Detection Error Probability (DEP)

• Detection Error Probability (DEP) is the probability of making an incorrect decision in testing, decoding, or detection systems, combining Type I and Type II errors.
• DEP underpins the design of distributed sensor networks and coding schemes by providing explicit bounds and guiding the trade-offs between detection accuracy and system complexity.
• DEP is critical in covert communication and quantum detection, where it helps optimize system performance under noise, uncertainty, and resource constraints.

Detection Error Probability (DEP) quantifies the likelihood of making an erroneous decision in hypothesis testing, decoding, or error detection protocols, with notable implications for distributed detection, communication systems, coding theory, and circuit fault analysis. DEP underpins the reliability of inference and communication processes, often serving as the figure of merit in rigorous system design and analysis.

1. Formal Definitions and Theoretical Foundations

Detection Error Probability is defined as the probability that a decision-making system (hypothesis test, decoder, or detector) produces an incorrect outcome, conditional on the underlying statistical model and system architecture. In binary hypothesis testing, DEP aggregates false alarm (Type I) and miss detection (Type II) probabilities, commonly as $$P_e = \pi_0\,P_{\mathrm{FA}} + \pi_1\,P_{\mathrm{MD}}$$, where $\pi_0$ and $\pi_1$ are hypothesis priors (Nandi et al., 2012, Bai et al., 2024, Bash et al., 2014). In decoding applications, DEP is identified as the undetected error probability, corresponding to the scenario where the decoder outputs an erroneous message without triggering a detection alarm or erasure (Sauter et al., 4 Mar 2025, Andrews et al., 2012).

In distributed detection architectures—such as relay trees—DEP is analyzed through recursion relations of local error probabilities, leading to explicit bounds on the global error rate at the fusion center (Zhang et al., 2011, Zhang et al., 2012, Zhang et al., 2011). Coding-theoretic treatments relate DEP to minimum distance and codeword enumerators for bounded-distance or rank-metric decoders [0610148], (0812.2379, Andrews et al., 2012, 0901.1762).

2. DEP in Distributed Detection and Fusion Architectures

Balanced binary and $M$-ary relay trees exemplify hierarchical fusion models, where DEP captures the probability of an incorrect global decision after successive local fusions. For balanced binary relay trees with $N$ identical sensors and leaves, the aggregation rule is likelihood-ratio fusion, and DEP at the fusion center is given by $P_e^{(H)} = \frac{1}{2}(\alpha_H + \beta_H)$, where $(\alpha_H,\beta_H)$ are terminal Type I/II errors after $H = \log_2 N$ fusion levels (Zhang et al., 2011).

For binary relay trees, error propagation is characterized by the recursion: \begin{align*} \alpha_{k+1} &= \begin{cases} 1-(1-\alpha_k)^2,\quad \beta_{k+1} = \beta_k^2, &\text{if } \alpha_k \le \beta_k \ \alpha_k^2,\quad \beta_{k+1} = 1-(1-\beta_k)^2, &\text{if } \alpha_k > \beta_k \end{cases} \end{align*} Global DEP decays sub-exponentially: $$c_1\cdot 2^{-\sqrt{N}}\le P_e(N)\le c_2\cdot 2^{-\sqrt{N}}$$ for fixed sensor quality, with explicit rates derived even for “crummy” sensors (with failure or near-random local performance) (Zhang et al., 2011, Zhang et al., 2011). For $M$-ary relay trees employing majority fusion, DEP at the root decays as $\exp(-c N^{\log_M \lambda})$, where $\lambda$ is determined by the branching factor, and convergence accelerates with larger $M$ (Zhang et al., 2012).

3. DEP in Coding: Classical, Rank, and Constant-Dimension Codes

DEP is intrinsic to code design for reliable transmission. In memoryless channels under bounded-distance decoding, DEP is the probability $P_u$ that an incorrect codeword is accepted as valid, with explicit integral bounds derived using the union bound and geometric analysis of high-dimensional noise spheres (Andrews et al., 2012). The undetected error probability can be tightly bounded in terms of code parameters: $$P_u(r_d) \le \sum_{w=d_{\min}}^{n} \frac{A_w}{2} \int_{\sqrt{w}}^{r_d} p_0(r) I_{1-w/r^2}\left(\tfrac{n-1}{2}, \tfrac{1}{2}\right) dr$$ where $A_w$ is the weight enumerator, $r_d$ the decoding radius, and $I$ the incomplete beta function.

