Accelerating Metropolis-Hastings algorithms by Delayed Acceptance

Marco Banterle; Clara Grazian; Anthony Lee; Christian P. Robert

doi:10.3934/fods.2019005

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June 2019, 1(2): 103-128. doi: 10.3934/fods.2019005

Accelerating Metropolis-Hastings algorithms by Delayed Acceptance

Marco Banterle ^1,, , Clara Grazian ^2, , Anthony Lee ^3, and Christian P. Robert ^4,

1.	Department of Medical Statistics, London School of Hygiene and Tropical Medicine, Keppel St, Bloomsbury, London WC1E 7HT, UK
2.	Dipartimento di Economia, Università degli Studi "Gabriele D'Annunzio", Viale Pindaro, 42, 65127 Pescara, Italy
3.	School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
4.	Department of Statistics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK

^* Corresponding author: Christian Robert

Published April 2019

MCMC algorithms such as Metropolis--Hastings algorithms are slowed down by the computation of complex target distributions as exemplified by huge datasets. We offer a useful generalisation of the Delayed Acceptance approach, devised to reduce such computational costs by a simple and universal divide-and-conquer strategy. The generic acceleration stems from breaking the acceptance step into several parts, aiming at a major gain in computing time that out-ranks a corresponding reduction in acceptance probability. Each component is sequentially compared with a uniform variate, the first rejection terminating this iteration. We develop theoretical bounds for the variance of associated estimators against the standard Metropolis--Hastings and produce results on optimal scaling and general optimisation of the procedure.

Keywords: Large Scale Learning and Big Data, MCMC algorithms, likelihood function, acceptance probability, mixtures of distributions, Jeffreys prior.

Mathematics Subject Classification: Primary: 68U20, 65C40; Secondary: 62C10.

Citation: Marco Banterle, Clara Grazian, Anthony Lee, Christian P. Robert. Accelerating Metropolis-Hastings algorithms by Delayed Acceptance. Foundations of Data Science, 2019, 1 (2) : 103-128. doi: 10.3934/fods.2019005

References:

