likelihood-free mcmc...about mcmc algorithms that realize markov chain we want the invariant...

Simple Monte CarloLikelihood-free Methods

Application to Financial DataConclusion

Likelihood-free MCMCBayesian inference for stable distributions with applications in

finance

Yuanwei XuDepartment of Mathematics

University of Leicester

September 2, 2011MSc project final presentation

Yuanwei Xu Likelihood-free MCMC



Outline

1 Simple Monte Carlo

2 Likelihood-free MethodsLikelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC

3 Application to Financial Data

4 Conclusion




Rejection sampling

1 Sample x (i) ∼ g (x)

2 Accept x (i) with

probability f (x(i))/Mg(x(i)),then go to 1




Importance sampling

A different way to view Ef [h (θ)]

Ef [h (θ)] =

ˆh (θ)

f (θ)

g (θ)g (θ) dθ = Eg

[f (θ)

g (θ)h (θ)

]≈ 1

m

m∑j=1

f(θ(j))

g(θ(j))h (θ(j)

)(1)

for θ(j)drawn from g (θ)

Importance sampling does not throw away samples, it givesdifferent weights(importance) f

(θ(j)|D

)/g(θ(j)).




Importance sampling

In Bayesian context with normalising constant not known:

Ef [h (θ)] = Eg

[f (θ|D)

g (θ)h (θ)

]=

1

P (D)Eg

[π (θ)P (D|θ)

g (θ)h (θ)

]=

∑mj=1 π

(θ(j))P(D|θ(j)

)h(θ(j))/g(θ(j))∑m

j=1 P(D|θ(j)

)π(θ(j))/g(θ(j))

where θ(j) ∼ g (θ)

This can also be used in general setting, i.e. use∑mj=1 h(θ(j))f (θ(j))/g(θ(j))∑m

j=1 f (θ(j))/g(θ(j))as an alternative to (1), with an

improvement in variance.




Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC

What if likelihoods are unavailable?

Approximating posterior, avoid likelihood evaluation—knownas approximate Bayesian computationSome early literatures

LF-RS

Tavare et al., 1997 Inferring Coalescence Times From DNASequence Data

replacing the full dataset with summary statistics.

Fu and Li, 1997 Estimating the age of the common ancestorof a sample of DNA sequences

simulating a new dataset, comparing with the observedone.

LF-MCMC

Marjoram et al., 2003 Markov Chain Monte Carlo withoutlikelihoods

MCMC approach generalized from LF-RS





Likelihood-free rejection sampling

The idea can be seen in the following algorithm.

LF-RS

1 Simulate from the prior θ ∼ π2 Generate D′ under the model with parameter θ

3 Accept θ if D = D′;go to 1

D : observed dataset D′: simulated dataset

In practice one replace D and D′ with corresponding summarystatistics S and S ′.The condition can be rewritten asρ (S , S ′) ≤ ε for some distance measure ρ (e.g. Euclidean).This will result in an approximate posterior f (θ|ρ (S ,S ′) ≤ ε).





LF-RS example

Example

Suppose y1, y2, . . . yn are observations from Exp (θ) with densityf (y |θ) = θe−θy , y > 0. The prior for θ is conjugate gammadistribution θ ∼ Gamma (α, β), then the posterior is gamma withaltered parameters θ|D ∼ Gamma (n + α, β/ (β

∑yi + 1)). Let

α = 3, β = 1 and n = 5 observations from Exp (2), and choosethe sample mean y as a sufficient statistic. We simulate theposterior distribution using LF-RS algorithm with ε = 1, and 0.1





LF-RS exampleSimulation results

Results for ε = 1(left) and ε = 0.1(right)





Markov Chain Monte Carlo(MCMC)

About MCMC

Algorithms that realize Markov chainWe want the invariant distribution of the chain to be ourtarget distributionSamples can be taken as drawn from the target distributionafter running the chain for a long time

MCMC History

Metropolis, et.al.(1953). Equations of state calculations byfast computing machines. J. Chem. Phys. 21 1087–1092.Hastings, W. (1970). Monte Carlo sampling methods usingMarkov chains and their application. Biometrika 57 97–109.Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-basedapproaches to calculating marginal densities. J. Amer. Statist.Assoc. 85 398–409.





