The Block-Correlated Pseudo Marginal Sampler for State Space Models

Figures & data

Table 1 Comparing different sorting methods for various combinations of N and d in terms of CPU time (in seconds): I: The fast Euclidean sorting algorithm, II: Choppala et al. (2016) sorting method, and III: Hilbert sorting method.

d	N	I	II	III	I actual
10	500	1.00	30.50	87.50	0.0002
	1000	1.00	66.67	115.00	0.0003
	2000	1.00	172.40	141.00	0.0005
30	500	1.00	39.67	141.00	0.0003
	1000	1.00	113.00	214.00	0.0004
	2000	1.00	289.80	335.00	0.0005

NOTE: The entries in columns I–III are multiples of the time in column I. The column headed “I actual” is the actual time for column I; for example the 30.5 in column II and row 1 means that I is 30.5 times faster than the Choppala et al. (2016) sorting. The corresponding entry for I actual is 0.0002.

Table 2 Comparing the variance of the log of the estimated likelihood for five different estimators of the likelihood: I: Averaging the likelihood (0% trimmed mean), II: Averaging the likelihood (5% trimmed mean), III: Averaging the likelihood (10% trimmed mean), IV: Averaging the likelihood (25% trimmed mean), and V: Averaging the likelihood (50% trimmed mean), for d = 10 dimensional linear Gaussian state space model with T = 300.

N	S	I	II	III	IV	V	VI	V actual
100	1						563.65
	20	3.26	2.02	1.48	1.11	1.00		29.30
	100	11.47	2.60	1.80	1.22	1.00		5.73
	1000	70.45	2.32	1.58	1.01	1.00		0.66
250	1						629.83
	20	3.60	2.03	1.67	1.12	1.00		15.75
	100	11.90	2.48	1.56	1.12	1.00		3.20
	1000	76.55	2.24	1.62	1.12	1.00		0.32
1000	1						620.74
	20	3.69	1.94	1.51	0.99	1.00		6.14
	100	13.75	2.50	1.79	1.12	1.00		1.16
	1000	82.18	2.26	1.59	1.08	1.00		0.13

NOTE: The variance of the log of the estimated likelihood of a single particle filter is reported in column VI. The results are based on 1000 independent runs. The entries in columns headed I–V are relative to the variance in column V. The entries in column “V actual” are the actual variances for estimator V. For example, the entry on row 4, column I is 70.45, which means the variance of the log of the likelihood estimate is 70.45 times the corresponding variance in column V. The entries in column VI are relative to column V with S = 1000 particles.

Table 3 Comparing the performance of different PMMH samplers with different number of particle filters S and different number of particles N in each particle filter for estimating the linear Gaussian state space model using a simulate dataset with T = 300 and d = 10 dimensions.

	I	II		III		IV		V		V actual
N	50,000	250	500	250	500	250	500	250	500	250	500
S	1	100	100	100	100	100	100	100	100	100	100
$\widehat{\text{IF}}$	50.73	88.09	4.59	3.03	1.21	1.40	1.10	1.00	0.90	9.98	9.00
CT	8.18	1.00	2.00	1.00	2.00	1.00	2.00	1.00	2.00	0.99	1.98
$\widehat{\text{RTNIF}}$	415.08	88.10	9.17	3.03	2.42	1.40	2.20	1.00	1.80	9.88	17.82

NOTE: Sampler I: Correlated PMMH of Deligiannidis, Doucet, and Pitt (2018). Sampler II: MPM with 5% trimmed mean. Sampler III: MPM with 10% trimmed mean. Sampler IV: MPM with 25% trimmed mean. Sampler V: MPM with 50% trimmed mean. Time denotes the time taken in seconds for one iteration of the method. The $\widehat{\text{IF}}$, CT, and $\widehat{\text{RTNIF}}$ entries in columns headed I to V are relative to the entries in column V (MPM with 50% trimmed mean with N = 250 and S = 100). The entries in column “V actual” are the actual $\widehat{\text{IF}}$, CT, and $\widehat{\text{RTNIF}}$ values for estimator V. For example, the entry on row 5, column I is 415.08, which means the $\widehat{\text{RTNIF}}$ is 415.08 times the corresponding $\widehat{\text{RTNIF}}$ in column V (MPM with 50% trimmed mean with N = 250 and S = 100).

