Below I discuss an algorithm to approximate a kernel of a posterior density which I routinely use in my research. It is called Mixture of \(t \) by Importance Sampling weighted Expectation Maximization (MitISEM) and for more details I refer you to the original paper by Hoogerheide et al. (2012).

### Introduction

### Illustration

As an illustration I chose the standard bivariate Gelman and Meng (1991) density function, with kernel given by \[ f(x_{1},x_{2}) = \exp\Big\{-0.5(Ax_{1}^{2}x_{2}^{2}+x_{1}^{2}+x_{2}^{2}-2Bx_{1}x_{2}-2C_{1}x_{1}-2C_{2}x_{2})\Big\} \]

with the function parameters set as follows: \(A=1 \), \(B=0 \), \(C_{1}=3 \) and \(C_{2}=3 \). I call it ‘standard’, as due to its non-elliptical shaper it is often used to demonstrate performance of various approximation algorithms.

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### References

Gelman, A. and X. L. Meng (1991), “A Note on Bivariate Distributions that are Conditionally Normal”, *The American Statistician*, **45**(2), 125-126.

Hoogerheide, L. F., A. Opschoor and H. K. van Dijk (2012), “A Class of Adaptive Importance Sampling Weighted EM Algorithms for Efficient and Robust Posterior and Predictive Simulation”, *Journal of Econometrics*, **171**(2), 101-120.

Peel, D. and G. McLachlan (2000), “Robust Mixture Modelling using the \(t \) -Distribution”, _Statistics and Computing, **10**, 339-348.