Criteria for Multiple Sample Detection of Binary Hypotheses

Each decision criterion—Bayes, MAP, Minimax, and Neyman-Pearson—can be effectively extended to handle multiple measurements by adapting their respective decision rules and thresholds to accommodate the higher-dimensional data involved.

Bayes Criterion

The Bayes criterion for handling multiple measurements extends naturally from its single-measurement counterpart.

Definition of Bayes Risk:

The Bayes risk is expressed as:

\[ \boxed{ r = [P_{00}C_{00} + P_{10}C_{10}]\pi_0 + [P_{01}C_{01} + P_{11}C_{11}]\pi_1 } \]

Here:

• \( P_{11} = 1 - P_{01} \)

• \( P_{10} = 1 - P_{00} \)

The probabilities \( P_{00} \) and \( P_{01} \) are defined by the integrals:

\[ P_{00} = \int_{R_0} p_0(\vec{y}) d\vec{y} \]

\[ P_{01} = \int_{R_0} p_1(\vec{y}) d\vec{y} \]

These integrals are similar to those in the single-measurement case but are extended to a \( k \)-dimensional vector \( \vec{y} \).

Simplified Bayes Risk:

The Bayes risk can be rewritten as:

\[ r = \pi_0 C_{01} + \pi_1 C_{11} + \int_{R_0} \left[ \pi_1 (C_{01} - C_{11}) p_1(\vec{y}) - \pi_0 (C_{10} - C_{00}) p_0(\vec{y}) \right] d\vec{y} \]

To minimize \( r \), we seek the volume \( R_0 \) in \( k \)-space that achieves this minimum.

Decision Threshold

\[\boxed{ \eta_B = \frac{\pi_0 (C_{10} - C_{00})}{\pi_1 (C_{01} - C_{11})} } \]

Decision Rule:

Choose \( R_0 \) such that:

\[ \pi_1 (C_{01} - C_{11}) p_1(\vec{y}) - \pi_0 (C_{10} - C_{00}) p_0(\vec{y}) < 0 \]

This leads to a likelihood ratio decision rule:

\[ L(\vec{y}) = \frac{p_1(\vec{y})}{p_0(\vec{y})} < \eta_B \]

In summary, the decision rule based on the combined notation is:

\[\begin{split} L(\vec{y}) \begin{cases} \ge \eta_B & \text{choose } D_1 \\ < \eta_B & \text{choose } D_0 \end{cases} \end{split}\]

MAP Criterion

When the costs associated with decisions are unknown, the Maximum A Posteriori (MAP) criterion is applied.

Equivalence to Cost Setting:

The MAP criterion is equivalent to assuming that the cost differences satisfy \( C_{10} - C_{00} = C_{01} - C_{11} \).

Threshold and Decision Rule:

Under MAP, the threshold is determined by:

\[ \boxed{ \eta_{MAP} = \frac{\pi_0}{\pi_1} } \]

The corresponding decision rule is:

\[\begin{split} \delta_{MAP}(\vec{y}) = \begin{cases} 1, & L(\vec{y}) \ge \eta_{MAP} \\ 0, & L(\vec{y}) < \eta_{MAP} \end{cases} \end{split}\]

This means that if the likelihood ratio \( L(\vec{y}) \) exceeds \( \eta_{MAP} \), decision \( D_1 \) is made; otherwise, \( D_0 \) is chosen.

Minimax Criterion

The minimax approach for multiple measurements mirrors the single-measurement scenario.

Extension to Multiple Measurements:

All principles applicable to single measurements in detection theory are valid for multiple measurements. The key difference is that the single-measurement likelihood ratio \( L(y) \) is replaced by the multiple-measurement ratio \( L(\vec{y}) \).

Minimax Decision Rule:

The minimax decision rule, which minimizes the maximum possible risk, is formulated as:

\[\begin{split} \delta_{mm}(\vec{y}) = \begin{cases} 1, & L(\vec{y}) > \eta_{mm} \\ \eta, & L(\vec{y}) = \eta_{mm} \\ 0, & L(\vec{y}) < \eta_{mm} \end{cases} \end{split}\]

Typically, a randomized decision is employed to determine the exact value of \( \eta \). The threshold \( \eta_{mm} \) is established using methodologies from single-measurement detection theory.

Neyman-Pearson Criterion

Similar to the minimax approach, the Neyman-Pearson criterion for multiple measurements builds directly on the single-measurement framework, ensuring that the false-alarm rate remains below a specified level \( \alpha_f \).

Adaptation to Multiple Measurements:

By replacing the single measurement \( y \) with the measurement vector \( \vec{y} \) and extending the integrals to \( k \)-dimensional spaces, the Neyman-Pearson derivation seamlessly applies to multiple measurements.

Neyman-Pearson Decision Rule:

The decision rule under the Neyman-Pearson criterion, which may involve randomized tests, is given by:

\[\begin{split} \delta_{NP}(\vec{y}) = \begin{cases} 1, & L(\vec{y}) > \eta_{NP} \\ \eta, & L(\vec{y}) = \eta_{NP} \\ 0, & L(\vec{y}) < \eta_{NP} \end{cases} \end{split}\]

This rule ensures that the probability of a false alarm does not exceed the predefined threshold \( \alpha_f \).