Bayesian detector

Test Your Knowledge

Quiz on Bayesian Detector

Instructions: Choose the best answer for each question.

1. What is the primary advantage of a Bayesian detector over a traditional threshold-based detector?

(a) It can be implemented with simpler hardware. (b) It is less computationally expensive. (c) It utilizes prior information about the signal and noise. (d) It is more resistant to noise.

2. What does the Bayesian detector calculate to make a decision?

(a) The likelihood of the signal being present. (b) The likelihood of the noise being present. (c) The posterior probability of the signal being present. (d) The prior probability of the signal being present.

3. What is the goal of a Bayesian detector in terms of error probabilities?

(a) Minimizing only the false alarm probability. (b) Minimizing only the miss probability. (c) Minimizing the sum of false alarm and miss probabilities. (d) Minimizing the average of false alarm and miss probabilities weighted by prior probabilities.

4. What is NOT an advantage of a Bayesian detector?

(a) Optimal decision-making. (b) Incorporation of prior information. (c) Simplicity of implementation. (d) Adaptability to changing conditions.

5. Which application is NOT typically suitable for a Bayesian detector?

(a) Radar systems. (b) Sonar systems. (c) Communication systems. (d) Image processing.

Exercise on Bayesian Detector

Scenario: A communication system transmits a binary signal (0 or 1) over a noisy channel. The signal is received as a voltage value (x). The prior probabilities of transmitting 0 and 1 are P(H0) = 0.6 and P(H1) = 0.4 respectively. The likelihood functions are:

• P(x|H0) = 0.5 * exp(-(x-1)^2/2) for signal 0
• P(x|H1) = 0.5 * exp(-(x-3)^2/2) for signal 1

Task:

1. Calculate the posterior probability of transmitting 1 (P(H1|x)) given a received voltage value x = 2.
2. Based on the posterior probability, would the Bayesian detector decide that signal 1 was transmitted? Use a threshold of 0.5.

Techniques

Chapter 1: Techniques of the Bayesian Detector

This chapter delves into the technical aspects of the Bayesian detector, exploring the fundamental principles and algorithms that underpin its operation.

1.1 Bayesian Framework and Bayes' Theorem

At the heart of the Bayesian detector lies Bayes' theorem, a cornerstone of probability theory. This theorem provides a framework for updating our belief about an event (signal presence in our case) based on new evidence (observed data). Bayes' theorem states:

P(H1|x) = [P(x|H1) * P(H1)] / [P(x|H1) * P(H1) + P(x|H0) * P(H0)]

where:

• P(H1|x) is the posterior probability of the signal being present (H1) given the observed data (x).
• P(x|H1) is the likelihood of observing the data (x) given the signal is present (H1).
• P(H1) is the prior probability of the signal being present.
• P(x|H0) is the likelihood of observing the data (x) given the signal is absent (H0).
• P(H0) is the prior probability of the signal being absent.

1.2 Likelihood Functions and Prior Distributions

The Bayesian detector relies heavily on two key components:

• Likelihood functions: These describe the probability of observing specific data under different hypotheses (signal present or absent). They are determined by the underlying signal and noise models.
• Prior distributions: These represent our prior knowledge about the signal and noise characteristics. They can be informed by domain expertise or previous observations.

1.3 Decision Rule and Thresholding

The Bayesian detector makes its decision based on the posterior probability calculated using Bayes' theorem. A threshold is set, and if the posterior probability exceeds this threshold, the detector concludes the signal is present. Otherwise, it declares the signal absent. The choice of the threshold can influence the trade-off between false alarms and missed detections.

1.4 Optimality of the Bayesian Detector

The Bayesian detector is considered optimal because it minimizes the average of false alarm and miss probabilities, weighted by the prior probabilities of each hypothesis. This optimality is based on minimizing the expected cost of making a wrong decision.

1.5 Types of Bayesian Detectors

Several variations of the Bayesian detector exist, each tailored to specific signal and noise characteristics. These include:

• Matched filter detector: Optimal for detecting a known signal in additive white Gaussian noise.
• Adaptive Bayesian detector: Adjusts its decision rule based on changing signal and noise conditions.
• Non-parametric Bayesian detector: Does not require explicit assumptions about the signal and noise distributions.

This chapter lays the foundation for understanding the techniques employed by Bayesian detectors, setting the stage for exploring specific models and applications in the following chapters.