[1]	C. Andrieu, A. Lee and M. Vihola, Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers, Bernoulli, 24 (2018), 842-872. doi: 10.3150/15-BEJ785. Google Scholar
[2]	E. Angelino, E. Kohler, A. Waterland, M. Seltzer and R. Adams, Accelerating MCMC via parallel predictive prefetching, UAI'14 Proceedings of the Thirtieth Conference on Uncertainty in Artificial Intelligence, (2014), 22-31. Google Scholar
[3]	R. Bardenet, A. Doucet and C. Holmes, On Markov chain Monte Carlo methods for tall data, The Journal of Machine Learning Research, 18 (2017), Paper No. 47, 43 pp. Google Scholar
[4]	A. Brockwell, Parallel Markov chain Monte Carlo simulation by pre-fetching, J. Comput. Graphical Stat., 15 (2006), 246-261. doi: 10.1198/106186006X100579. Google Scholar
[5]	J. Christen and C. Fox, Markov chain Monte Carlo using an approximation, Journal of Computational and Graphical Statistics, 14 (2005), 795-810. doi: 10.1198/106186005X76983. Google Scholar
[6]	O. F. Christensen, G. O. Roberts and J. S. Rosenthal, Scaling limits for the transient phase of local Metropolis-Hastings algorithms, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67 (2005), 253-268. doi: 10.1111/j.1467-9868.2005.00500.x. Google Scholar
[7]	R. Cornish, P. Vanetti, A. Bouchard-Côté, G. Deligiannidis and A. Doucet, Scalable Metropolis-Hastings for exact Bayesian inference with large datasets, arXiv preprint, arXiv: 1901.09881. Google Scholar
[8]	L. Devroye, Nonuniform Random Variate Generation, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8643-8. Google Scholar
[9]	J. Diebolt and C. P. Robert, Estimation of finite mixture distributions by Bayesian sampling, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 56 (1994), 363-375. doi: 10.1111/j.2517-6161.1994.tb01985.x. Google Scholar
[10]	C. Fox and G. Nicholls, Sampling conductivity images via MCMC, The Art and Science of Bayesian Image Analysis, (1997), 91-100. Google Scholar
[11]	A. Gelfand and S. Sahu, On Markov chain Monte Carlo acceleration, J. Comput. Graph. Statist., 3 (1994), 261-276. doi: 10.2307/1390911. Google Scholar
[12]	M. Girolami and B. Calderhead, Riemann manifold Langevin and Hamiltonian Monte Carlo methods, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73 (2011), 123-214. doi: 10.1111/j.1467-9868.2010.00765.x. Google Scholar
[13]	A. Golightly, D. Henderson and C. Sherlock, Delayed acceptance particle MCMC for exact inference in stochastic kinetic models, Statistics and Computing, 25 (2015), 1039-1055. doi: 10.1007/s11222-014-9469-x. Google Scholar
[14]	H. Jeffreys, Theory of Probability, 1st ed. The Clarendon Press, Oxford, 1939. Google Scholar
[15]	A. Korattikara, Y Chen and M. Welling, Austerity in MCMC land: Cutting the Metropolis-Hastings budget, In ICML 2014, International Conference on Machine Learning, (2014), 181-189. Google Scholar
[16]	G. MacLachlan and D. Peel, Finite Mixture Models, John Wiley, New York, 2000. doi: 10.1002/0471721182. Google Scholar
[17]	K.L. Mengersen and R. Tweedie, Rates of convergence of the Hastings and Metropolis algorithms, Ann. Statist., 24 (1996), 101-121. doi: 10.1214/aos/1033066201. Google Scholar
[18]	P. Neal and G. O. Roberts, Optimal scaling of random walk Metropolis algorithms with non-Gaussian proposals, Methodology and Computing in Applied Probability, 13 (2011), 583-601. doi: 10.1007/s11009-010-9176-9. Google Scholar
[19]	R. Neal, Markov chain Monte Carlo methods based on 'slicing' the density function, Tech. rep., University of Toronto, 1997. Google Scholar
[20]	W. Neiswanger, C. Wang and E. Xing, Asymptotically exact, embarrassingly parallel MCMC, arXiv preprint, 2013, arXiv: 1311.4780. Google Scholar
