Blackwell Garbling and Decision Risk

Worked Example

Consider a setup where the hidden state H has prior probability P(H=1) = 0.5. A signal S provides accuracy a = 0.8. We garble this signal with error e = 0.25, and evaluate decisions with false-positive cost c = 2.

1. Effective Accuracy: The garbled accuracy b is computed as e + (1 - 2e) * a = 0.25 + (1 - 0.5) * 0.8 = 0.65.
2. Original Risk: With a = 0.8, the joint probabilities for s=1 are P(H=1, s=1) = 0.4 and c * P(H=0, s=1) = 2 * 0.1 = 0.2. The minimum risk is 0.2. For s=0, P(H=1, s=0) = 0.1 and c * P(H=0, s=0) = 2 * 0.4 = 0.8, with minimum risk 0.1. The total Bayes risk is 0.3.
3. Garbled Risk: With b = 0.65, for s=1 we have min(0.325, 2 * 0.175) = 0.325. For s=0 we have min(0.175, 2 * 0.325) = 0.175. Total Bayes risk is 0.5.
4. Result: Garbling increased the risk from 0.3 to 0.5 (a risk increase of 0.2).

Experiment

You can interact with this model in the Blackwell Garbling and Decision Risk module.

1. Set Signal accuracy to 1 (perfect signal) and observe the original risk drop to 0.
2. Increase the Garbling error to 0.5. The effective accuracy drops to 0.5, representing a completely uninformative signal, causing the risk to match the "No Signal Risk".
3. Change the Prior state one to 0.9. At extreme priors, the optimal decision rule might ignore the signal entirely, meaning garbling does not strictly increase risk.

Limitations

This module considers a simplified environment with a binary state and a symmetric binary channel.

• Binary Restrictions: In reality, signals and states can be continuous or have multiple categories. The general Blackwell theorem applies to arbitrary state and signal spaces, but visualization becomes difficult.
• Symmetric Costs: Only the false-positive cost is a free variable (with false-negative cost fixed at 1). Asymmetric false-negative costs would require an extra parameter but wouldn't change the fundamental result.
• Equality Case: When prior probability is highly skewed (e.g., 0.9) or cost is extreme, the optimal decision may be to always choose the same action regardless of the signal. In this boundary case, garbling the signal does not increase the risk, as the signal was already being ignored.

References

Primary sources and educational material covering Blackwell's theorem and decision risk.

• Blackwell's Theorem (Karni, 2021) - Detailed theoretical background on how degrading signals cannot improve Bayes risk for any decision maker.