Introduction

This edition focuses on a practical application of statistical decision theory in a medical context. We explore how a doctor can apply a randomized decision rule to minimize potential loss when diagnosing a disease and prescribing treatment.

We will investigate the concepts of:

  • Bayes decision rule and Bayes risk

  • Minimax rule and equalizer rule

  • Game-theoretic interpretation

  • Importance of randomized rules

Problem Statement

A doctor is faced with a diagnostic problem involving three possible diseases:

  • \(\theta_1\)
  • \(\theta_2\)
  • \(\theta_3\)

The doctor can prescribe one of three treatments:

  • \(d_1\)
  • \(d_2\)
  • \(d_3\)

After thorough analysis, the doctor assigns the following loss function, where rows correspond to diseases ($\theta$) and columns to decisions ($d$):

\(d_1\)\(d_2\)\(d_3\)
\(\theta_1\)713
\(\theta_2\)016
\(\theta_3\)120

The proposed randomized decision rule is:

$$
\delta_\star = \frac{3}{49} d_1 + \frac{39}{49} d_2 + \frac{7}{49} d_3
$$

a) Show that \(\delta_\star\) is a Bayes and Minimax rule

Let the prior distribution be:

$$
\pi_\star = \left(\frac{7}{49}, \frac{10}{49}, \frac{32}{49} \right)^T
$$

Let \(\mathcal{E}\) denote the set of all randomized rules:

$$
\mathcal{E} = \left{ \delta = p_1 d_1 + p_2 d_2 + p_3 d_3 ; | ; p_i \geq 0, ; \sum p_i = 1 \right}
$$

Compute Risk Function

For a generic rule \(\delta = p_1 d_1 + p_2 d_2 + p_3 d_3\), the risk function is:

$$
\begin{aligned} R(\theta_1, \delta) &= 7p_1 + p_2 + 3p_3 \\ R(\theta_2, \delta) &= p_2 + 6p_3 \\ R(\theta_3, \delta) &= p_1 + 2p_2 \end{aligned}
$$

Bayes Risk

The Bayes risk under \(\pi_\star\) is:

$$
\begin{aligned} B(\pi_\star, \delta) &= \frac{7}{49} R(\theta_1, \delta) + \frac{10}{49} R(\theta_2, \delta) + \frac{32}{49} R(\theta_3, \delta) \\ &= \frac{81}{49}(p_1 + p_2 + p_3) = \frac{81}{49} \end{aligned}
$$

Since this is constant for all \(\delta \in \mathcal{E}\), all such rules are Bayes rules. In particular, \(\delta_\star\) is a Bayes rule.

Constant Risk

Compute the risk of \(\delta_\star\):

$$
\begin{aligned} R(\theta_1, \delta_\star) &= \frac{3}{49}\cdot 7 + \frac{39}{49}\cdot 1 + \frac{7}{49}\cdot 3 = \frac{81}{49} \\ R(\theta_2, \delta_\star) &= \frac{39}{49} + \frac{7}{49}\cdot 6 = \frac{81}{49} \\ R(\theta_3, \delta_\star) &= \frac{3}{49} + \frac{39}{49}\cdot 2 = \frac{81}{49} \end{aligned}
$$

Since risk is the same for all states, \(\delta_\star\) is also an equalizer rule, hence Minimax.

b) Game-Theoretic Interpretation

We can interpret this decision problem as a two-player zero-sum game:

  • Player 1 (Doctor) chooses a strategy \(\delta\) from \(\mathcal{E}\) to minimize loss.
  • Player 2 (Nature) chooses a state \(\theta \in \{\theta_1, \theta_2, \theta_3\}\) to maximize loss.

The game value is:

$$
\bar{V} = \inf_{\delta \in \mathcal{E}} \sup_{\theta} R(\theta, \delta) = \sup_{\pi} \inf_{\delta} B(\pi, \delta) = V = \frac{81}{49}
$$

The randomized strategy \(\delta_\star\) guarantees the doctor a maximum expected loss of \(\frac{81}{49}\), even under the worst-case state. Conversely, \(\pi_\star\) guarantees Nature at least this value regardless of the doctor’s choice.

Hence, \((\delta_\star, \pi_\star)\) is a saddle point of the game.

c) Why Randomized Rules Matter

Suppose the doctor uses a non-randomized Minimax rule, say \(d_2\). The risk is:

$$
R(\theta_3, d_2) = 2 > \frac{81}{49}
$$

This gives Nature an opportunity to exploit the strategy by choosing \(\theta_3\). The randomized rule spreads the risk and avoids such exploitation.

Key Insight

  • Randomized rules help hedge against adversarial conditions.

  • Equalizer property guarantees robust performance in worst-case scenarios.

  • Game theory provides intuitive understanding of optimality and robustness.

Conclusion

This application bridges abstract theory and practical decision-making under uncertainty. Concepts like Bayes risk, Minimax rules, and game theory illustrate how statistical decisions can be optimized in real-world scenarios such as medicine.