[21]	P. Peskun, Optimum Monte Carlo sampling using Markov chains, Biometrika, 60 (1973), 607-612. doi: 10.1093/biomet/60.3.607. Google Scholar
[22]	M. Plummer, N. Best, K. Cowles and K. Vines, CODA: Convergence diagnosis and output analysis for MCMC, R News, 6 (2006), 7-11. Google Scholar
[23]	C. P. Robert, The Bayesian Choice, 2nd ed. Springer-Verlag, New York, 2001. Google Scholar
[24]	C. P. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4145-2. Google Scholar
[25]	C. P. Robert and D. M. Titterington, Reparameterisation strategies for hidden Markov models and Bayesian approaches to maximum likelihood estimation, Statistics and Computing, 8 (1998), 145-158. Google Scholar
[26]	G. O. Roberts, A. Gelman and W. R. Gilks, Weak convergence and optimal scaling of random walk Metropolis algorithms, Ann. Appl. Probab., 7 (1997), 110-120. doi: 10.1214/aoap/1034625254. Google Scholar
[27]	G. O. Roberts and J. S. Rosenthal, Optimal scaling for various Metropolis-Hastings algorithms, Statist. Science, 16 (2001), 351-367. doi: 10.1214/ss/1015346320. Google Scholar
[28]	G. O. Roberts and J. S. Rosenthal, Coupling and ergodicity of adaptive MCMC, J. Applied Proba., 44 (2005), 458-475. doi: 10.1239/jap/1183667414. Google Scholar
[29]	G. O. Roberts and O. Stramer, Langevin diffusions and Metropolis-Hastings algorithms, Methodology and Computing in Applied Probability, 4 (2002), 337-357. doi: 10.1023/A:1023562417138. Google Scholar
[30]	G. O. Roberts and R. Tweedie, Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms, Biometrika, 83 (1996), 95-110. doi: 10.1093/biomet/83.1.95. Google Scholar
[31]	K. Roeder and L. Wasserman, Practical Bayesian density estimation using mixtures of Normals, J. American Statist. Assoc., 92 (1997), 894-902. doi: 10.1080/01621459.1997.10474044. Google Scholar
[32]	S. Scott, A. Blocker, F. Bonassi, M. Chipman, E. George and R. McCulloch, Bayes and big data: The consensus Monte Carlo algorithm, EFaBBayes 250 Conference, 16 (2013). Google Scholar
[33]	C. Sherlock, A. Golightly and D. A. Henderson, Adaptive, delayed-acceptance MCMC for targets with expensive likelihoods, Journal of Computational and Graphical Statistics, 26 (2017), 434-444. doi: 10.1080/10618600.2016.1231064. Google Scholar
[34]	C. Sherlock and G. O. Roberts, Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets, Bernoulli, 15 (2009), 774-798. doi: 10.3150/08-BEJ176. Google Scholar
[35]	C. Sherlock, A. Thiery and A. Golightly, Efficiency of delayed-acceptance random walk Metropolis algorithms, arXiv preprint, 2015, arXiv: 1506.08155. Google Scholar
[36]	C. Sherlock, A. Thiery, G. O. Roberts and J. S. Rosenthal, On the efficiency of pseudo-marginal random walk Metropolis algorithms, The Annals of Statistics, 43 (2015), 238-275. doi: 10.1214/14-AOS1278. Google Scholar
[37]	A. Y. Shestopaloff and R. M. Neal, MCMC for non-linear state space models using ensembles of latent sequences, arXiv preprint, 2013, arXiv: 1305.0320. Google Scholar
[38]	M. Stephens, Bayesian Methods for Mixtures of Normal Distributions, Ph.D. thesis, University of Oxford, 1997. Google Scholar
[39]	I. Strid, Efficient parallelisation of Metropolis-Hastings algorithms using a prefetching approach, Computational Statistics & Data Analysis, 54 (2010), 2814-2835. doi: 10.1016/j.csda.2009.11.019. Google Scholar
[40]	L. Tierney, A note on Metropolis-Hastings kernels for general state spaces, Ann. Appl. Probab., 8 (1998), 1-9. doi: 10.1214/aoap/1027961031. Google Scholar
[41]	L. Tierney and A. Mira, Some adaptive Monte Carlo methods for Bayesian inference, Statistics in Medicine, 18 (1998), 2507-2515. Google Scholar
[42]	X. Wang and D. Dunson, Parallel MCMC via Weierstrass sampler, arXiv preprint, 2013, arXiv: 1312.4605. Google Scholar

