Tractable skew-normal approximations via matching

Figures & data

Table 1. Mean marginal $L^1$ accuracies (Acc.) (expressed as a percentage) and times (in seconds) across probit regression simulations with independent covariate data. The highest accuracies for each value of p appear in bold.

	p = 2		p = 4		p = 8		p = 16		p = 24		p = 40
Method	Acc.	Time	Acc.	Time	Acc.	Time	Acc.	Time	Acc.	Time	Acc.	Time
$\mathbf{n}\mathbf{=}\mathbf{2}\mathbf{p}$
Laplace	76.7	0.01	75.0	0.00	72.1	0.00	68.8	0.00	68.3	0.00	62.2	0.01
DMVB	75.9	0.01	71.9	0.01	64.5	0.04	64.2	0.19	63.1	0.52	58.7	3.01
GVB	86.2	0.04	86.3	0.07	88.6	0.21	89.0	0.83	90.6	1.50	88.7	2.91
EP	86.0	0.01	89.5	0.00	92.8	0.01	94.3	0.01	95.8	0.04	96.0	0.10
SM	78.7	0.53	80.7	0.88	78.3	1.56	74.8	2.94	74.2	4.37	67.3	7.68
MM	96.4	2.13	94.8	2.78	93.7	3.00	92.1	3.41	90.4	3.85	83.0	4.96
DM	83.7	0.04	79.7	0.06	78.1	0.11	73.5	0.22	76.5	0.32	69.1	0.53
MMH-Jensen	90.9	0.04	89.7	0.08	88.2	0.16	85.5	0.38	85.9	0.60	80.7	1.52
MMH-IL	92.4	0.07	91.7	0.19	91.4	0.50	90.5	1.36	90.5	2.76	86.5	11.09
MMH-IS	94.3	2.12	93.6	2.78	93.3	2.99	91.7	3.43	90.1	3.85	85.6	4.99
MMC	96.5	2.13	95.0	2.78	94.8	3.00	92.6	3.41	90.5	3.81	83.0	4.96
$\mathbf{n}\mathbf{=}\mathbf{4}\mathbf{p}$
Laplace	78.6	0.00	83.3	0.00	86.2	0.00	90.3	0.00	91.4	0.00	92.7	0.01
DMVB	78.4	0.01	85.9	0.01	92.1	0.03	97.1	0.19	98.2	0.67	98.7	3.92
GVB	85.8	0.05	91.7	0.10	95.5	0.31	98.1	0.84	98.7	1.24	98.9	3.35
EP	85.6	0.01	93.2	0.01	96.4	0.01	98.5	0.02	98.9	0.05	99.2	0.13
SM	80.9	0.53	90.5	0.86	92.6	1.54	94.9	2.88	95.6	4.32	96.0	7.68
MM	97.0	2.11	97.4	2.72	98.0	2.97	98.4	3.43	98.5	3.94	98.3	5.19
DM	86.9	0.04	87.9	0.06	88.7	0.11	93.3	0.22	93.9	0.32	94.8	0.52
MMH-Jensen	93.9	0.05	95.6	0.08	96.6	0.18	97.8	0.43	98.2	0.82	98.3	2.55
MMH-IL	94.8	0.08	95.9	0.21	96.8	0.51	97.6	1.48	97.9	3.39	98.1	14.33
MMH-IS	95.9	2.11	96.2	2.72	96.9	2.97	97.8	3.44	98.0	3.94	98.0	5.25
MMC	96.7	2.11	97.3	2.72	98.0	2.97	98.6	3.42	98.6	3.91	98.4	5.22

Figure 1. Contour plots for the posterior and across various approximation methods, for a small probit regression example with p = 2 and n = 4.

Table 2. Mean marginal $L^1$ accuracies (Acc.) (expressed as a percentage) across probit regression benchmarks. The highest accuracies for each benchmark dataset appear in bold.

	O-rings		Liver		Diabetes		Glass		Breast		Heart		German		Ion.
Method	Acc.	Time	Acc.	Time	Acc.	Time	Acc.	Time	Acc.	Time	Acc.	Time	Acc.	Time	Acc.	Time
Laplace	92.5	0.17	98.0	0.00	98.3	0.00	92.4	0.00	95.3	0.05	95.7	0.00	97.9	0.11	86.0	0.03
DMVB	95.5	0.14	99.1	0.19	99.2	0.61	96.9	0.33	97.1	1.97	99.1	0.74	99.2	31.00	93.2	10.85
GVB	96.5	0.33	99.1	0.09	99.2	0.06	97.3	2.16	98.2	2.12	99.0	0.09	99.2	0.46	97.3	11.54
EP	96.6	0.32	99.1	0.14	99.2	0.14	97.8	0.08	98.4	0.30	99.1	0.11	99.2	0.67	98.6	0.35
SM	96.9	0.91	99.2	1.14	99.0	1.62	96.2	2.03	97.9	1.81	98.5	2.61	99.0	4.47	90.5	6.05
MM	99.1	2.41	99.1	3.39	99.1	4.03	98.2	3.23	98.6	5.36	98.9	3.64	99.0	7.64	96.5	5.95
DM	95.7	0.31	98.3	0.11	98.4	0.12	52.9	0.14	96.2	0.17	97.1	0.21	95.3	0.42	86.4	0.46
MMH-Jensen	98.5	0.36	99.3	0.37	99.2	0.93	97.3	0.42	98.4	2.20	99.1	1.06	99.1	21.46	96.0	5.75
MMH-IL	98.6	0.53	99.3	2.19	99.2	6.01	97.5	3.64	98.4	21.26	99.0	4.90	99.1	125.89	97.1	39.86
MMH-IS	98.5	2.45	99.2	3.45	99.2	4.06	97.6	3.31	98.4	5.42	98.9	3.70	99.0	7.69	96.7	6.04
MMC	98.8	2.61	99.2	3.40	99.2	4.02	98.1	3.23	98.6	5.35	98.9	3.63	99.0	7.65	96.6	5.95

