Ties and Simultaneous Jumps in Likelihoods

Why the Meaning of a Tie Matters

Two event records can carry the same timestamp for quite different reasons. Their underlying events may have happened at different times that were rounded to the same minute, or the events may have occurred simultaneously as one shared process jump. These situations are not interchangeable in likelihood calculations. A no-common-jump likelihood treats distinct event types as separate jumps; a model allowing a shared jump needs a likelihood contribution for the joint event.

The distinction also matters for inference based on counting-process martingales. If two component processes never jump together, their martingale increments have zero cross predictable variation. A genuine common-jump mechanism contributes to that variation. A tie in a file of rounded timestamps, by itself, establishes neither conclusion.

Definition: Let \(N_j(t)\), for \(j=1,\ldots,d\), count events of type \(j\) by time \(t\). A genuine simultaneous jump of components \(j\) and \(k\) at time \(t\) means \(\Delta N_j(t)=\Delta N_k(t)=1\). A recorded tie means only that the recorded timestamps coincide; it does not establish that the underlying event times coincide.

Likelihood When There Are No Common Jumps

Suppose each component has a predictable intensity \(\lambda_j(t)\), so its compensator is \(\int_0^t\lambda_j(u)\,du\). Assume the intensities are locally integrable, the observation interval has no unmodelled fixed-time atoms, and the components have no common jumps: almost surely, at most one component jumps at any time. Conditional on the observed past, the likelihood for the event history over \([0,\tau]\), up to factors not depending on the model parameters, is

L(\tau)= \exp\left\{-\int_0^\tau\sum_{j=1}^d\lambda_j(u)\,du\right\} \prod_{j=1}^d\prod_{0<t\leq\tau:\,\Delta N_j(t)=1}\lambda_j(t).

Each event contributes its component’s intensity at its event time. The exponential term accounts for the integrated total intensity over the observation interval, including times with no events. If the intensity depends on a parameter, both parts generally contribute to the likelihood.

The no-common-jump condition is a statement about the underlying process, not the formatting of the data. In continuous time, separate event times can have identical rounded records even when exact simultaneous events have probability zero. In that case, the exact-time likelihood is not obtained by treating the shared timestamp as a common jump.

Worked Example: Distinct Events With the Same Recorded Minute

Suppose two event types follow independent Poisson processes with rates \(1.2\) and \(0.8\) events per hour. A recording system rounds times to the nearest ten minutes. In one ten-minute bin, the record shows one event of each type, both stamped with the same minute. Calculate the probability of exactly one event of each type in the bin under this model, with no other events of either type.

The bin width is \(\Delta=1/6\) hour. Under the independent-process model, the probability is

P(N_1(\Delta)=1,N_2(\Delta)=1) = e^{-(1.2+0.8)\Delta}(1.2\Delta)(0.8\Delta).

Substituting the rates and bin width gives

e^{-2/6}\left(\frac{1.2}{6}\right)\left(\frac{0.8}{6}\right) = e^{-0.3333}(0.2000)(0.1333) \approx 0.01911.

The calculation concerns a bin count, not two events at an identical exact time. Independent Poisson processes have no simultaneous jumps with probability one, yet their events can receive the same rounded timestamp. For exact event times, the separate event factors are \(1.2\) per hour and \(0.8\) per hour at their respective times, alongside the integrated-intensity factor.

When timestamps are coarsened, an appropriate likelihood for the recorded data uses probabilities of the observed bin counts or integrates over possible event times within each bin. It must account for the recording rule and the process model. Simply inserting a shared recorded time into the exact-time likelihood can misrepresent the data; imposing a no-common-jump model does not, by itself, recover unobserved event times.

Likelihoods That Allow Genuine Simultaneous Jumps

To represent common jumps explicitly, let \(z=(z_1,\ldots,z_d)\) denote a nonzero binary vector describing the components that jump together. For each possible mark \(z\), let \(\nu_z(t)\) be the predictable intensity of jumps with exactly that mark. Thus a jump with mark \(z\) increments \(N_j\) if \(z_j=1\), and otherwise leaves \(N_j\) unchanged. Assume these marked jump types have locally integrable intensities and no unmodelled fixed-time atoms. The likelihood is

L(\tau)= \exp\left\{-\int_0^\tau\sum_{z\ne 0}\nu_z(u)\,du\right\} \prod_{\substack{0<t\leq\tau\\\text{a jump with mark }z\text{ occurs}}} \nu_z(t).

The marginal intensity of component \(j\) is \(\lambda_j(t)=\sum_{z:z_j=1}\nu_z(t)\). These marginal intensities do not generally determine the joint jump rates. In particular, multiplying marginal intensities at a tied time does not substitute for specifying the intensity of the shared mark.

Worked Example: A Shared-Shock Likelihood Contribution

Consider two event components. Type 1 alone has jump intensity \(\alpha=0.7\) per hour, type 2 alone has intensity \(\beta=0.3\) per hour, and a shared jump of both components has intensity \(\gamma=0.5\) per hour. Assume these three marked event processes have constant intensities and no other jump types. The marginal component intensities are \(1.2\) and \(0.8\) per hour. If exactly one shared jump occurs at time \(0.4\) hours in a one-hour observation, with no other jumps, its likelihood density is

\gamma\exp\{-(\alpha+\beta+\gamma)\tau\} = 0.5e^{-1.5(1)} \approx 0.1116.

The event factor is the shared-mark intensity \(0.5\), not the product \(1.2\times0.8\) of marginal intensities. The survival factor includes all three possible jump types, so the total rate is \(0.7+0.3+0.5=1.5\) per hour. The contribution is a density for the jump time, with units per hour.