Constructing MCMC algorithms

Ergodic Theorem gaurantees convergence.

From Markov chain theory, general balance implies(in discretesetting)

f ∗P = f ∗ (2)

f ∗: invariant distribution P: transition matrix with elementsPij = P (xt+1 = j |xt = i) := P (i → j). Sum over each row is one.

Detailed balance

P(x → x ′

)f ∗ (x) = P

(x ′ → x

)f ∗(x ′)

Summing both sides over x , we get (2).





Metropolis-Hastings algorithm

Metropolis-Hastings

1 If now at θ , propose a move to θ′ according to a proposaldistribution q (θ → θ′)

2 Accept θ′ with probability A (θ, θ′) = min{1, f (θ′)q(θ′→θ)f (θ)q(θ→θ′) }

3 Go to 1 until desired number of iterations





Theorem

The invariant distribution of the chain is f (θ).

Proof. We show that detailed balance is satisfied.The M-H transition probability is

P(θ → θ′

)= q

(θ → θ′

)A(θ, θ′

)Choose(w.l.o.g)

f (θ′) q (θ′ → θ)

f (θ) q (θ → θ′)≤ 1

Then

P(θ → θ′

)f (θ) = q

(θ → θ′

) f (θ′) q (θ′ → θ)

f (θ) q (θ → θ′)f (θ)

= f(θ′)q(θ′ → θ

)A(θ′, θ

)= P

(θ′ → θ

)f(θ′)





Likelihood-free MCMC

(Marjoram et al.,2003) proposed a MCMC method withoutlikelihood evaluation

LF-MCMC

1.If now at θ , propose a move to θ′ according to a proposaldistribution q (θ → θ′)2.Generate D′ under model with θ′

3.If D′ = D, go to 4; otherwise return to 1

4.Accept θ′ with probability A (θ, θ′) = min{1, π(θ′)q(θ′→θ)π(θ)q(θ→θ′) }, then

go to 1

One can proof the invariant distribution is f (θ|D).

Approximate posterior: replacing D′ = D with ρ (S , S ′) ≤ ε





Stable distributions

No closed form densities in general.

4 parameters: α ∈ (0, 2] determins tail behavior, β ∈ [−1, 1]the skewness, γ > 0 the scale and δ ∈ R the location.

Special cases:

Cauchy (α = 1, β = 0)Normal (α = 2, β = 0)Levy (α = 1/2, β = 1)

Infinite variance(except α = 2), mean is existed only if1 < α ≤ 2.

Generalized CLT





Some literaturesBayesian inference for stable models

Buckle, D.J., 1995. Bayesian inference for stable distributions.Journal of the American Statistical Association 90, 605–613.

Auxiliary variable Gibbs sampler

Lombardi, M.J., 2007. Bayesian inference for alpha stabledistributions: a random walk MCMC approach.Computational Statistics & Data Analysis 51, 2688–2700.

Evaluating likelihood via inverse Fourier transform combinedwith a series expansion

Peters, G.W., Sisson, S.A., Fan, Y., 2010. Likelihood-freeBayesian inference for α-stable models. ComputationalStatistics and Data Analysis. doi:10.1016/j.csda.2010.10.004

Likelihood-free sequential Monte Carlo sampler





Before implementing LF-MCMC

Assumptions of simulation:

Estimate one parameter with the other three parametersfixed.*

Use flat prior for the parameter to be estimated.

Use a Gaussian transition kernel centered at current state. Ifthe parameter is within some interval, simply truncate thosevalues that are outside the interval.

