Raftery-Akman (1986) Bayesian Analysis of a Poisson Process with a Change-Point

Summary

A short, influential paper giving a fully Bayesian treatment of a Poisson process with a single change-point at an unknown time: the event rate is $\lambda_1$ before the change and $\lambda_2$ after, and the change occurs at unknown $t$ . Raftery and Akman derive the joint posterior of $(\lambda_1,\lambda_2,t)$ under Gamma priors, and — the paper's methodological hook — construct a Bayes factor to test whether a change-point exists at all (constant-rate model $M_0$ vs. change-point model $M_1$ ), including a device for making the test well-defined under vague/improper priors. The classic coal-mining disasters series is the illustration, locating the change around 1890.

Key Claims

Model. On $[0,T]$ with $n$ events at times $t_1,\dots,t_n$ , the rate is $\lambda(s)=\lambda_1$ for $s\le t$ and $\lambda_2$ for $s>t$ ; conjugate Gamma priors on the rates give a closed-form posterior for $(\lambda_1,\lambda_2)$ given $t$ , and the marginal posterior of the change-point $t$ (a discrete-plus-continuous object over the inter-event intervals).
Magnitude of change. The posterior of $\beta=\lambda_2/\lambda_1$ (the ratio of rates) is obtained; it summarizes how big the change was, separately from when.
Bayes factor for existence of a change. Testing is framed as model comparison: $M_0$ (constant rate) vs. $M_1$ (one change-point), with the Bayes factor $B_{01}=p(t\mid M_0)/p(t\mid M_1)$ built from the two integrated likelihoods.
Improper-prior fix (imaginary observation). With vague priors the Bayes factor contains an arbitrary ratio of prior constants $c_0/c_1$ and is not well-defined. Raftery-Akman adopt the Spiegelhalter-Smith (1982) boundary-condition device: fix the constant so that the smallest data set maximally favouring $M_0$ (a single event halfway through $[0,T]$ ) yields a Bayes factor just above one. Under Jeffreys-type priors ( $b_k=-\tfrac12$ ) this pins down $c_{01}(T)$ and gives an operational, scale-invariant test.
Coal-mining example. On Jarrett's (1979) corrected British coal-mining disaster series (1851–1962), the change-point posterior has mode 10 March 1890, median 27 August 1890, 95% region $[$ 15 May 1887, 3 August 1895 $]$ ; the rate-ratio $\beta$ has posterior mean 3.41 (95% HPD $[2.48, 4.46]$ ) — a roughly 3.4-fold drop in disaster rate after 1890.

Concepts Introduced or Extended

Change-Point Analysis

Entities Mentioned

Quotes

"A Bayesian approach to estimation and hypothesis testing for a Poisson process with a change-point is developed, and an example given."

"The boundary condition is that a data set which involves the smallest possible sample size permitting a comparison of $M_0$ and $M_1$ and provides maximum possible support for $M_0$ should yield a Bayes factor which is only very slightly greater than one."

My Take

This is a small paper that punches above its length: it fixes the coal-mining change-point as the canonical worked example that Carlin-Gelfand-Smith (1992), Barry-Hartigan (1993), and Green's (1995) reversible-jump treatment all return to, and it gets a clean answer (a ~3.4× rate drop near 1890) from conjugate Gamma-Poisson algebra alone — no MCMC needed for a single change-point. Its genuinely reusable idea is the improper-prior Bayes factor problem and the Spiegelhalter-Smith imaginary-observation calibration: because a diffuse prior leaves an undetermined constant in the marginal likelihood, "does a change exist?" is ill-posed until you anchor that constant, and their boundary-condition trick is an early, concrete instance of the intrinsic/fractional-Bayes-factor ideas that later formalized model comparison under vague priors. For the wiki it is the single-change-point Bayesian precedent underneath the change-point concept, the fixed point the trans-dimensional methods generalize.