The order of the small-time probabilities makes the distinction clear. For a short interval of width \(h\), the probability of a shared jump is \(\gamma h+o(h)\). Under a model with only independent component jumps, the probability of at least one jump of each type in that interval is of order \(h^2\), not order \(h\).

Deriving the Likelihood and the Orthogonality Consequence

A short-interval argument explains both the likelihood factors and why common jumps affect covariance calculations. Given the history just before time \(t\), a marked jump of type \(z\) has probability \(\nu_z(t)h+o(h)\) over an interval of width \(h\). The probability of no jump of any mark is

1-h\sum_{z\ne0}\nu_z(t)+o(h).

Multiplying the no-jump probabilities across small intervals and taking the limit yields the exponential of minus the integrated total rate. At observed jump times, the corresponding mark intensity supplies the event factor. This gives the marked likelihood above; when only singleton marks are allowed, it reduces to the no-common-jump likelihood.

Define the component martingales \(M_j(t)=N_j(t)-\int_0^t\lambda_j(u)\,du\). Their jumps equal those of \(N_j\), because the compensators are continuous. The predictable cross variation, which is the compensator of the product of their jumps, is therefore

\langle M_j,M_k\rangle(t) = \int_0^t \sum_{z\ne0} z_jz_k\nu_z(u)\,du.

For \(j\ne k\), if only singleton marks occur then \(z_jz_k=0\) for every possible mark, so \(\langle M_j,M_k\rangle(t)=0\). If a mark can include both components, its rate contributes to the cross variation. This is the relevant orthogonality calculation: zero cross predictable variation follows from the absence of common jumps, not merely from two event types having different labels.

Worked Example: Cross Variation Under Shared Jumps

Use the marked model with \(\alpha=0.7\), \(\beta=0.3\), and \(\gamma=0.5\) events per hour from the preceding example. The only mark containing both components is the shared mark, whose rate is \(\gamma\). Over one hour,

\langle M_1,M_2\rangle(1) = \int_0^1\gamma\,du = 0.5(1) = 0.5.

The cross predictable variation is \(0.5\) events squared in the counting-process scale. In this constant-rate Poisson construction, the independent component-only counts have zero covariance with each other and with the shared count. Since the shared count enters both \(N_1\) and \(N_2\), their one-hour count covariance is also \(0.5\). Setting that cross term to zero would omit the shared source of variation.

Diagnostics, Alternatives, and Limits

A defensible analysis begins by checking what the time field means. Determine its resolution and rounding rule, whether events belong to one individual or to different individuals, and whether the scientific mechanism permits a shared jump. Process design and event definitions matter: a single subject occupying one state at a time may rule out some simultaneous transitions, while a shared external shock may make common jumps plausible. A timestamp match alone is not enough to decide.

When the exact times are observed and common jumps are ruled out by the model, use the component-wise likelihood. When the model allows a joint event, specify the joint-mark intensity and use its event factor and total-rate survival term. When only coarsened times are available, use a likelihood for the observed coarsened data, summing or integrating over latent event histories consistent with the recording process. These approaches answer different observation and modeling problems; no tie correction can turn one into another without assumptions.

For orthogonality, inspect whether the model permits joint marks and whether the definition of each counting process assigns a shared event to multiple components. If the data are grouped into intervals, covariance between interval count increments is not automatically the same object as common-jump predictable variation. Separate events in a bin can create nonzero dependence in grouped data under some models, while a true shared-jump process has a joint event mechanism at the process level.

Failure mode: treating every identical timestamp as a shared jump. This can add a joint likelihood factor unsupported by rounded data.
Failure mode: multiplying marginal intensities for a genuine common event. A shared jump has probability of order \(h\), whereas two separate jumps in a short interval have probability of order \(h^2\) under regular intensities.
Failure mode: setting cross predictable variation to zero because event types have different names. The correct check is whether any allowed jump mark increments both components.
Limitation: if the recording mechanism or joint-event mechanism is unknown, the observed ties alone may not identify which explanation generated them.

Key takeaway: A shared timestamp is a feature of the recorded data; a common jump is a feature of the process model. The likelihood and martingale covariance calculations must follow the latter, while accounting for the former through the observation mechanism.

Check Your Understanding

Answer each question, then compare with the brief solutions.

Two continuously timed event processes have no common jumps, but their timestamps are rounded to the nearest hour. Can their recorded times tie? What does that imply about exact simultaneous jumps?
In a marked model, singleton jump rates for two components are \(0.4\) and \(0.6\) per day, and their shared-jump rate is \(0.2\) per day. What is the total rate in the no-jump likelihood factor, and what is the one-day cross predictable variation?
Why is the probability of a shared jump in a short interval generally of order \(h\), while the probability of two separate jumps is of order \(h^2\)?
What likelihood principle should be used if two events are reported in the same interval but their exact times are unobserved?

Solutions. (1) Yes. Distinct exact times can round to the same hour; a recorded tie does not imply a common jump. (2) The total rate is \(0.4+0.6+0.2=1.2\) per day. The shared mark gives cross variation \(0.2(1)=0.2\) over one day. (3) A shared mark has probability \(\gamma h+o(h)\) by its intensity. With regular finite component intensities, two separate events require two event occurrences, whose joint probability is of order \(h^2\). (4) Use the likelihood for the observed interval data, accounting for the recording rule and summing or integrating over compatible latent event histories; do not treat the shared interval label as an exact common jump.