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why is it the case that $N(x_i|\mu_k,\Sigma_k)$ is not a conditional probability but can still be used in Bayes' theorem ? $N(x_i|\mu_k,\Sigma_k)$ is a conditional probability density function. It is conditional on cluster assignment $z_i = k$. High-level formula is $$p(k | X) \propto p(X|k)\, p(k)$$ Is the numerator in Bayes' theorem a distribution or a ...

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For a Gaussian mixture, the functions $p(x)$ and $p(x|Z=k)$ are probability densities, not probability mass functions. In case the mixture model has parameters $\theta$ like $\mu_k$ and $\sigma_k$, the likelihood function is the product of the $p_\theta(x_i)$'s $$\ell(\theta) =\prod_{i=1}^n p_\theta(x_i)\tag{1}$$ seen as a function of $\theta$ for a given ...

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Let me answer a few of the questions Mixture models are typically weighted sum of individual probability density functions (or mass functions) from well known families such as Gaussian, Poisson, etc. So, as the individual components of MLR are not densities, it may not be appropriate to consider MLR as a mixture model. Probability density functions are ...

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Both fitting a mixture of Gaussians to the transformed data and transforming the data prior to fitting appear to be valid approaches, but you may get very different results. Depending on the purpose of your analysis, this may be a problem. Let's say we are interested in determining the order of a Gaussian mixture. In the example below, we will see that log ...

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There isn't a "correct" order of the model. There is only a "most likely" order of the model, and what "most likely" means depends on the criterion you use to evaluate likelihood. There is a good paper that overviews the methods for determining model order, but it won't help unless you understand the fundamentals of Bayesian ...

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