show all references

References:

[1]	C. Andrieu, A. Lee and M. Vihola, Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers, Bernoulli, 24 (2018), 842-872. doi: 10.3150/15-BEJ785. Google Scholar
[2]	E. Angelino, E. Kohler, A. Waterland, M. Seltzer and R. Adams, Accelerating MCMC via parallel predictive prefetching, UAI'14 Proceedings of the Thirtieth Conference on Uncertainty in Artificial Intelligence, (2014), 22-31. Google Scholar
[3]	R. Bardenet, A. Doucet and C. Holmes, On Markov chain Monte Carlo methods for tall data, The Journal of Machine Learning Research, 18 (2017), Paper No. 47, 43 pp. Google Scholar
[4]	A. Brockwell, Parallel Markov chain Monte Carlo simulation by pre-fetching, J. Comput. Graphical Stat., 15 (2006), 246-261. doi: 10.1198/106186006X100579. Google Scholar
[5]	J. Christen and C. Fox, Markov chain Monte Carlo using an approximation, Journal of Computational and Graphical Statistics, 14 (2005), 795-810. doi: 10.1198/106186005X76983. Google Scholar
[6]	O. F. Christensen, G. O. Roberts and J. S. Rosenthal, Scaling limits for the transient phase of local Metropolis-Hastings algorithms, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67 (2005), 253-268. doi: 10.1111/j.1467-9868.2005.00500.x. Google Scholar
[7]	R. Cornish, P. Vanetti, A. Bouchard-Côté, G. Deligiannidis and A. Doucet, Scalable Metropolis-Hastings for exact Bayesian inference with large datasets, arXiv preprint, arXiv: 1901.09881. Google Scholar
[8]	L. Devroye, Nonuniform Random Variate Generation, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8643-8. Google Scholar
[9]	J. Diebolt and C. P. Robert, Estimation of finite mixture distributions by Bayesian sampling, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 56 (1994), 363-375. doi: 10.1111/j.2517-6161.1994.tb01985.x. Google Scholar
[10]	C. Fox and G. Nicholls, Sampling conductivity images via MCMC, The Art and Science of Bayesian Image Analysis, (1997), 91-100. Google Scholar
[11]	A. Gelfand and S. Sahu, On Markov chain Monte Carlo acceleration, J. Comput. Graph. Statist., 3 (1994), 261-276. doi: 10.2307/1390911. Google Scholar
[12]	M. Girolami and B. Calderhead, Riemann manifold Langevin and Hamiltonian Monte Carlo methods, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73 (2011), 123-214. doi: 10.1111/j.1467-9868.2010.00765.x. Google Scholar
[13]	A. Golightly, D. Henderson and C. Sherlock, Delayed acceptance particle MCMC for exact inference in stochastic kinetic models, Statistics and Computing, 25 (2015), 1039-1055. doi: 10.1007/s11222-014-9469-x. Google Scholar
[14]	H. Jeffreys, Theory of Probability, 1st ed. The Clarendon Press, Oxford, 1939. Google Scholar
[15]	A. Korattikara, Y Chen and M. Welling, Austerity in MCMC land: Cutting the Metropolis-Hastings budget, In ICML 2014, International Conference on Machine Learning, (2014), 181-189. Google Scholar
[16]	G. MacLachlan and D. Peel, Finite Mixture Models, John Wiley, New York, 2000. doi: 10.1002/0471721182. Google Scholar
[17]	K.L. Mengersen and R. Tweedie, Rates of convergence of the Hastings and Metropolis algorithms, Ann. Statist., 24 (1996), 101-121. doi: 10.1214/aos/1033066201. Google Scholar
[18]	P. Neal and G. O. Roberts, Optimal scaling of random walk Metropolis algorithms with non-Gaussian proposals, Methodology and Computing in Applied Probability, 13 (2011), 583-601. doi: 10.1007/s11009-010-9176-9. Google Scholar
[19]	R. Neal, Markov chain Monte Carlo methods based on 'slicing' the density function, Tech. rep., University of Toronto, 1997. Google Scholar
[20]	W. Neiswanger, C. Wang and E. Xing, Asymptotically exact, embarrassingly parallel MCMC, arXiv preprint, 2013, arXiv: 1311.4780. Google Scholar
