Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Experimental Design for Partially Observable Pure Birth Processes

Ali Eshragh, Matthew P. Skerritt, Bruno Salvy, Thomas McCallum

TL;DR

This work tackles optimal experimental design for partially observable pure birth processes by maximizing Fisher information about the birth rate $\lambda$ when each observation reveals a binomial sample with fixed probability $p$. The authors introduce a generating-function framework that yields a recursive form for the likelihood and its FI, enabling efficient computation for higher numbers of observations than prior methods. They implement a parallel, slice-based algorithm in C++ with Maple-assisted symbolic precomputation and use NLPSolve to optimize observation times under order constraints, demonstrating detailed numerical results for $n=2,3,4$. The study reveals non-smooth FI landscapes driven by drop values $\mathfrak{D}_i(\lambda)$, analyzes the trade-offs between $\lambda$, $p$, and $n$, and provides a publicly available implementation to facilitate practical design of POPBP experiments.

Abstract

We develop an efficient algorithm to find optimal observation times by maximizing the Fisher information for the birth rate of a partially observable pure birth process involving $n$ observations. Partially observable implies that at each of the $n$ observation time points for counting the number of individuals present in the pure birth process, each individual is observed independently with a fixed probability $p$, modeling detection difficulties or constraints on resources. We apply concepts and techniques from generating functions, using a combination of symbolic and numeric computation, to establish a recursion for evaluating and optimizing the Fisher information. Our numerical results reveal the efficacy of this new method. An implementation of the algorithm is available publicly.

Optimal Experimental Design for Partially Observable Pure Birth Processes

TL;DR

This work tackles optimal experimental design for partially observable pure birth processes by maximizing Fisher information about the birth rate when each observation reveals a binomial sample with fixed probability . The authors introduce a generating-function framework that yields a recursive form for the likelihood and its FI, enabling efficient computation for higher numbers of observations than prior methods. They implement a parallel, slice-based algorithm in C++ with Maple-assisted symbolic precomputation and use NLPSolve to optimize observation times under order constraints, demonstrating detailed numerical results for . The study reveals non-smooth FI landscapes driven by drop values , analyzes the trade-offs between , , and , and provides a publicly available implementation to facilitate practical design of POPBP experiments.

Abstract

We develop an efficient algorithm to find optimal observation times by maximizing the Fisher information for the birth rate of a partially observable pure birth process involving observations. Partially observable implies that at each of the observation time points for counting the number of individuals present in the pure birth process, each individual is observed independently with a fixed probability , modeling detection difficulties or constraints on resources. We apply concepts and techniques from generating functions, using a combination of symbolic and numeric computation, to establish a recursion for evaluating and optimizing the Fisher information. Our numerical results reveal the efficacy of this new method. An implementation of the algorithm is available publicly.
Paper Structure (16 sections, 4 theorems, 31 equations, 38 figures, 2 tables)

This paper contains 16 sections, 4 theorems, 31 equations, 38 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Optimal Experimental Design
  3. Partially Observable Pure Birth Process
  4. Generating Functions for the Likelihood
  5. Experimental Methodology
  6. Parallelization
  7. Implementation Considerations
  8. Optimization Method
  9. Experimental Results
  10. Optimal Observation Times
  11. Two Observations ($n=2$)
  12. Three Observations ($n=3$)
  13. Four Observations ($n=4$)
  14. Optimal Fisher Information
  15. Timings
  16. ...and 1 more sections

Key Result

Proposition 3.3

If $(t_1^*,\dots,t_n^*)$ constitutes an optimal experimental design for a POPBP with parameters $(\lambda,p)$ and a time-horizon of 1, then for any fixed $\tau>0$, the scaled design $(\tau t_1^*,\dots,\tau t_n^*)$ forms the corresponding optimal experimental design for a POPBP with parameters $(\lam

Figures (38)

  • Figure 1: Fisher information comparison of Becker & Kersting's approximate $t_i^*$ and direct numeric optimization ($s_i^*$)
  • Figure 2: Fisher information and calculated optimal observation time comparison of Becker & Kersting's approximate $t_i^*$ and direct numeric optimization ($s_i^*$)
  • Figure 3: Optimal observation times for $n=2,\lambda=0.5$
  • Figure 4: Optimal observation times for $n=2,\lambda=0.8$
  • Figure 5: Optimal observation times for $n=2,\lambda=1$
  • ...and 33 more figures

Theorems & Definitions (19)

  • Definition 2.1: Fisher Information
  • Definition 2.2: Pure Birth Process, PBP
  • Definition 3.1: Partially Observable Pure Birth Process, POPBP Bean
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Lemma 4.1
  • proof
  • Remark 4.2
  • ...and 9 more