in which one of the elements is 1 (specifying the identity of the observation) and all other elements are 0. is the Gaussian distribution, in this case specifically the multivariate Gaussian distribution. The interpretation of the above variables is as follows: is the set of data points, each of which is a -dimensional vector distributed according to a multivariate Gaussian distribution. is a set of latent variables, one per data point, specifying which mixture component the corresponding data point belongs to, using a "one-of-K" vector representation with components for , as described above. is the mixing proportions for the mixture components. and specify the parameters (mean and precision) associated with each mixture component. The joint probability of all variables can be rewritten as where the individual factors are where Assume that . Then where we have defined Exponentiating both sides of the formula for yields Requiring that this be normalized ends up requiring that the sum to 1 over all values of , yielding where In other words, is a product of single-observation multinomial distributions, and factors over each individual , which is distributed as a single-observation multinomial distribution with parameters for . Furthermore, we note that which is a standard result for categorical distributions. Now, considering the factor , note that it automatically factors into due to the structure of the graphical model defining our Gaussian mixture model, which is specified above. Then, Taking the exponential of both sides, we recognize as a Dirichlet distribution where where Finally Grouping and reading off terms involving and , the result is a Gaussian-Wishart distribution given by given the definitions Finally, notice that these functions require the values of , which make use of , which is defined in turn based on , , and . Now that we have determined the distributions over which these expectations are taken, we can derive formulas for them: These results lead to These can be converted from proportional to absolute values by normalizing over so that the corresponding values sum to 1. Note that: The update equations for the parameters , , and of the variables and depend on the statistics , , and , and these statistics in turn depend on . The update equations for the parameters of the variable depend on the statistic , which depends in turn on . The update equation for has a direct circular dependence on , ,