[21]	P. Peskun, Optimum Monte Carlo sampling using Markov chains, Biometrika, 60 (1973), 607-612. doi: 10.1093/biomet/60.3.607. Google Scholar
[22]	M. Plummer, N. Best, K. Cowles and K. Vines, CODA: Convergence diagnosis and output analysis for MCMC, R News, 6 (2006), 7-11. Google Scholar
[23]	C. P. Robert, The Bayesian Choice, 2nd ed. Springer-Verlag, New York, 2001. Google Scholar
[24]	C. P. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4145-2. Google Scholar
[25]	C. P. Robert and D. M. Titterington, Reparameterisation strategies for hidden Markov models and Bayesian approaches to maximum likelihood estimation, Statistics and Computing, 8 (1998), 145-158. Google Scholar
[26]	G. O. Roberts, A. Gelman and W. R. Gilks, Weak convergence and optimal scaling of random walk Metropolis algorithms, Ann. Appl. Probab., 7 (1997), 110-120. doi: 10.1214/aoap/1034625254. Google Scholar
[27]	G. O. Roberts and J. S. Rosenthal, Optimal scaling for various Metropolis-Hastings algorithms, Statist. Science, 16 (2001), 351-367. doi: 10.1214/ss/1015346320. Google Scholar
[28]	G. O. Roberts and J. S. Rosenthal, Coupling and ergodicity of adaptive MCMC, J. Applied Proba., 44 (2005), 458-475. doi: 10.1239/jap/1183667414. Google Scholar
[29]	G. O. Roberts and O. Stramer, Langevin diffusions and Metropolis-Hastings algorithms, Methodology and Computing in Applied Probability, 4 (2002), 337-357. doi: 10.1023/A:1023562417138. Google Scholar
[30]	G. O. Roberts and R. Tweedie, Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms, Biometrika, 83 (1996), 95-110. doi: 10.1093/biomet/83.1.95. Google Scholar
[31]	K. Roeder and L. Wasserman, Practical Bayesian density estimation using mixtures of Normals, J. American Statist. Assoc., 92 (1997), 894-902. doi: 10.1080/01621459.1997.10474044. Google Scholar
[32]	S. Scott, A. Blocker, F. Bonassi, M. Chipman, E. George and R. McCulloch, Bayes and big data: The consensus Monte Carlo algorithm, EFaBBayes 250 Conference, 16 (2013). Google Scholar
[33]	C. Sherlock, A. Golightly and D. A. Henderson, Adaptive, delayed-acceptance MCMC for targets with expensive likelihoods, Journal of Computational and Graphical Statistics, 26 (2017), 434-444. doi: 10.1080/10618600.2016.1231064. Google Scholar
[34]	C. Sherlock and G. O. Roberts, Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets, Bernoulli, 15 (2009), 774-798. doi: 10.3150/08-BEJ176. Google Scholar
[35]	C. Sherlock, A. Thiery and A. Golightly, Efficiency of delayed-acceptance random walk Metropolis algorithms, arXiv preprint, 2015, arXiv: 1506.08155. Google Scholar
[36]	C. Sherlock, A. Thiery, G. O. Roberts and J. S. Rosenthal, On the efficiency of pseudo-marginal random walk Metropolis algorithms, The Annals of Statistics, 43 (2015), 238-275. doi: 10.1214/14-AOS1278. Google Scholar
[37]	A. Y. Shestopaloff and R. M. Neal, MCMC for non-linear state space models using ensembles of latent sequences, arXiv preprint, 2013, arXiv: 1305.0320. Google Scholar
[38]	M. Stephens, Bayesian Methods for Mixtures of Normal Distributions, Ph.D. thesis, University of Oxford, 1997. Google Scholar
[39]	I. Strid, Efficient parallelisation of Metropolis-Hastings algorithms using a prefetching approach, Computational Statistics & Data Analysis, 54 (2010), 2814-2835. doi: 10.1016/j.csda.2009.11.019. Google Scholar
[40]	L. Tierney, A note on Metropolis-Hastings kernels for general state spaces, Ann. Appl. Probab., 8 (1998), 1-9. doi: 10.1214/aoap/1027961031. Google Scholar
[41]	L. Tierney and A. Mira, Some adaptive Monte Carlo methods for Bayesian inference, Statistics in Medicine, 18 (1998), 2507-2515. Google Scholar
[42]	X. Wang and D. Dunson, Parallel MCMC via Weierstrass sampler, arXiv preprint, 2013, arXiv: 1312.4605. Google Scholar

