Bayesian Changepoint Detection

The core mechanism tracks the run length—the time elapsed since the last changepoint—as a latent variable. At each time step, the algorithm computes a recursive posterior distribution over possible run lengths using Bayes' theorem. This creates a change point probability that spikes when the predictive likelihood of new data under the current model drops sharply, indicating a structural break. The formulation elegantly handles online, real-time detection without requiring a fixed window size.

Unlike frequentist methods that yield a single point estimate, Bayesian changepoint detection produces a complete posterior distribution over changepoint locations. This allows practitioners to set credible intervals on when a change occurred and to compute the marginal probability of a change at each time step. For mission-critical RF applications, this uncertainty quantification enables risk-adjusted decision-making—a spectrum regulator can choose to act only when the posterior probability exceeds 0.99, minimizing false alarms.

The framework accommodates arbitrary generative models for the data within each segment. Common choices include:

• Gaussian with changing mean: Detects shifts in signal power or DC offset
• Gaussian with changing variance: Identifies onset of noise-like jamming
• Poisson models: For discrete event streams like pulse trains
• Linear regression models: Detects changes in trend or slope This flexibility allows the same algorithmic core to address diverse RF detection tasks by simply swapping the observation model.

The hazard rate—the prior probability of a changepoint at any given time step—encodes domain knowledge about expected event frequency. A constant hazard rate assumes changepoints occur at a fixed average interval, suitable for random burst transmissions. A time-varying hazard function can incorporate external context, such as known transmission schedules or geolocation-based threat levels. This explicit prior specification makes the detector's assumptions transparent and auditable.

The algorithm operates in constant time per observation using a recursive message-passing scheme. At each step t, it computes the posterior over run lengths using only the previous step's posterior and the current observation's predictive likelihood. This O(n) complexity makes it suitable for real-time streaming RF applications on edge hardware. The recursive structure also enables pruning of low-probability run lengths to maintain a bounded computational footprint over arbitrarily long sequences.

While the online algorithm provides real-time detection, a backward smoothing pass can refine changepoint estimates after observing future data. This forward-backward algorithm computes the marginal posterior probability that a changepoint occurred at each historical time step, conditioned on the entire sequence. For forensic spectrum analysis, this yields the maximum a posteriori (MAP) segmentation—the single most probable partition of the signal into homogeneous segments.

A fundamental sequential hypothesis testing framework that Bayesian changepoint detection generalizes. SPRT continuously evaluates the log-likelihood ratio of incoming observations against two competing hypotheses, making a decision as soon as sufficient evidence accumulates rather than waiting for a fixed sample size.

• Key advantage: Minimizes average sample number (ASN) for a given Type I and Type II error constraint
• Connection to changepoint detection: The CUSUM algorithm is a repeated SPRT, restarting whenever a decision boundary is crossed
• Bayesian extension: Replaces fixed thresholds with prior probabilities and cost functions for optimal decision-making under uncertainty

A non-Bayesian changepoint detection algorithm that accumulates the log-likelihood ratio of observations relative to a pre-change distribution. When the cumulative sum exceeds a predefined threshold, a changepoint is declared.

• Core mechanism: Maintains a running statistic Sₖ = max(0, Sₖ₋₁ + log(p₁(xₖ)/p₀(xₖ)))
• Burst transmission detection: Rapidly identifies the onset of a sudden signal in previously vacant spectrum
• Limitation: Requires known pre-change and post-change distributions, unlike Bayesian methods that can maintain uncertainty over parameters
• Typical use case: Detecting the start of a radar pulse or frequency-hopping transmission in electronic warfare receivers

The canonical Bayesian framework for real-time changepoint inference, introduced by Adams and MacKay (2007). BOCPD maintains a posterior distribution over the run length—the time since the most recent changepoint—and updates it recursively with each new observation.

• Recursive message-passing: Computes P(rₖ|x₁:ₖ) using the product of the predictive probability and the changepoint prior hazard function
• Key output: Produces both a point estimate of changepoint locations and a full posterior over run lengths, quantifying detection uncertainty
• Underlying predictive model (UPM): Can use any conjugate exponential family model (Gaussian, Poisson, etc.) for efficient closed-form updates
• Spectrum application: Detects the abrupt appearance of a burst transmission while simultaneously estimating its power spectral density parameters

The hazard function h(r) defines the prior probability that a changepoint occurs at run length r, given that no changepoint has occurred before. It is the critical design parameter that encodes domain knowledge about expected event frequencies.

• Constant hazard rate: Assumes memoryless, geometrically distributed segment lengths—appropriate when burst arrivals follow a Poisson process
• Periodic hazard functions: Encode expected transmission schedules or time-slotted protocol behavior
• Nonparametric priors: Dirichlet process or gamma process priors allow the data to infer the hazard structure when transmission patterns are unknown
• Impact on detection latency: Higher hazard rates produce faster detection but increase false alarm probability

The mathematical engine enabling efficient closed-form Bayesian updates in changepoint detection. By selecting a conjugate prior for the observation model, the posterior distribution remains in the same parametric family, allowing recursive computation without numerical integration.

• Normal-inverse-gamma prior: For Gaussian observations with unknown mean and variance—models thermal noise floor with sudden signal power changes
• Beta-binomial conjugacy: For detecting changes in discrete spectrum occupancy states (occupied/vacant)
• Dirichlet-multinomial: For changepoints in categorical modulation type sequences
• Computational benefit: O(1) update per observation versus O(t) for particle filter methods, critical for real-time wideband spectrum monitoring

A critical distinction in spectrum monitoring architectures. Changepoint detection identifies persistent regime shifts in the statistical properties of a signal stream, while anomaly detection flags transient outliers that deviate from a baseline but do not represent a sustained change.

• Changepoint example: A cognitive radio user begins transmitting, permanently altering the noise floor statistics in a band
• Anomaly example: A single lightning strike causing a brief wideband impulse that immediately returns to baseline
• Architectural integration: Changepoint detectors trigger spectrum reallocation; anomaly detectors trigger interference logging and forensic analysis
• Combined approach: Bayesian changepoint models with heavy-tailed observation likelihoods (Student-t) can simultaneously handle both persistent changes and outlier robustness