This was basically one question that kept unfolding for 40 minutes.
Frame the problem as a binary hypothesis test with Gaussian noise, derive the optimal likelihood ratio test, and show how to combine independent node observations via sufficient statistics. Then discuss threshold selection based on the false-positive rate and analyze power scaling with the number of nodes.
Pro tip: Emphasize that the optimal fusion rule is a weighted sum of node observations (matched filter), and that under equal noise variance, averaging is optimal. Also note that power increases with the square root of the number of nodes for a fixed false-positive rate, which is a key insight for system design.
Assume each node i observes x_i = s + n_i, where s is the true signal (zero if absent, A if present) and n_i ~ N(0, σ^2) is independent Gaussian noise. State the null hypothesis H0: s=0 vs. alternative H1: s=A>0.
Use the Neyman-Pearson lemma to derive the likelihood ratio test. Show that the optimal test statistic is the sum of observations (or average), which is a sufficient statistic. For known σ and A, the test reduces to comparing the sample mean to a threshold.
Since nodes are independent and identically distributed, the optimal fusion is to average the observations (or sum them). This maximizes the signal-to-noise ratio (SNR) of the combined statistic. If noise variances differ, use a weighted sum with weights inversely proportional to variance.
Define the test statistic T = (1/N) Σ x_i. Under H0, T ~ N(0, σ^2/N); under H1, T ~ N(A, σ^2/N). For a given false-positive rate α, set threshold τ = z_{1-α} σ/√N, where z_{1-α} is the (1-α) quantile of the standard normal.
Power = P(T > τ | H1) = 1 - Φ( z_{1-α} - A√N/σ ). As N increases, the argument increases, so power approaches 1. For fixed α, the required signal A scales as 1/√N to maintain a given power, meaning detection improves with more nodes.
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