Use quantiles and Kolmogorov-Smirnov statistic as summarystatistics.

Use a fixed ε value during computation.*

*: These assumptions will be dropped later.





Simulation resultsFix parameters β, γ, δ

Simulation results for α based on 200 observations fromStable (1.5, 0.5, 10, 10) using a fixed ε = 25.(Left) Sample path of α, true value is 1.5. (Right) Trace of sampleaverage.





Simulation resultsFix parameters β, γ, δ

Sample path and ergodic average plot for α. Top: ε = 15 ,acceptance rate: 1.3% Bottom: ε = 50 acceptance rate:34.2%





Modified LF-MCMC

Motivation

Modification

dynamically define εt as a monotonically decreasing sequence:

εt =

{max{εmin,min{ε′, εt−1}} if accept θ′

εt−1 otherwise

ε0 = ρ (S , S0), ε′ = ρ (S , S ′), where S0 : summary statisticsfor the dataset generated by the intital value and εmin: targetε value.Before: compare with the target ε value(global comparison)Now: compare with the previous ε value(local comparison)adaptively change the variance of the proposal distributionaccelerate/control chain mixing





Simulation resultsAll four parameters unknown

Simulation results for Stable (α, β, γ, δ) based on 500 observationsfrom Stable (1.5, 0.5, 10, 10), using 10000 iterations and εmin = 15.





LF-MCMC for variance gamma (VG) distribution

For comparison, we apply the method to VG distribution.

VG process (Madan and Seneta, 1990) (Madan, Carr and Chang, 1998)

X(VG)t = θGt + σWGt

Gt is a gamma process with mean rate unity and variance rate ν, Wt isthe standard Brownian motion.

Unit period distribution—VG (σ, ν, θ)

pdf can be written in terms of modified Bessel function of thesecond kind

VG distribution has finite moments of all order.





LF-MCMC for variance gamma (VG) distributionSimulation results

Simulation results for VG (σ, ν, θ, µ) based on 500 observations fromVG (0.8, 1, 0.5, 10), using 10000 iterations and εmin = 1. Added summary statistics:mean and variance.




Application to financial dataFit stable distribution to real financial data.

The data is the S&P 500 index from the period of January2009 to July 2011, with 629 daily log returns and the pricesare adjusted close price.

Implement 10000 iterations of LF-MCMC, discard first 2000iterations, averaging over the samples gave the values ofposterior estimates: α: 1.3542 β: 0.0741 γ: 0.0070 δ: 0.0019


blue=stable fit, green=smoothed data

The figure is produced using J.P. Nolan’s STABLE program, available athttp://academic2.american.edu/∼jpnolan

http://academic2.american.edu/~jpnolan



Concluding remarks

Our results:

apply LF-MCMC to the inference for stable models

make the method applicable to general casesrelatively low computational cost

Pitfalls:

need to specify a proper target ε valuedon’t know when convergence will happen, need moreiterationschoice of summary statistics can crucially affect samplerperformance




Concluding remarks

Our results:

apply LF-MCMC to the inference for stable modelsmake the method applicable to general cases

relatively low computational cost

Pitfalls:





Concluding remarks

Our results:

apply LF-MCMC to the inference for stable modelsmake the method applicable to general casesrelatively low computational cost

Pitfalls:





Concluding remarks

Our results:


Pitfalls:

need to specify a proper target ε value

don’t know when convergence will happen, need moreiterationschoice of summary statistics can crucially affect samplerperformance




Concluding remarks

Our results:


Pitfalls:

need to specify a proper target ε valuedon’t know when convergence will happen, need moreiterations

choice of summary statistics can crucially affect samplerperformance




Concluding remarks

Our results:


Pitfalls:





Acknowledgement

Thanks to

My supervisorDr. Ray Kawai


likelihood-free mcmc...about mcmc algorithms that realize markov chain we want the invariant...

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