Obtaining first place estimates for Model 1 James Ritchie (and - - PowerPoint PPT Presentation

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Obtaining first place estimates for Model 1 James Ritchie (and - - PowerPoint PPT Presentation

Nov 09, 2023 324 likes •573 views

Obtaining first place estimates for Model 1 James Ritchie (and Iain Murray) Bayesian Inference Bayes' rule: Prior Likelihood Posterior Bayesian Inference Approximate with S samples drawn from Priors Need to choose Read the

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Obtaining first place estimates for Model 1

James Ritchie

(and Iain Murray)

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Bayesian Inference Bayes' rule: Prior Likelihood Posterior

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Bayesian Inference Approximate with S samples drawn from

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Priors — Need to choose — Read the paper!1 — Types — Unconstrained parameters, e.g. — Constrained positive/negative: e.g. — Ordered parameters, e.g.

1 Simitev, R.D. and Biktashev, V.N., 2011. Asymptotics of Conduction Velocity Restitution in Models of Electrical Excitation in

the Heart. Bulletin of Mathematical Biology, 73(1), pp.72-115.

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Priors on ordered parameters — E.g. — Parameterise as ) — Normal prior on as before — Log-normal prior on as before

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Likelihood — Run the provided solver for given — Get outputs for each signal over the timeseries — Gaussian log-likelihood

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Error handling

Warning: Failure at t=3.000000e+01. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.136868e-13) at time t.

Just return a likelihood of 0!

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Posterior — Could pass this to an MCMC tool — First we need to find a starting point...

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Maximum A Posteriori (MAP) Solution — Not differentiable2. — Use Powell's method with multiple restarts — Not Bayesian

2 At least, not in provided implementation.

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Markov Chain Monte Carlo (MCMC) Methods — Generate samples from posterior — Use MCMC methods — Strong correlations in posterior — Use emcee3

3 dfm.io/emcee

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emcee — Open source Python package — Implements affine-invariant sampling4 — Run many MCMC chains in parallel — Propose new samples based on other chains — Ran for 10,000 steps with 100 chains each — Discard first half of chain

4 Goodman, J. and Weare, J., 2010. Ensemble Samplers With Affine Invariance. Communications in Applied Mathematics and

Computational Science, 5(1), pp.65-80.

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Procedure checking

  • 1. Check convergence5
  • 2. Check we can recover example

parameters

  • 3. Check other parameter

settings

  • 4. Check scaled residuals,

5 Gelman, A., Stern, H.S., Carlin, J.B., Dunson, D.B., Vehtari, A. and Rubin,

D.B., 2013. Bayesian Data Analysis. Chapman and Hall/CRC.

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Competition Entry Tempting to submit MAP estimate Instead submit sample mean Evaluate sample covariance similarly Submit the mean output of the ODE, , not !

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Problems

  • 1. Unstable Covariance
  • 2. Slow
  • 3. Can't handle multi-modal posteriors
  • 4. Won't scale to high dimensional
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Potential Improvements

  • 1. Rescale the parameters
  • 2. Don't evaluate the likelihood directly
  • 3. Use Parallel-Tempered Ensemble Sampling
  • 4. Use more scalable MCMC algorithms
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Recommendations

  • 1. Choose appropriate parameterisation
  • 2. Find a good initialisation
  • 3. Use tuning-free algorithms
  • 4. Start with generic methods
  • 5. Think about expectations you need