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Anytime-Valid Continuous-Time Confidence Processes for Inhomogeneous Poisson Processes

Michael Lindon, Nathan Kallus

Abstract

Motivated by monitoring the arrival of incoming adverse events such as customer support calls or crash reports from users exposed to an experimental product change, we consider sequential hypothesis testing of continuous-time inhomogeneous Poisson point processes. Specifically, we provide an interval-valued confidence process $C^α(t)$ over continuous time $t$ for the cumulative arrival rate $Λ(t) = \int_0^t λ(s) \mathrm{d}s$ with a continuous-time anytime-valid coverage guarantee $\mathbb{P}[Λ(t) \in C^α(t) \, \forall t >0] \geq 1-α$. We extend our results to compare two independent arrival processes by constructing multivariate confidence processes and a closed-form $e$-process for testing the equality of rates with a time-uniform Type-I error guarantee at a nominal $α$. We characterize the asymptotic growth rate of the proposed $e$-process under the alternative and show that it has power 1 when the average rates of the two Poisson process differ in the limit. We also observe a complementary relationship between our multivariate confidence process and the universal inference $e$-process for testing composite null hypotheses.

Anytime-Valid Continuous-Time Confidence Processes for Inhomogeneous Poisson Processes

Abstract

Motivated by monitoring the arrival of incoming adverse events such as customer support calls or crash reports from users exposed to an experimental product change, we consider sequential hypothesis testing of continuous-time inhomogeneous Poisson point processes. Specifically, we provide an interval-valued confidence process over continuous time for the cumulative arrival rate with a continuous-time anytime-valid coverage guarantee . We extend our results to compare two independent arrival processes by constructing multivariate confidence processes and a closed-form -process for testing the equality of rates with a time-uniform Type-I error guarantee at a nominal . We characterize the asymptotic growth rate of the proposed -process under the alternative and show that it has power 1 when the average rates of the two Poisson process differ in the limit. We also observe a complementary relationship between our multivariate confidence process and the universal inference -process for testing composite null hypotheses.

Paper Structure

This paper contains 26 sections, 13 theorems, 57 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Inhomogeneous Poisson Point Processes
  4. Confidence Processes for the Intensity Measure $\Lambda(t)$
  5. Confidence Processes for Dual Intensity Measures
  6. $e$- and sequential $p$-processes for hypothesis testing
  7. Comparisons to Other Methods
  8. anytimecounts
  9. asymptoticcs
  10. Application: Software A/B Test
  11. Conclusion
  12. Proofs
  13. Useful lemmas
  14. Proofs for section \ref{['sec:one_process']}
  15. Proof of Theorem \ref{['thm:simplemartingale']}
  16. ...and 11 more sections

Key Result

Theorem 3.1

Suppose $\mathcal{T} \sim \mathcal{P}(\lambda_0)$, then for any $t >0$, $\delta > 0$, and $\lambda_1 : \mathbb{R}\rightarrow \mathbb{R}_{\geq 0}$ The likelihood ratio process satisfies the following time-uniform bound

Figures (13)

  • Figure 1: The intensity measure $\Lambda(t) = \mathbb{E}[N(t)]$ (red), the observed counting process $N(t)$ (blue), the $0.95$ confidence process $C^\alpha(t)$ (shaded blue), and the realized point process (black ticks).
  • Figure 2: The time-average of intensity $\frac{1}{t}\Lambda(t)$ (red), the estimated time-average of intensity (blue), the $0.95$ confidence process (shaded blue), and the realized point process (black ticks).
  • Figure 3: (Left) $1-\alpha$ Confidence set $C^\alpha(t)$ for $(\Lambda^A(t), \Lambda^B(t))$. (Right) $1-\alpha$ confidence set $T(C^\alpha(t))$, where $T : (x,y) \rightarrow (y-x, y+x)$, for $(\Lambda^B(t)-\Lambda^A(t), \Lambda^B(t)+\Lambda^A(t))$. Parameters: $\alpha=0.05$, $\phi = 1$, $N^A = 40$ and $N^B = 100$.
  • Figure 4: $1-\alpha$ simultaneous confidence processes on $\Lambda^B(t)$ and $\Lambda^A(t)$
  • Figure 5: $1-\alpha$ confidence processes on $\Lambda^B(t)/t$ and $\Lambda^A(t)/t$
  • ...and 8 more figures

Theorems & Definitions (23)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Example 3.5
  • Theorem 4.1
  • Corollary 4.2
  • Example 4.3
  • Theorem 5.1
  • Theorem 5.2
  • ...and 13 more