Sequential Experimental Design for Predator-Prey Functional Response Experiments

1 Introduction

• Functional-response models quantify prey consumption as a function of prey density and are relevant to predator–prey prediction and ecosystem management.The introduction links these models to invasions, extinction, biological control, ecosystem management, and predator–prey dynamics.Source: §1 · p. 2
• Functional-response experiments record prey attacked from a fixed number available to a predator over a set period, and multiple trials are needed for inference.The volume of experiments carries costs in time, money, and animal sacrifice.Source: §1 · p. 2
• Earlier optimal-design work considered static designs for precise frequentist estimation under one assumed model, while model uncertainty can be substantial.The paper argues that accounting for model uncertainty and discriminating among models is desirable.Source: §1 · p. 2
• Sequential design updates information about model structure and parameter values as observations arrive, unlike static designs chosen before experimentation.The authors present it as the first sequential experimental design approach for predator–prey functional-response experiments.Source: §1 · p. 2
• The approach uses sequential Monte Carlo to estimate utility functions for parameter estimation, model discrimination, or their combination via total entropy.The total-entropy criterion is described as a dual-purpose utility function.Source: §1 · p. 3

2 Background on Functional Response Models and Predator-Prey Functional Response Experiments

• In the type II model, attack rate is per-capita consumption at low prey density and handling time is time spent subduing, pursuing, and eating prey.The model is presented as a mechanistic equation linked to consumption probability over time.Source: §2 · pp. 3, 5
• Holling type II consumption rises at a decelerating rate to a plateau, whereas type III rises sigmoidally, accelerating at low and decelerating at high prey density.The paper focuses on these popular, simple models while noting that its method can accommodate any functional-response model.Source: §2 · pp. 3–4
• Figure 1 provides an example dataset from a predator–prey experiment with varying initial prey densities.The example dataset is from Papanikolaou et al. (2016).Source: §2 · p. 4, Figure 1 · p. 4
• The experimenter controls initial prey density across independent runs and records the number consumed during a fixed time period.Mechanistic model solutions determine expected proportions consumed in the probabilistic models.Source: §2 · p. 4
• Prey consumed can be modelled with a binomial distribution or, when data indicate overdispersion, a beta-binomial distribution.Overdispersion is noted as common in predator–prey functional-response data.Source: §2 · pp. 4–5

3 Illustration of the Sequential Design Process

• The illustration is intended to make the sequential-design methodology more accessible to practitioners, with a more technical account provided later.Source: §3 · p. 6
• The illustration assumes a Holling type II beta-binomial model with a = 0.5, Th = 0.7, and λ = 0.5, and targets parameter estimation and model discrimination.The design is myopic: experiments run one at a time, with results informing later choices.Source: §3 · p. 6
• The four selected initial prey densities are 299, 38, 20, and 30, chosen by maximizing a utility function for the current experiment.Figure 2 plots utility against experimental design and marks each curve’s optimum with an arrow.Source: Figure 2 · p. 6, §3 · p. 7
• The marginal posterior for log(Th) becomes more precise over four sequential experiments, while the true model probability fluctuates from a prior of 0.25 to 0.45, 0.62, 0.52, and 0.47.Figure 3 compares the marginal posteriors with the prior and marks the true parameter value.Source: §3 · p. 7, Figure 3 · p. 7
• After each experiment, the observed prey count updates the posterior, which becomes the prior for choosing the next design.The example uses sequential Monte Carlo, though the text says other posterior sampling or approximation methods could be adopted.Source: §3 · p. 7

4.1 General Notation

• The notation distinguishes candidate models, their parameters, accumulated observations and design points, and the proposed design's utility given current data.For functional response models, observations are prey consumed and design points are initial prey densities.Source: §4.1 · p. 8

4.2 Sequential Optimal Design

• The next design point maximizes expected utility over possible responses and models, with the goal of meeting experimental objectives in as few runs as possible.The expectation is taken over the response and model space.Source: §4.2 · p. 9

4.3 Utility Functions

• The paper uses three utility functions for parameter estimation, model discrimination, and their combination, with Kullback-Leibler divergence as a common information measure.The divergence represents information gain about the true data-generating process.Source: §4.3 · p. 9

4.3.1 Parameter Estimation Utility

• For parameter estimation, the utility measures the Kullback-Leibler divergence between current and updated parameter posteriors.The criterion is used to design experiments for parameter estimation across the candidate models.Source: §4.3.1 · p. 10

4.3.2 Model Discrimination Utility

• For model discrimination, utility is based on mutual information between the model indicator and the predicted observation.This mutual information is equivalent to a Kullback-Leibler divergence involving their joint and marginal distributions.Source: §4.3.2 · p. 10

4.3.3 Dual-Purpose Utility

• The dual-purpose utility adds the parameter-estimation and model-discrimination utilities.The total entropy criterion combines both experimental goals.Source: §4.3.3 · pp. 10–11

4.4 Sequential Monte Carlo

• SMC carries weighted particles through sequential target distributions using reweighting, resampling, and move steps as observations arrive.When effective sample size falls below a threshold, resampling restores it, while a move step diversifies the duplicated particles.Source: §4.4 · pp. 11–12

4.5 Estimation of Utility Functions

• The utility-estimation procedure uses particle values and weights to approximate utilities and posterior model probabilities at proposed design points.After selecting a design point and collecting its response, the algorithm updates weights and model evidences before proceeding.Source: §4.5 · pp. 13–14

