Student Notes: Detection Theory Examples
These notes rewrite the April 17 student notes into a clean reference for the main detection-theory examples.
Hypotheses
A detection problem starts with two competing descriptions of the world:
\[H_0: \text{regular or null state}, \qquad H_1: \text{alternative state}.\]Given data $X_1,\ldots,X_n$, the goal is to decide which hypothesis better explains the observations.
A decision rule is a function
\[\delta(x_1,\ldots,x_n)\in\{H_0,H_1\}.\]Once a rule is chosen, its performance is judged under each possible truth:
\[P_{\mathrm{FA}}=P(\delta(X)=H_1\mid H_0), \qquad P_M=P(\delta(X)=H_0\mid H_1),\]and
\[P_D=1-P_M=P(\delta(X)=H_1\mid H_1).\]Aircraft Detection Example
A simple radar model can be written as:
\[H_0: \text{no aircraft}, \qquad H_1: \text{aircraft present}.\]If aircraft presence changes the mean amplitude of the received signal, a mean-based detector is natural.
Mean Change Example
Suppose the observed signal is
\[X_i = \epsilon_i, \qquad \epsilon_i\sim N(0,\sigma^2)\]under $H_0$, and
\[X_i = \mu+\epsilon_i\]under $H_1$.
The sample mean
\[\bar{X}=\frac{1}{n}\sum_{i=1}^n X_i\]is informative because it shifts under $H_1$.
Specifically,
\[\bar{X}\mid H_0\sim N\left(0,\frac{\sigma^2}{n}\right), \qquad \bar{X}\mid H_1\sim N\left(\mu,\frac{\sigma^2}{n}\right).\]The two distributions have the same spread but different centers. A threshold test therefore compares $\bar{X}$ with a cutoff.
Likelihood Ratio
The likelihood ratio is
\[\Lambda(x) = \frac{p(x\mid H_1)}{p(x\mid H_0)}.\]If $\Lambda(x)$ is large, the data are more likely under $H_1$ than under $H_0$.
The Neyman-Pearson test compares this ratio to a threshold:
\[\Lambda(x)>\eta.\]Taking logarithms is usually easier:
\[\log\Lambda(x)>\log\eta.\]Because the logarithm is monotone, this does not change the decision rule. It only turns products of densities into sums of log densities.
Threshold Selection
The threshold should be chosen to control false alarms:
\[P_{\mathrm{FA}} =P(\text{decide }H_1\mid H_0).\]For Gaussian mean-shift problems, this threshold can be translated into a threshold on $\bar{X}$ using the normal CDF.
For the one-sample example
\[H_0:X\sim N(0,1), \qquad H_1:X\sim N(1,1),\]the likelihood ratio is
\[\Lambda(x) = \frac{\exp[-(x-1)^2/2]}{\exp[-x^2/2]} = \exp\left(x-\frac{1}{2}\right).\]Thus $\Lambda(x)>\eta$ is equivalent to
\[x>\log\eta+\frac{1}{2}.\]The threshold on $x$ is then chosen to make the false alarm probability equal to the desired value.
For example, if the target false alarm probability is $\alpha$, then
\[P_{\mathrm{FA}} = P(X>\gamma\mid H_0) = P(Z>\gamma) = Q(\gamma) = \alpha,\]so
\[\gamma=Q^{-1}(\alpha).\]The detection probability is
\[P_D = P(X>\gamma\mid H_1) = P(X>\gamma\mid X\sim N(1,1)) = Q(\gamma-1).\]For $n$ samples with known positive shift $\mu$, the same calculation gives
\[\gamma = \frac{\sigma}{\sqrt{n}}Q^{-1}(\alpha)\]for a threshold on $\bar{X}$, and
\[P_D = Q\left(Q^{-1}(\alpha)-\frac{\mu\sqrt{n}}{\sigma}\right).\]Variance Change Example
If the hypotheses differ in variance rather than mean, a natural statistic is the energy:
\[Y=\sum_{i=1}^n X_i^2.\]Under a Gaussian null model,
\[\frac{Y}{\sigma_0^2}\sim \chi_n^2.\]This makes chi-squared quantiles useful for setting thresholds.
For $n$ samples, the false alarm equation has the form
\[P\left(\chi_n^2>\frac{\gamma}{\sigma_0^2}\right)=\alpha\]when the rejection region is $Y>\gamma$. Solving this equation gives the detector threshold.
The inequality direction depends on the alternative. If $H_1$ has larger variance than $H_0$, large energy is suspicious. If $H_1$ has smaller variance than $H_0$, unusually small energy is suspicious. The likelihood ratio determines which tail to use.
For the larger-variance case, after choosing $\gamma$ from the null distribution,
\[P_D = P(Y>\gamma\mid H_1) = P\left(\chi_n^2>\frac{\gamma}{\sigma_1^2}\right).\]This expression makes the role of separation clear: the same fixed threshold $\gamma$ is easier to exceed when $\sigma_1^2$ is larger.
Change-Point Interpretation
Detection theory can also be used for change-point problems. Here the question is whether the data continue to follow a baseline distribution or whether the distribution has changed.
The hypotheses might be:
\[H_0: X_i \sim p_0, \qquad H_1: X_i \sim p_1.\]The likelihood ratio compares how well the two distributions explain the observed sequence.
For independent observations with a known change distribution, the likelihood ratio is
\[\Lambda(x_1,\ldots,x_n) = \prod_{i=1}^n \frac{p_1(x_i)}{p_0(x_i)}.\]Taking logs gives the cumulative evidence statistic
\[\log\Lambda(x_1,\ldots,x_n) = \sum_{i=1}^n \log\frac{p_1(x_i)}{p_0(x_i)}.\]This sum is useful because each observation contributes an evidence increment. In online detection, one can update the statistic as new data arrive instead of recomputing the full likelihood ratio from scratch.
Quick Detector Checklist
- Write $p(x\mid H_0)$ and $p(x\mid H_1)$.
- Form the likelihood ratio $\Lambda(x)=p(x\mid H_1)/p(x\mid H_0)$.
- Simplify the ratio to a statistic such as $\bar{X}$ or $\sum_i X_i^2$.
- Determine which direction favors $H_1$.
- Choose the threshold from the null distribution to control $P_{\mathrm{FA}}$.
- Compute $P_D$ from the alternative distribution.
Student Takeaways
- A detector converts data into a decision.
- The likelihood ratio measures relative evidence for $H_1$ versus $H_0$.
- Thresholds control the tradeoff between missed detections and false alarms.
- Mean changes lead to average-based detectors; variance changes lead to energy-based detectors.
- Always set thresholds using the null distribution when the goal is to control false alarms.