Post on 27-Jul-2015
Florian Hartig Department of Biometry and Environmental System Analysis
Florian Hartig Department of Biometry and Environmental System Analysis
Consistency of Bayesian and maximum likelihood inference in state-space models of ecological systems with strongly nonlinear dynamics
Florian Hartig, Carsten F. Dormann
University of Freiburg, Department of Biometry and Environmental System Analysis
http://florianhartig.wordpress.com/ ISEC 2014, Montpellier,
Figures by Ernst Haeckel, scans by Kurt Stüber, MPI Köln
Florian Hartig Department of Biometry and Environmental System Analysis
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Introduction: a strange result …
Claim: population models, fit in a Bayesian state-space framework to data produced by themselves (no model error), lead to worse forecasts than a non-parametric forecasting method; for chaotic dynamics, # data >> # parameters
Florian Hartig Department of Biometry and Environmental System Analysis
Worrying, given that state-space models widely advertised as state-of-the-art
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Florian Hartig Department of Biometry and Environmental System Analysis
1: Population model – logistic map
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Florian Hartig Department of Biometry and Environmental System Analysis
2: Process error on population dynamics
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Florian Hartig Department of Biometry and Environmental System Analysis
3: Observation error on those dynamics
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Florian Hartig Department of Biometry and Environmental System Analysis
4: The final observations (red triangles)
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Florian Hartig Department of Biometry and Environmental System Analysis
State-space model to recover parameter estimates from those observations
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Population model
Observation model
Observed data
SSM: calculate P(Observations|Parameter) by summing over all possible „latent“ trajectories (state space), find parameters that have the highest likelihood to „produce“ the observations
Florian Hartig Department of Biometry and Environmental System Analysis
Growth rate estimates for increasing true growth rates
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Model estimated with JAGS, median posterior values shown
No bias line
Florian Hartig Department of Biometry and Environmental System Analysis
Hypothesis I
Why? Imagine you are the „statistical model“
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Observations
Hypothesis II Stable dynamics ------, All variability from observation error
Hypothesis I Chaotic pop dynamics, Medium observation error
Hypothesis II
Florian Hartig Department of Biometry and Environmental System Analysis
Solution: chopping up the data
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Fit small chunks of the data independently, Optimize joint likelihood / posterior Pisarenko & Sornette (2004), Phys. Rev. E
Florian Hartig Department of Biometry and Environmental System Analysis
Suddenly, all is fine
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Hartig, F. & Dormann, C. F. (2013) Does model-free forecasting really outperform the true model? PNAS, 110, E3975.
Florian Hartig Department of Biometry and Environmental System Analysis
Conclusions / implications
► MLE / Bayesian inference can be asymptotically inconsistent for nonlinear dynamical systems ► This conditions may readily occur in more complex predator-prey / food
web / host-parasitoid systems
► When / why? ► Asymptotic inconsistency formally proven by Judd (2007), Phys.
Rev. E for SSM + chaotic + observation error only ► Our conclusion (without formal proof): remains the same for
process << observation error, we think this is what happens here.
► Additional consideration: if observation error sufficiently rigid, likelihoods might get extremely ragged, problems for the samplers, see Wood (2010) Nature, Wood & Fasiolo (plenary).
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Florian Hartig Department of Biometry and Environmental System Analysis
Recommendations
► Be aware that parameter estimates in a state-space framework can be massively biased if the dynamics are strongly nonlinear.
► Remedies: ► Chopping up the data Pisarenko & Sornette (2004), Phys.
Rev. E ► Diagnose by comparing model / data with summary
statistics Judd (2007), Phys. Rev. E ► Use of ABC / synthetic likelihoods? Wood (2010) Nature,
Hartig et al. (2011), Ecol. Lett. ► Get strong data on observation models!
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Florian Hartig Department of Biometry and Environmental System Analysis
Thank you!
Hartig, F. & Dormann, C. F. (2013) Does model-free forecasting really outperform the true model? PNAS, 110, E3975.
Available at http://arxiv.org/abs/1305.3544 , code
https://github.com/florianhartig/NonlinearOrChaoticBayesianStateSpaceModels
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