5 Simulation Study

• The study evaluates 40 true model/parameter settings, repeating each design method 30 times per setting with 25 observations per run.The settings comprise four true models and ten parameter settings per model; priors are described as reasonably vague and give little support to unrealistic parameter regions.Source: §5 · p. 15

5.1 Results

• Sequential design outperforms static and random designs for parameter estimation across all true models.Source: §5.1 · p. 16, Figure 4 · p. 16
• Sequential design also outperforms static and random designs for model discrimination across the true models.Source: §5.1 · p. 16, Figure 5 · p. 17
• For dual-purpose designs, sequential design outperforms static design on parameter precision and model probabilities across all true models.Source: §5.1 · p. 16, Figure 6 · p. 18, Figure 7 · p. 19

6 Discussion

• When fitting an incorrect model, SMC weight updates can reduce effective sample size sharply, leaving few unique particles and poor utility estimates.The authors suggest intermediate target distributions as a possible way to avoid these computational issues.Source: §6 · p. 20
• The myopic design considers only one observation ahead, which the authors state is not optimal.They suggest investigating two-step-ahead dynamic programming to assess its effect on results.Source: §6 · p. 20
• Simulations indicate the algorithm can support conclusions about model and parameter uncertainty earlier than random or static designs.Source: §6 · p. 20

Funding

• HM received a QUT Master of Philosophy Award, and MH was funded by the Austrian Science Fund grant J3959-N32.Source: Funding · p. 21
• NEP received a postdoctoral research grant from the Foundation for Education and European Culture, and CD was supported by Australian Research Council Discovery Project DP200102101.Source: Funding · p. 21

Appendix A

• Appendix A describes optimal static design methodology for experiments targeting parameter estimation and/or model discrimination.Source: Appendix A · p. 24

• Bayesian optimal static design uses prior information to select multidimensional designs for a specified experimental goal.Source: Static Optimal Design Methodology · p. 24
• Finding such a design is computationally challenging because it requires maximizing an analytically intractable utility over a high-dimensional design space.Source: Static Optimal Design Methodology · p. 24
• The methodology of Overstall et al. (2018) is used to find a multidimensional near-optimal static design.Source: Static Optimal Design Methodology · p. 24

General Methodology

• Expected utility is maximized over the design space, with its expectation over models and responses approximated by Monte Carlo draws when no closed form is available.The approximation requires sampling from the joint distribution of model and responses and evaluating the chosen utility for each sample.Source: General Methodology · p. 24

Utility Functions

• Parameter-estimation utility uses KLD between parameter distributions; sequential designs compare the current posterior with the posterior after one additional observation.Model-discrimination utility uses the Box and Hill criterion, while total entropy sums parameter-estimation and model-discrimination utilities.Source: Utility Functions · p. 25

Approximating Posterior Distributions and Other Quantities

• Laplace approximation uses the posterior mode and Fisher information to estimate marginal likelihoods and posterior model probabilities.The posterior mode and Hessian can be estimated numerically, with the Hessian obtained by finite differencing.Source: Approximating Posterior Distributions… · p. 26
• Normal approximations to parameter priors and posteriors provide an explicit KLD expression, simplifying parameter-estimation utility calculations.Expected utility is then estimated by Monte Carlo integration over joint model and response samples.Source: Approximating Posterior Distributions… · p. 27

ACE Algorithm

• ACE cyclically optimizes each design point and uses a Gaussian-process emulator to reduce expected-utility calculations.The algorithm is used to maximize expected utility in a high-dimensional design space.Source: ACE Algorithm · p. 27

Appendix B

• The study explores the benefits of different utility functions for experimental goals within sequential design.Source: Appendix B · p. 28

Simulation Study Details

• Four selection methods are compared: random design points and optimal designs for each of three utility functions.The design points are selected over the discrete design space.Source: Simulation Study Details · p. 28
• The study uses 40 true model-and-parameter configurations, repeats each algorithm 30 times, and collects 25 observations per run.The configurations comprise 10 posterior-based true parameter settings for each of four models.Source: Simulation Study Details · p. 28
• With 24-hour exposure, design points from 1 to 300, and equal prior probabilities for four models, the study compares parameter precision and model discrimination across methods.Parameter precision is assessed using Bayesian D-posterior precision; posterior model probabilities are used to assess discrimination.Source: Simulation Study Details · p. 28

Results

• Parameter-estimation designs perform best for parameter precision, model-discrimination designs perform best for discrimination, and total entropy is only marginally less effective on both goals.Source: Results · pp. 28, 30, Figure 8 · p. 29, Figure 9 · p. 31
• Random and parameter-estimation designs struggle to discriminate between beta-binomial models when one is true, and between binomial models when one is true.Source: Results · p. 30, Figure 9 · p. 31
• Dual-purpose allocations appear to combine features of parameter-estimation and model-discrimination designs, while parameter-estimation allocations are less concentrated in the lower third.Source: Results · p. 30, Figure 12 · p. 34
• Precision gains across true models appear to decay exponentially as experiments accumulate under parameter-estimation design.Figure 10 tracks log Bayesian D-posterior precision across SMC iterations.Source: Results · p. 30, Figure 10 · p. 32
• After 25 experiments, Models 3 and 4 can be identified with high probability under model-discrimination design.Source: Results · p. 30, Figure 11 · p. 33
• The proposed designs are predominantly bimodal at design-space boundaries; discrimination allocations differ by model family, with beta-binomial designs concentrated low and binomial designs placing more mass high.Source: Results · p. 30, Figure 12 · p. 34