Bayesian Posterior with Two Binary Signals
A binary event $E \in \{0, 1\}$ has prior $P(E = 1) = 0.40$. You receive two independent analyst signals $S_1, S_2 \in \{0, 1\}$, each with the same accuracy:
$$P(S_i = 1 \mid E = 1) = 0.70, \quad P(S_i = 1 \mid E = 0) = 0.30$$
The signals are conditionally independent given $E$. You observe $(S_1, S_2) = (1, 0)$.
Compute the posterior probability $P(E = 1 \mid S_1 = 1, S_2 = 0)$.
Hints
- The signals are conditionally independent given $E$, so the joint likelihood factors: $P(S_1, S_2 \mid E) = P(S_1 \mid E) \cdot P(S_2 \mid E)$.
- Compute the likelihood of $(S_1, S_2) = (1, 0)$ under both $E = 1$ and $E = 0$. Notice anything about the ratio?
- When one signal agrees and one disagrees, and the signal accuracy is symmetric ($P(S=1|E=1) = P(S=0|E=0)$), the combined likelihood ratio is exactly 1, so the posterior equals the prior.
Worked Solution
How to Think About It: You have a prior belief that the event happens with probability 0.40, and you get two analyst signals with 70% accuracy. One signal says "yes" ($S_1 = 1$) and one says "no" ($S_2 = 0$). Since the signals disagree, they partially cancel out, and the posterior should not move dramatically from the prior. But they are symmetric in accuracy, so the "yes" signal and "no" signal have equal information content -- intuitively, the posterior should stay near the prior. The slight asymmetry comes from the prior itself.
Quick Estimate: With symmetric signal accuracy (70/30 split), one "yes" and one "no" should roughly cancel. The likelihood ratio for $(1, 0)$ under $E=1$ vs. $E=0$ is $\frac{0.7 \times 0.3}{0.3 \times 0.7} = 1$. So the posterior should equal the prior! Let's verify.
Approach: Direct application of Bayes' theorem with conditionally independent signals.
Formal Solution:
By Bayes' theorem:
$$P(E = 1 \mid S_1 = 1, S_2 = 0) = \frac{P(S_1 = 1, S_2 = 0 \mid E = 1) \cdot P(E = 1)}{P(S_1 = 1, S_2 = 0)}$$
Numerator (likelihood $\times$ prior for $E = 1$):
By conditional independence: $$P(S_1 = 1, S_2 = 0 \mid E = 1) = P(S_1 = 1 \mid E = 1) \cdot P(S_2 = 0 \mid E = 1) = 0.70 \times 0.30 = 0.21$$
$$\text{Numerator} = 0.21 \times 0.40 = 0.084$$
Denominator (total probability of observed signals):
$$P(S_1 = 1, S_2 = 0 \mid E = 0) = P(S_1 = 1 \mid E = 0) \cdot P(S_2 = 0 \mid E = 0) = 0.30 \times 0.70 = 0.21$$
$$P(S_1 = 1, S_2 = 0) = 0.21 \times 0.40 + 0.21 \times 0.60 = 0.21 \times 1.00 = 0.21$$
Posterior:
$$P(E = 1 \mid S_1 = 1, S_2 = 0) = \frac{0.084}{0.21} = 0.40$$
Answer: $P(E = 1 \mid S_1 = 1, S_2 = 0) = 0.40$
The posterior equals the prior. This is not a coincidence -- when one signal agrees and one disagrees, and both signals have symmetric accuracy ($P(S_i = 1 \mid E = 1) = P(S_i = 0 \mid E = 0) = 0.70$), the likelihood ratio is exactly 1, so the data provides no net information.
Verification via likelihood ratio: The likelihood ratio for each signal is $\Lambda_i = P(S_i \mid E=1) / P(S_i \mid E=0)$. For $S_1 = 1$: $\Lambda_1 = 0.7/0.3 = 7/3$. For $S_2 = 0$: $\Lambda_2 = 0.3/0.7 = 3/7$. Combined: $\Lambda = \Lambda_1 \cdot \Lambda_2 = (7/3)(3/7) = 1$. A likelihood ratio of 1 means no update -- the posterior odds equal the prior odds.
Intuition
This problem beautifully illustrates the likelihood ratio framework for Bayesian updating. Each signal has a likelihood ratio of $7/3$ when it matches the event and $3/7$ when it does not. With one matching and one not matching, the ratios multiply to 1 -- perfect cancellation. The posterior equals the prior because the evidence is exactly as likely under either hypothesis.
This is a fundamental concept in quantitative trading. When you receive conflicting signals of equal quality, they wash out -- your belief should not change. Market makers encounter this constantly: one model says buy, another says sell, and if both have equal track records, the net signal is zero. The key insight is that you should track likelihood ratios (not just "how many signals agree") because that is what determines how beliefs should update. Two strong agreeing signals are much more informative than ten weak signals that split 6-4.