Table 4 Comparing the performance of different PMMH samplers with different number of particle filters S and different number of particles N in each particle filter for estimating multivariate GARCH diffusion-driven volatility in mean model using a real dataset with T = 100 and d = 30.

Param.	I			II
N	250	250	500	250	250
S	100	200	100	100	200
ψ	183.559	124.258	227.685	151.745	80.073
μ	74.070	49.250	384.767	74.485	45.554
a	46.159	45.662	126.968	32.721	14.753
${\tau }^2$	102.234	70.255	218.866	78.171	42.075
$\widehat{{\text{IF}}_{\mathrm{\text{MAX}}}}$	183.559	124.258	384.767	151.745	80.073
${\widehat{\text{TNIF}}}_{\mathrm{\text{MAX}}}$	523.143	877.262	3351.321	432.473	565.315
${\widehat{\text{RTNIF}}}_{\mathrm{\text{MAX}}}$	1	1.677	6.406	0.827	1.081
$\widehat{{\text{IF}}_{\text{MEAN}}}$	101.505	72.356	239.571	84.280	45.614
${\widehat{\text{TNIF}}}_{\text{MEAN}}$	289.289	510.833	2077.953	240.198	322.035
${\widehat{\text{RTNIF}}}_{\text{MEAN}}$	1	1.766	7.183	0.830	1.113
Time	2.85	7.06	8.71	2.85	7.06

NOTE: The model parameters are ψ, μ, a, and ${\tau }^2$. Sampler I: MPM with 25% trimmed mean. Sampler II: MPM with 50% trimmed mean. Time is the time in seconds for one iteration of the method. The RTNIF is the TNIF relative to the MPM with 25% trimmed mean with S = 100 and N = 250.

Table 5 Comparing the performance of different PMMH samplers using different number of particle filters S and different number of particles N in each particle filter for estimating the small scale DSGE model using a real dataset with T = 124 observations.

	I	II	III	IV	V	VI	VII	VIII	IX	VI actual
S	100	100	100	100	100	100	100	100	1	100
N	250	100	100	100	100	100	100	250	20000	100
${\widehat{\text{IF}}}_{\mathrm{\text{MAX}}}$	3.73	0.25	0.22	0.25	0.28	1.00	0.39	NA	2.42	604.72
${\widehat{\text{RTNIF}}}_{\mathrm{\text{MAX}}}$	36.27	$\mathbf{1.26}$	$\mathbf{1.11}$	$\mathbf{1.25}$	1.54	$\mathbf{1.00}$	1.96	NA	114.37	66.52
${\widehat{\text{IF}}}_{\text{MEAN}}$	5.88	0.49	0.50	0.49	0.61	1.00	0.59	NA	3.32	190.74
${\widehat{\text{TNIF}}}_{\text{MEAN}}$	57.23	$\mathbf{2.47}$	$\mathbf{2.49}$	$\mathbf{2.44}$	3.41	$\mathbf{1.00}$	2.93	NA	157.02	20.98
Time	9.73	5.00	5.00	5.00	5.55	1.00	5.00	6.82	47.27	0.11

NOTE: Sampler I: MPM (0% trimmed mean, ${\rho }_u=0.9$, disturbance sorting, ADPF). Sampler II: MPM (10% trimmed mean, ${\rho }_u=0.9$, disturbance sorting, ADPF). Sampler III: MPM (25% trimmed mean, ${\rho }_u=0.9$, disturbance sorting, ADPF). Sampler IV: MPM (25% trimmed mean, ${\rho }_u=0$, disturbance sorting, ADPF). Sampler V: MPM (25% trimmed mean, ${\rho }_u=0.9$, state sorting, ADPF). Sampler VI: DA-MPM (25% trimmed mean, ${\rho }_u=0.9$, disturbance sorting, ADPF). Sampler VII: MPM (50% trimmed mean, ${\rho }_u=0.9$, disturbance sorting, ADPF). Sampler VIII: MPM (0% trimmed mean, ${\rho }_u=0.9$, disturbance sorting, bootstrap). Sampler IX: Correlated PMMH of Deligiannidis, Doucet, and Pitt (2018). Time is the time taken in seconds for one iteration of the method. The ${\widehat{\text{IF}}}_{\mathrm{\text{MAX}}},{\widehat{\text{RTNIF}}}_{\mathrm{\text{MAX}}},{\widehat{\text{IF}}}_{\text{MEAN}}$, and ${\widehat{\text{RTNIF}}}_{\text{MEAN}}$ entries in columns headed I to IX are relative to the entries in column VI. The entries in column “VI actual” are the actual ${\widehat{\text{IF}}}_{\mathrm{\text{MAX}}},{\widehat{\text{RTNIF}}}_{\mathrm{\text{MAX}}},{\widehat{\text{IF}}}_{\text{MEAN}}$, and ${\widehat{\text{RTNIF}}}_{\text{MEAN}}$ for estimator VI. The numbers in bold are for the sampler with similar values of the ${\widehat{\text{RTNIF}}}_{\mathrm{\text{MAX}}}$ and ${\widehat{\text{RTNIF}}}_{\text{MEAN}}$.

