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
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Classical Monte Carlo Integration
Bayes formula
f (θ|D) ∝ π (θ)P (D|θ)
f (θ|D): posteriorπ (θ): priorP (D|θ):likelihood
Evaluating integrals
NormalisationZ =
´π (θ)P (D|θ) dθ
Marginalisationf (θ|D) =
´f (θ, x |D) dx
ExpectationEf [h (θ)] =
´h (θ) f (θ|D) dθ
Suppose we can draw samplesθ(j) ∼ f (θ|D) , j = 1, . . . ,mEf [h (θ)] ≈ 1
m
∑mj=1 h
(θ(j))
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Rejection sampling
1 Sample x (i) ∼ g (x)
2 Accept x (i) with
probability f (x(i))/Mg(x(i)),then go to 1
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
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)).
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
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.
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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).
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
LF-RS exampleSimulation results
Results for ε = 1(left) and ε = 0.1(right)
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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.
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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.
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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).
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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′)
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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.
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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.
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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%
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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.
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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.
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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.
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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.
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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.
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Likelihood-free Rejection Sampling(LF-RS)Basic notions of MCMCLikelihood-free MCMCIllustrations of LF-MCMC
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.
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
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
Yuanwei Xu Likelihood-free MCMC
blue=stable fit, green=smoothed data
The figure is produced using J.P. Nolan’s STABLE program, available athttp://academic2.american.edu/∼jpnolan
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Concluding remarks
Our results:
apply LF-MCMC to the inference for stable modelsmake the method applicable to general cases
relatively 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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Concluding remarks
Our results:
apply LF-MCMC to the inference for stable modelsmake 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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Concluding remarks
Our results:
apply LF-MCMC to the inference for stable modelsmake the method applicable to general casesrelatively low computational cost
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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Concluding remarks
Our results:
apply LF-MCMC to the inference for stable modelsmake 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 moreiterations
choice of summary statistics can crucially affect samplerperformance
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Concluding remarks
Our results:
apply LF-MCMC to the inference for stable modelsmake 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
Yuanwei Xu Likelihood-free MCMC
Simple Monte CarloLikelihood-free Methods
Application to Financial DataConclusion
Acknowledgement
Thanks to
My supervisorDr. Ray Kawai
Yuanwei Xu Likelihood-free MCMC