Gaussian mixture model of the form with new parameters and that are updated using the EM algorithm. Although EM-based parameter updates are well-established, providing the initial estimates for these parameters is currently an area of active research. Note that this formulation yields a closed-form solution to the complete posterior distribution. Estimations of the random variable may be obtained via one of several estimators, such as the mean or maximum of the posterior distribution. Such distributions are useful for assuming patch-wise shapes of images and clusters, for example. In the case of image representation, each Gaussian may be tilted, expanded, and warped according to the covariance matrices . One Gaussian distribution of the set is fit to each patch (usually of size 8x8 pixels) in the image. Notably, any distribution of points around a cluster (see k-means) may be accurately given enough Gaussian components, but scarcely over K=20 components are needed to accurately model a given image distribution or cluster of data. Categorical mixture model A typical non-Bayesian mixture model with categorical observations looks like this: as above as above as above dimension of categorical observations, e.g., size of word vocabulary probability for component of observing item vector of dimension composed of must sum to 1 The random variables: A typical Bayesian mixture model with categorical observations looks like this: as above as above as above dimension of categorical observations, e.g., size of word vocabulary probability for component of observing item vector of dimension composed of must sum to 1 shared concentration hyperparameter of for each component concentration hyperparameter of The random variables: Examples A financial model Financial returns often behave differently in normal situations and during crisis times. A mixture model for return data seems reasonable. Sometimes the model used is a jump-diffusion model, or as a mixture of two normal distributions. See and for further context. House prices Assume that we observe the prices of N different houses. Different types of houses in different neighborhoods will have vastly different prices, but the price of a particular type of house in a particular neighborhood (e.g., three-bedroom house in moderately upscale neighborhood) will tend to cluster fairly closely around the mean. One possible model of such prices would be to assume that the prices are accurately described by a mixture model with K different components, each distributed as a normal distribution with unknown mean and variance, with each component specifying a