Table 6 Comparing the performance of different PMMH samplers using different numbers of particle filters S and different numbers of particles N in each particle filter for estimating the medium scale DSGE model using the U.S. quarterly dataset from 1983Q1 to 2007Q4.

	I	II	I actual	I	II	I actual
${\widehat{\text{IF}}}_{\mathrm{\text{MAX}}}$	1.00	2.74	796.82	1.00	1.48	383.31
${\widehat{\text{RTNIF}}}_{\mathrm{\text{MAX}}}$	1.00	26.51	1338.66	1.00	14.37	643.96
${\widehat{\text{IF}}}_{\text{MEAN}}$	1.00	2.78	428.71	1.00	0.96	275.34
${\widehat{\text{RTNIF}}}_{\text{MEAN}}$	1.00	26.92	720.22	1.00	9.31	462.57
Time	1.00	9.68	1.68	1.00	9.68	1.68

NOTE: Sampler I: MPM (50% trimmed mean, ${\rho }_u=0.9$, disturbance sorting, ADPF) with S = 250 and N = 100. Sampler II: Correlated PMMH with $N=50,000$. The first three columns are for the case when the measurement error variances are estimated. The ${\widehat{\text{IF}}}_{\mathrm{\text{MAX}}},{\widehat{\text{RTNIF}}}_{\mathrm{\text{MAX}}},{\widehat{\text{IF}}}_{\text{MEAN}}$, and ${\widehat{\text{RTNIF}}}_{\text{MEAN}}$ entries in columns headed I–II are relative to the entries in column I. The entries in column “I actual” are the actual ${\widehat{\text{IF}}}_{\mathrm{\text{MAX}}},{\widehat{\text{RTNIF}}}_{\mathrm{\text{MAX}}},{\widehat{\text{IF}}}_{\text{MEAN}}$, and ${\widehat{\text{RTNIF}}}_{\text{MEAN}}$ for estimator I. The last three columns are for the case when the measurement error variances are fixed at 25% of the sample variance of the observables. The ${\widehat{\text{IF}}}_{\mathrm{\text{MAX}}},{\widehat{\text{RTNIF}}}_{\mathrm{\text{MAX}}},{\widehat{\text{IF}}}_{\text{MEAN}}$, and ${\widehat{\text{RTNIF}}}_{\text{MEAN}}$ entries in columns headed I to II are relative to the entries in column I. The entries in column “I actual” are the actual ${\widehat{\text{IF}}}_{\mathrm{\text{MAX}}},{\widehat{\text{RTNIF}}}_{\mathrm{\text{MAX}}},{\widehat{\text{IF}}}_{\text{MEAN}}$, and ${\widehat{\text{RTNIF}}}_{\text{MEAN}}$ for estimator I.

Fig. 1 Kernel density estimates of the posterior densities of some parameters of the medium scale DSGE model for the U.S. quarterly dataset from 1983Q1 to 2007Q4 estimated using: (a) MPM with 50% trimmed mean (${\rho }_u=0.9$, disturbance sorting, ADPF) with S = 250 and N = 100; (b) correlated PMMH with $N=50,000$. The measurement error variances are fixed to 25% of the variance of the observables.

Choppala, P., Gunawan, D., Chen, J., Tran, M. N., and Kohn, R. (2016), “Bayesian Inference for State-Space Models Using Block and Correlated Pseudo Marginal Methods,” arXiv:1612.07072v1.

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Deligiannidis, G., Doucet, A., and Pitt, M. (2018), “The Correlated Pseudo-Marginal Method,” Journal of Royal Statistical Society, Series B, 80, 839–870. DOI: 10.1111/rssb.12280.

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