Associations, chance coincidences and backgrounds

Closeness becomes evidence through a comparison

Two events near each other in time or reconstructed sky position invite an association hypothesis: perhaps they came from one physical source. They also invite a competing explanation: unrelated events can sometimes look close. A meaningful comparison must describe what “close” means and how often the background could satisfy that same definition.

Specify the event selections, energy and direction conditions, time windows, instrument exposure and source or catalogue search. A rarer selected background can make a match more informative, but a selection chosen after inspecting the most attractive pair needs its own accounting.

The background need not consist only of detector malfunctions. Atmospheric particles, unrelated astronomical transients and an otherwise real astrophysical event can all be background to a particular common-source test. Background is defined relative to the question being asked.

The following invented search is deliberately simple enough to calculate exactly. It is not an empirical alert rate, a real event significance or a substitute for an observatory’s full association analysis.

Count the union of the searched windows

Condition on 20 fixed event centres from instrument A, at times 10, 20, 30, …, 200 s in a declared common clock. Before inspecting instrument B, choose the interval from 2 s before to 2 s after each A centre. Use half-open intervals so endpoint conventions are unambiguous; isolated endpoints do not affect the continuous-time probability.

Each interval has duration 4 s and the intervals do not overlap. Their union therefore has duration T = 20 × 4 = 80 s. We ask whether at least one B background event falls anywhere in that whole union. We are not asking only about whichever one window later contains the closest event.

Stipulate that unrelated B arrivals form an independent homogeneous Poisson process of rate r = 0.005 s⁻¹, with the stated observing coverage over all windows. Homogeneity and independent increments are model assumptions, not consequences of knowing a long-term average rate.

For the union of these disjoint fixed windows, the count K is Poisson with mean λ = rT = 0.4. Thus P(K = k) = e^(−λ)λ^k/k! for nonnegative integers k. The parameter λ is dimensionless expected count; it is not itself a probability.

The zero-count complement gives the requested probability

The probability of no B arrivals in the full union is P(K = 0) = e^(−0.4) ≈ 0.670320046. Therefore P(K ≥ 1) = 1 − e^(−0.4) ≈ 0.329679954. A match somewhere in this supplied 20-window search would not be especially rare under this null model.

Now define a separate optional spatial selection before examining coincidences. Stipulate that each B background event independently passes with probability f = 0.1, with that probability the same throughout these windows and independent of its arrival time. Poisson thinning gives selected mean λ_sel = frT = 0.04.

For that fully specified selected search, P(K_sel = 0) = e^(−0.04) ≈ 0.960789439 and P(K_sel ≥ 1) = 1 − e^(−0.04) ≈ 0.039210561. The selection acts on the background process and its mean. Simply multiplying the original at-least-one probability by f would not give the exact answer.

The assumed 0.1 acceptance is not a real-sky probability map. Exposure can vary with direction and time; localisation uncertainties and source catalogues can be nonuniform and incomplete. In such cases the background and selection model must reflect those conditions rather than multiplying by an unexplained angular-area fraction.

The interesting window is part of a larger search

The calculation already includes all 20 supplied windows through their union. It would be incorrect to treat that result as if it described one window, or to apply an additional factor of 20 for those same windows. Every trials correction must refer to the actual statistic and search it adjusts.

If searched windows overlap, the same B event can lie in several of them. For an at-least-one-event question, use the duration of the union rather than summing duplicated exposure. A different question, such as the number of matched pairs, can count that event more than once and needs a different statistic.

Searching several window widths, source catalogues, energy cuts or physical models gives further opportunities to find a striking result. A narrow window chosen around a pair after seeing it does not have the same interpretation as that identical window selected before looking. Report which choices were planned and which were exploratory.

The 2018 IceCube and multi-observatory study of the neutrino event IceCube-170922A and a flaring blazar provides a concrete methodological example. Its association tests used source and detector weights, alternative emission connections and a correction for the alerts examined. We use that dated example to illustrate search dependence, without importing its p-values, event census or any current observatory performance.

A null probability is not the probability that the source is false

An expected background count describes a mean number under a model. A false-alarm probability describes the probability of a statistic at least as extreme under a specified null search. A false-alarm rate additionally has units such as events per observing year and needs a defined exposure and search procedure.

The posterior probability of a physical association asks a different question. It needs alternatives, their prior probabilities and the likelihood of the observed evidence under them. The supplied P(K ≥ 1) is a null-tail probability for this count criterion, not P(unrelated | observed data) and not one minus the probability of a shared source.

A very small null probability also does not identify a unique emission mechanism. A background model can be inadequate, calibration can be wrong or several source interpretations can remain plausible. Preserve the event statistic and its assumptions alongside the physical conclusions.

The GW170817 gamma-ray association study likewise compares specified chance and physical-association reasoning rather than using proximity alone. Its time, direction and source assumptions are part of that analysis. They should not be erased when its result is used as an example of multi-messenger inference.

Different instruments can still share assumptions

Two detectors can have different hardware while their analyses use the same host catalogue, a shared clock reference, related selection cuts or a common source-emission assumption. Treating two such constraints as independent can count the same information twice.

Mathematically, multiplying independent evidence terms is justified only within the stated joint or conditional model. The covariance Foundation describes shared variation in a joint uncertainty budget, while the conditional-probability Foundation reminds us that independence is a model property. A zero covariance alone is not a general proof of independence.

For a real candidate, retain a readable association record: which events and revisions were compared, what windows and sky information were used, how background was estimated, which trials were included and which alternative interpretations remain. This allows a later reader to assess the claim rather than inherit only a significance label.

The next lesson studies what combining messengers can constrain about physical parameters. It will make the missing assumptions visible by exhibiting two different emission-and-propagation explanations that fit the same arrival delays.

Worked examples

Try the reasoning

1. What are the units of r, T and λ = rT in the supplied model?

   Show the answer and reasoning

   r is in s⁻¹, T in s, and λ is a dimensionless expected count.

   The mean count and the probability of at least one event are different quantities.

2. For an at-least-one background-event test, should overlapping windows be counted separately in the exposure total?

   Show the answer and reasoning

   Use the duration of their union.

   Adding every interval duration can count the same exposure more than once; a matched-pair statistic would be a different question.

3. Does selecting a very narrow window around a pair after inspecting the data justify the same search interpretation as preplanning it?

   Show the answer and reasoning

   No.

   The exploratory choices and opportunities for a striking match belong in the analysis of the full search.

4. Is the calculated null-tail probability the posterior probability that a candidate has no shared source?

   Show the answer and reasoning

   No.

   A posterior association probability additionally requires competing models and prior information; the null probability answers a different conditional question.

What to carry forward

• Define the background and the entire searched selection before interpreting a coincidence.
• For the supplied Poisson model, use union exposure and the exact zero-count complement.
• Search trials, shared assumptions and physical alternatives remain part of a defensible association claim.