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In the study of probabilistic graphical models (PGMs), the loopy belief update propagation (LBUP) message passing algorithm requires the division of unnormalised probability distributions. If the dividend and divisor are both mixtures of Gaussians, $$ f (\mathbf{x}) = \sum_{i=1}^{N} \alpha_{i} \cdot \mathcal{N} (\mathbf{x} | \boldsymbol{\mu}_{i}, \boldsymbol{\Sigma}_{i} ) $$ and $$ g (\mathbf{x}) = \sum_{j=1}^{M} \gamma_{j} \cdot \mathcal{N} (\mathbf{x} | \boldsymbol{\nu}_{j}, \mathbf{S}_{j} ) $$ respectively, is there a principled method of approximating the quotient, $$ h(\mathbf{x}) = \frac{f(\mathbf{x})}{ g (\mathbf{x}) } \text{?} $$

It was suggested to me that it might be possible to "data augment" the mixture model with allocation variables, placing it in a conjugate exponential family and thus making the division tractable. Unfortunately, I don't know anything about data augmentation; is this a reasonable suggestion? If it is reasonable, how would I proceed?

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