Abstract
Approximate Bayesian Computation (ABC) is a popular approach for Bayesian modeling, when these models exhibit an intractable likelihood. However, during each proposal of ABC, a great number of simulators are required and each simulation is always time-consuming. The overall goal of this work is to avoid inefficient computational cost of ABC. A pre-judgment rule (PJR) is proposed, which mainly aims to judge the acceptance condition using a small fraction of simulators instead of the whole simulators, thus achieving less computational complexity. In addition, it provided a theoretical study of the error bounded caused by PJR Strategy. Finally, the methodology was illustrated with various examples. The empirical results show both the effectiveness and efficiency of PJR compared with the previous methods.
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- 1.
\(\rho (\mathbb {S}(x),\mathbb {S}(y))\) is replaced by \(\rho (x,y)\), similar with Step \(\mathbb {C}\)3.
- 2.
E.g., \(\zeta \)-kernel can be chosen as \(\pi _\zeta (x_1\vert x_2) = (1/\sqrt{2\pi }\zeta )\exp (-\Vert x_1-x_2\Vert ^2/2\zeta ^2)\).
- 3.
In Step A2, \(z_0\) is more complex. Checking the acceptance condition is equivalent to judging \(\frac{z}{c} > u\), where c is defined in Eq. 2 and \(u\sim \text {Uniform}(0,1)\).
- 4.
The total variation distance between two distribution P and Q, absolutely continuous w.r.t. measure \(\varOmega \), is defined as \(d_v(P,Q) \triangleq 1/2\int \nolimits _{\theta } \vert f_P(\theta ) - f_Q(\theta ) \vert d\varOmega (\theta )\), where \(f_P(\cdot )\) and \(f_Q(\cdot )\) are their respective densities.
- 5.
Measured in term of number of simulator.
- 6.
Measured in term of TVD with the true posterior distribution.
- 7.
Note that in experiment the total variational distance is estimated empirically owing to the absence of explicit formulae.
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Acknowledgement
This study was funded by Scientific research fund of North University of China (No. XJJ201803).
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Wang, Y., Yu, X., Qin, P., Chai, R., Qiao, G. (2020). Improving Approximate Bayesian Computation with Pre-judgment Rule. In: Zeng, J., Jing, W., Song, X., Lu, Z. (eds) Data Science. ICPCSEE 2020. Communications in Computer and Information Science, vol 1257. Springer, Singapore. https://doi.org/10.1007/978-981-15-7981-3_15
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