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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
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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
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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
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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
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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
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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
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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.
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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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