) + 0.35 \times (-1) = +0.30$
- $a = -1$ (sell): $E[\text{payoff}] = 0.65 \times (-1) + 0.35 \times (+1) = -0.30$
- $a = 0$ (nothing): $E[\text{payoff}] = 0$
The optimal action when $S = +1$ is $a = +1$ (buy), with expected payoff $+0.30$.
By symmetry, when $S = -1$, the optimal action is $a = -1$ (sell), also with expected payoff $+0.30$.
*Step 3: Overall expected payoff.*
Since each signal is equally likely ($P(S = +1) = P(S = -1) = 0.5$):
$$E[\text{payoff}] = 0.5 \times 0.30 + 0.5 \times 0.30 = 0.30$$
Answer: The Bayes-optimal rule is to follow the signal: buy when $S = +1$, sell when $S = -1$, never do nothing. The overall expected payoff is $0.30$. More generally, for signal accuracy $q > 0.5$ on a symmetric $\pm 1$ payoff, the edge from following the signal is $2q - 1$.
Intuition
This problem illustrates the fundamental principle behind every signal-based trading strategy: you trade when your edge exceeds your cost. Here the cost of being wrong is symmetric with the reward for being right ($\pm 1$), so any signal accuracy above 50% gives you a positive expected value from trading. The edge is exactly $2q - 1$ where $q$ is the signal accuracy -- a formula worth memorizing. At $q = 0.65$, the edge is $0.30$ per trade, which is substantial.
In practice, this is the simplest version of a problem that gets much harder when you add position sizing, transaction costs, or asymmetric payoffs. If there were a cost $c$ to trade, the optimal rule would switch to "do nothing" when $2q - 1 < c$. If the payoffs were asymmetric (say $+1$ for a correct buy but $-2$ for a wrong buy), you would need the posterior to clear a higher threshold before buying. The key lesson: always compare the conditional expected payoff of acting to the payoff of doing nothing, and only act when the edge is positive.