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I am using scikit-learn to fit a gaussian mixture on a non-parametric multivariate distribution with three variables $ \mathbf{X} = (X_1, X_2, X_3) $

I want to sample from that distribution given specific values for $ X_1, X_2 $, i.e. I want to sample from the conditional distribution of $ X_3 $ given $X_1$ and $X_2$. Similarly, I also want to sample values $X_1,X_2$ given $X_3$.

Is this possible with the GaussianMixture class? If not, how would I go about writing a function like conditional_sample(X,size) where X is a vector of fixed values?

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One way to do this is to sample the cluster assignment (call it $z$) from the not-quite-a-full-posterior $P(Z|X_1, X_2)$ and then sample $X_3$ from $P(X_3 | Z=z, X_1, X_2)$. The second task involves just one cluster, and it's covered extensively elsewhere -- look for "multivariate Gaussian conditional distributions". The first task is the same as the usual prediction task for "which cluster is $X$ from", but you get the cluster probabilities ignoring $X_3$. This is how I would do it. Bayes' Rule says that if you have a prior $P(Z)$ on your cluster assignments, then

$$P(Z=z | X_{-k}) = \frac{P(X_{-k} | Z=z)P(Z=z)}{\sum_z P(X_{-k} | Z=z)P(Z=z)}$$

. If cluster $z$ has mean $\mu$ and covariance $\Sigma$ (paying attention to only variables 1 & 2), then you need to compute $$P(X_1, X_2 | Z=z) = \frac{1}{\sqrt{\det(2\pi\Sigma)}}\exp[\frac{-(x-\mu)^T\Sigma^{-1}(x-\mu)}{2}]$$.

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