Efficient parameter estimation with the MCMC toolbox

Marko Lainemarko.laine@fmi.fi

Finnish Meteorological Institute

DTU – MCMC lectures, part II – 17.12.2018

Statistical analysis by simulation

• We are interested in the uncertainty distribution of the unknown modelparameter vector θ given the observational data y and the model: p(θ|y, M).• This distribution is typically analytically intractable.• We can still simulate observations / realizations from p(y|θ, M).

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• Statistical inference is used to define what is a good fit. Parameters that areconsistent with the data and the modelling uncertainty are accepted.

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MCMC - Markov chain Monte Carlo

• Simulate the model while sampling theparameters from a proposal distribution.• Weight the (or accept) the parameters

according to a suitable goodness-of-fitcriteria depending on prior information anderror statistics.• The resulting chain (i.e. a sample of

parameters) is a sample from the Bayesianposterior distribution of parameteruncertainty.

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Random walk in probability distribution

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Posterior distributions

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predictive distribution of model

dY/dt = (µθT−20−ρ−Q/V−pZ)Y

95% predictive envelope

95% envelope for the observations

• Posterior distribution of model parameters.• Posterior distribution of model predictions.• Posterior distribution for model comparison.

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Terminology

The model in general form is

y = f (x|θ) + e,observations = model + error.

Likelihood function for Gaussian errors corresponds to a quadratic cost function,with

p(y|θ) ∝ exp{−1

2∑ni (yi − f (xi|θ))

2

σ2

}

= exp{−1

2SS(θ)

σ2

},

where SS(θ) = −2 log(p(y|θ)) is the -2 times log-likelihood in "sum-of-squares"cost function format. For the posterior, we also need to accountSSpri(θ) = −2 log(p(θ)), prior "sum-of-squares".

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Metropolis-Hastings algorithm

Random walk Metropolis-Hastings algorithm with Gaussian proposal distribution(and Gaussian likelihood).

• Propose new parameter value θprop = θcurr + ξ, where ξ ∼ N(0, Σprop) isdrawn from the proposal distribution.• Accept θprop with probability α,

α(θcurr, θprop) = 1∧ exp{− 1

2

(SS(θprop)− SS(θcurr)

σ2

)− 1

2

(SSpri(θprop)− SSpri(θcurr)

)}• Efficient proposal distribution⇒ adaptive tuning of Σprop, AM, DRAM

algorithms.

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Convergence diagnostics

• 1d and 2d plots of the chain• Mixing, rejection rate.• Monte Carlo error of the estimates, chain autocorrelation.• Efficient number of simulations, integrated autocorrelation time, batch mean

standard error.

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Short chains and adaptation• Important to make short chains as efficient as possible.• Efficient: produce estimates with small Monte Carlo error.

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mean of the 1. parameter

mhamdramram

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195% quantile

simulation index

mhamdramram

Short MCMC chain run 1000 times with different algorithms.M.Laine: Efficient parameter estimation with the MCMC toolbox 8/13

Chains have not mixed yet

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2 first dimensions of the chain with 50% and 95% target contoursc50 = 78.7%, c95 = 98.8%, τ=95.8

>> chainstats(chain,names)MCMC statistics, nsimu = 1000

mean std MC_err tau geweke---------------------------------------------------------------------theta_1 0.037146 0.29164 0.054374 67.086 0.017371theta_2 -0.98318 0.39323 0.082073 112.74 0.43098theta_3 0.0077533 0.37183 0.076218 101.3 0.013535theta_4 0.058239 0.34665 0.069839 101.88 0.014742---------------------------------------------------------------------

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Chains that have better mixing

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2 first dimensions of the chain with 50% and 95% target contoursc50 = 58.8%, c95 = 98.3%, τ=44.0

>> chainstats(chain,names)MCMC statistics, nsimu = 1000

mean std MC_err tau geweke---------------------------------------------------------------------theta_1 0.047117 0.42071 0.068536 43.274 0.027867theta_2 -1.1139 0.49858 0.088196 47.542 0.60531theta_3 0.050454 0.43527 0.069026 41.525 0.023323theta_4 0.033922 0.42226 0.067013 43.516 0.038749---------------------------------------------------------------------

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MCMC Toolbox for Matlab

https://github.com/mjlaine/mcmcstat

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https://github.com/mjlaine/mcmcstat

MCMC toolbox for matlab

%% MCMC toolboxmodel.ssfun = @mycostfun

data = load(’datafile.dat’);

parameters = {{’par1’, 2.3 }{’par2’, 1.2 }

};

options.nsimu = 5000;options.method = ’am’;

[results,chain] = mcmcrun(model,data,parameters,options);

mcmcplot(chain,[],results)

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MCMC demos

Matlab source code: https://github.com/mjlaine/mcmcstatDocumentation: https://mjlaine.github.io/mcmcstat/Installation by git:

git clone https://github.com/mjlaine/mcmcstat.git

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https://github.com/mjlaine/mcmcstathttps://mjlaine.github.io/mcmcstat/