For rank-metric codes, notably Maximum Rank Distance (MRD) codes, DEP under uniform-rank error models decays exponentially with the square of the decoding radius: $\exp(-\Omega(t^2))$ for $t = \lfloor (d-1)/2 \rfloor$ [0610148], (0812.2379). MRD codes maximize DEP among all codes of the same parameters, up to a scalar factor $H_q$, and explicit closed-form bounds and formulas are available. For constant-dimension codes utilized in network coding, DEP is analyzed over Grassmannian spaces, yielding precise subspace- and injection-metric bounds as functions of code distance distributions (0812.2379).

LT codes over the erasure channel admit a closed-form estimate of DEP via the Kovalenko rank distribution, giving the probability that the decoding matrix is rank-deficient and enabling optimization of code overhead and robustness (0901.1762).

4. DEP in Covert Communication and Quantum Detection

DEP serves as the central metric for information-theoretic covertness, quantifying the warden’s (adversary’s) ability to reliably distinguish between the null and transmission hypotheses. In covert communication protocols, the theoretical framework expresses DEP through the average of false alarm and missed detection probabilities, with explicit dependence on channel state information (CSI) estimation error, feedback delay, transmit power, and noise variance (Bai et al., 2024, Bash et al., 2014): $$\xi = P(H_0) P_{\mathrm{FA}} + P(H_1) P_{\mathrm{MD}}$$ In quantum channels, the minimum achievable DEP at the warden’s detector is lower-bounded by $P_e^{(w)} \ge \frac{1}{2} - \epsilon$, where $\epsilon$ is governed by trace distance (quantum relative entropy) between signal and noise hypotheses (Bash et al., 2014). The square-root law asserts that covert capacity scales as $\mathcal{O}(\sqrt{n})$ bits over $n$ channel uses while maintaining DEP arbitrarily close to $1/2$, provided there is any nonzero excess noise.

5. Decision-Theoretic, Stochastic, and Learning-Based Perspectives

DEP in decision-theoretic contexts encompasses both classic hypothesis tests (Neyman–Pearson, Bayesian) and stochastic decoding rules. Under stochastic decision-making, single-sample stochastic decisions have DEP at most twice that of deterministic maximum posterior (MAP) decisions: $P_e^\text{stoch} \le 2 P_e^\text{MAP}$ Moreover, multi-sample decision protocols exponentially drive DEP toward the MAP optimum as sample size grows (Muramatsu et al., 2017). For distributed detection with quantized sensor outputs, the MAPDEP is bounded below by $\sum_u \min \{ \pi_0 p(u|H_0), \pi_1 p(u|H_1) \}$, with the minimum achieved by identical quantizers optimizing Chernoff information across the network (Guo et al., 2024). Model-driven deep learning architectures leverage this structure, minimizing empirical MAPDEP via neural training, resulting in near-optimal performance confirmed through numerical analysis.

6. DEP in Reversible Circuits and Fault Detection

DEP for reversible circuits is solely a function of the error’s support size $k$, not the global circuit size. For a single reversible error affecting $k$ wires, the detection probability with random simulation input is at least $2^{-(k-1)}$, facilitating highly efficient error screening (Burgholzer et al., 2020). For wireless sensor network fault detection, DEP is incorporated in Bayesian and Neyman–Pearson event detection tests with explicit formulas for sensor error rates, prior probabilities, and communication faults (Nandi et al., 2012).

7. Finite Blocklength Regimes, Bounds, and Practical Implications

In short blocklength coding, DEP (undetected error probability) and total block error probability are characterized via new finite-blocklength bounds—outer-code (CRC-style) and threshold-based (e.g., Forney’s rule) (Sauter et al., 4 Mar 2025). Analytical and simulation results reveal that threshold-based detection is superior in matched CSIs and short blocks, while CRC-based approaches are more robust under CSI uncertainty. These results enable precise trade-off analysis for URLLC and other reliability-constrained short-packet applications.

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In summary, Detection Error Probability is central to the analysis and design of inference, coding, and detection systems across a wide range of architectures—distributed sensor networks, communication channels (classical, quantum, and covert), error-correcting codes, and circuit-level fault detection. DEP bounds and asymptotics dictate system reliability, inform architectural choices (e.g., branching factor in relay trees, quantizer uniformity in distributed detection, code design for desired undetected error rates), and motivate the development of physically realizable, theoretically grounded detection and error-control schemes (Zhang et al., 2011, Zhang et al., 2012, Zhang et al., 2011, Andrews et al., 2012), [0610148], (0812.2379, 0901.1762, Bai et al., 2024, Bash et al., 2014, Muramatsu et al., 2017, Guo et al., 2024, Sauter et al., 4 Mar 2025, Burgholzer et al., 2020, Nandi et al., 2012).