Table 3. Mean total marginal $L^1$ improvement/deterioration of accuracies (expressed as a percentage) across probit regression post-hoc simulations with independent covariate data. The column ‘+’ indicates a mean total improvement in marginal $L^1$ accuracy, while the column ‘−’ indicates the mean total decrease in marginal $L^1$ accuracy across simulations. For example, if the marginal improvements in $L^1$ accuracy for a p = 4 example were $(5,-2,1,-3)$, then the total marginal $L^1$ improvement and deterioration are 6 and 5 respectively. Pairs where there is an overall improvement are highlighted in bold. Long dashes indicate no data.

	p = 2		p = 4		p = 8		p = 16		p = 24		p = 40
Adjust.	+	−	+	−	+	−	+	−	+	−	+	−
$\mathbf{n}\mathbf{=}\mathbf{2}\mathbf{p}$
DMVB
MMH	22.1	1.5	57.7	1.2	153.4	0.9	271.3	3.7	395.9	5.9	644.1	16.1
MMC	3.2	0.4	5.4	0.1	4.5	0.0	7.0	0.0	4.9	0.0	–	–
GVB
MMH	16.6	0.7	33.4	3.1	48.6	7.3	76.3	14.6	74.6	26.4	104.4	66.1
MMC	3.8	0.1	3.2	0.3	2.8	0.2	1.5	0.2	–	–	–	–
EP
MMH	7.8	0.5	12.5	5.6	21.7	17.4	27.1	40.4	15.6	82.1	11.5	209.6
MMC	4.2	0.4	7.0	0.3	8.4	0.0	7.2	0.0	–	–	–	–
$\mathbf{n}\mathbf{=}\mathbf{4}\mathbf{p}$
DMVB
MMH	25.9	0.5	31.2	0.3	30.1	1.5	14.1	7.0	7.1	13.2	4.2	26.3
MMC	4.7	0.2	7.3	0.0	8.0	0.0	7.0	0.0	6.1	0.0	4.3	0.0
GVB
MMH	20.2	0.1	19.6	1.4	14.6	2.7	5.6	9.6	3.8	15.6	2.5	27.9
MMC	3.6	0.2	4.2	0.0	3.8	0.2	2.8	0.2	2.1	0.3	1.1	0.3
EP
MMH	11.7	0.2	12.3	2.1	9.2	4.0	3.7	13.4	2.4	20.6	1.2	35.5
MMC	2.7	1.1	5.4	0.4	5.7	0.1	4.6	0.1	3.6	0.1	2.0	0.1

Table 4. Success rates, i.e. rates where a solution exists to Algorithm 4, (expressed as a percentage) averaged over base approximations for the mean-mode-covariance based skewness adjustment. The mean-mode-Hessian based adjustments had a 100% success rate.

Multiplier	p = 2	p = 4	p = 8	p = 16	p = 24	p = 40
2	92.0	79.3	48.7	17.3	0.7	0.0
4	93.3	94.0	93.3	97.3	96.7	96.7

Table 5. Mean total marginal improvement/deterioration in $L^1$ accuracies (expressed as a percentage) across probit regression post-hoc benchmarks. The ‘+’ columns indicates the total increases in marginal $L^1$ accuracy, while the ‘−’ column indicates the total decrease in marginal $L^1$ accuracy. For example, if the marginal improvements in $L^1$ accuracy for a p = 4 example were $(5,-2,1,-3)$, then the total marginal $L^1$ improvement and deterioration are 6 and 5 respectively. Pairs where there is an overall improvement are highlighted in bold. Long dashes indicate no data.

	O-rings		Liver		Diabetes		Glass		Breast		Heart		German		Ion.
Adjust.	+	–	+	–	+	–	+	–	+	–	+	–	+	–	+	–
DMVB
MMH	7.4	0.4	1.1	0.0	0.6	0.2	6.8	0.5	11.4	0.2	1.1	1.6	1.0	2.8	89.2	3.4
MMC	7.1	0.0	1.2	0.0	0.5	0.0	0.4	0.6	0.9	0.0	1.6	0.0	0.8	1.8	–	–
GVB
MMH	6.3	0.4	1.3	0.0	0.7	0.2	4.6	2.0	1.9	0.3	1.2	0.8	1.0	3.4	24.1	14.1
MMC	3.4	0.0	1.1	0.0	0.4	0.0	0.5	0.0	0.9	0.0	1.0	0.0	0.8	1.8	–	–
EP
MMH	6.4	0.4	1.3	0.0	0.7	0.3	1.8	3.4	1.4	1.2	1.0	1.0	1.0	3.0	1.7	34.0
MMC	4.1	0.0	1.2	0.0	0.4	0.0	0.5	0.4	0.9	0.0	1.0	0.0	0.8	1.9	–	–