Figure 1. Fit of a two-step Metropolis-Hastings algorithm applied to a normal-normal posterior distribution $ \mu|x\sim N(x/(\{1+\sigma_\mu^{-2}\}, 1/\{1+\sigma_\mu^{-2}\}) $ when $ x = 3 $ and $ \sigma_\mu = 10 $, based on $ T = 10^5 $ iterations and a first acceptance step considering the likelihood ratio and a second acceptance step considering the prior ratio, resulting in an overall acceptance rate of 12%

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Figure 2. (left) Fit of a multiple-step Metropolis-Hastings algorithm applied to a Beta-binomial posterior distribution $ p|x\sim Be(x+a, n+b-x) $ when $ N = 100 $, $ x = 32 $, $ a = 7.5 $ and $ b = .5 $. The binomial $ \mathcal{B}(N, p) $ likelihood is replaced with a product of $ 100 $ Bernoulli terms and an acceptance step is considered for the ratio of each term. The histogram is based on $ 10^5 $ iterations, with an overall acceptance rate of 9%; (centre) raw sequence of successive values of $ p $ in the Markov chain simulated in the above experiment; (right) autocorrelogram of the above sequence

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Figure 3. Two top panels: behaviour of $\ell^*(\delta)$ and $\alpha^*(\delta)$ as the relative cost varies. Note that for $\delta >> 1$ the optimal values converges towards the values computed for the standard Metropolis--Hastings (dashed in red). Two bottom panels: close--up of the interesting region for $0 < \delta < 1$.

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27] is met for $\delta = 1$.">Figure 4. Optimal acceptance rate for the DA-MALA algorithm as a function of $\delta$. In red, the optimal acceptance rate for MALA obtained by ^[27] is met for $\delta = 1$.

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Figure 5. Comparison between geometric MALA (top panels) and geometric MALA with Delayed Acceptance (bottom panels): marginal chains for two arbitrary components (left), estimated marginal posterior density for an arbitrary component (middle), 1D chain trace evaluating mixing (right).

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Table 1. Comparison between MH and MH with Delayed Acceptance on a logistic model. ESS is the effective sample size, ESJD the expected square jumping distance, time is the computation time

Algorithm	rel. ESS (av.)	rel. ESJD (av.)	rel. Time (av.)	rel. gain (ESS)(av.)	rel. gain (ESJD)(av.)
DA-MH over MH	1.1066	12.962	0.098	5.47	56.18

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Algorithm	rel. ESS (av.)	rel. ESJD (av.)	rel. Time (av.)	rel. gain (ESS)(av.)	rel. gain (ESJD)(av.)
DA-MH over MH	1.1066	12.962	0.098	5.47	56.18

Table 2. Comparison between standard geometric MALA and geometric MALA with Delayed Acceptance, with ESS the effective sample size, ESJD the expected square jumping distance, time the computation time and a the observed acceptance rate

Algorithm	ESS (av.)	(sd)	ESJD (av.)	(sd)	time (av.)	(sd)	a(aver.)	ESS/time (aver.)	ESJD/time (aver.)
MALA	7504.48	107.21	5244.94	983.47	176078	1562.3	0.661	0.04	0.03
DA-MALA	6081.02	121.42	5373.253	2148.76	17342.91	6688.3	0.09	0.35	0.31

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Algorithm	ESS (av.)	(sd)	ESJD (av.)	(sd)	time (av.)	(sd)	a(aver.)	ESS/time (aver.)	ESJD/time (aver.)
MALA	7504.48	107.21	5244.94	983.47	176078	1562.3	0.661	0.04	0.03
DA-MALA	6081.02	121.42	5373.253	2148.76	17342.91	6688.3	0.09	0.35	0.31

Table 3. Comparison using different performance indicators in the example of mixture estimation, based on 100 replicas of the experiments according to model (9) with a sample size $ n = 500 $, $ 10^5 $ MH simulations and $ 500 $ samples for the prior estimation. ("ESS" is the effective sample size, "time" is the computational time). The actual averaged gain ($ \frac{ESS_{DA}/ESS_{MH}}{time_{DA}/time_{MH}} $) is $ 9.58 $, higher than the "double average" that the table above suggests as being around $ 5 $

Algorithm	ESS (av.)	(sd)	ESJD (av.)	(sd)	time (av.)	(sd)
MH	1575.96	245.96	0.226	0.44	513.95	57.81
MH + DA	628.77	87.86	0.215	0.45	42.22	22.95

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Algorithm	ESS (av.)	(sd)	ESJD (av.)	(sd)	time (av.)	(sd)
MH	1575.96	245.96	0.226	0.44	513.95	57.81
MH + DA	628.77	87.86	0.215	0.45	42.22	22.95

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American Institute of Mathematical Sciences

Accelerating Metropolis-Hastings algorithms by Delayed Acceptance

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American Institute of Mathematical Sciences

Accelerating Metropolis-Hastings algorithms by Delayed Acceptance

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