In statistics, a mixture model is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation belongs. Formally a mixture model corresponds to the mixture distribution that represents the probability distribution of observations in the overall population. However, while problems associated with "mixture distributions" relate to deriving the properties of the overall population from those of the sub-populations, "mixture models" are used to make statistical inferences about the properties of the sub-populations given only observations on the pooled population, without sub-population identity information. Mixture models are used for clustering, under the name model-based clustering, and also for density estimation.
Mixture models should not be confused with models for compositional data, i.e., data whose components are constrained to sum to a constant value (1, 100%, etc.). However, compositional models can be thought of as mixture models, where members of the population are sampled at random. Conversely, mixture models can be thought of as compositional models, where the total size reading population has been normalized to 1.
Structure
General mixture model A typical finite-dimensional mixture model is a hierarchical model consisting of the following components:
N random variables that are observed, each distributed according to a mixture of K components, with the components belonging to the same parametric family of distributions (e.g., all normal, all Zipfian, etc.) but with different parameters. However, it is also possible to have a finite mixture model where each component belongs to a different parametric family of distributions, for example, a mixture of a multivariate normal distribution and a generalized hyperbolic distribution.
N random latent variables specifying the identity of the mixture component of each observation, each distributed according to a K-dimensional categorical distribution A set of K mixture weights, which are probabilities that sum to 1. A set of K parameters, each specifying the parameter of the corresponding mixture component. In many cases, each "parameter" is actually a set of parameters. For example, if the mixture components are Gaussian distributions, there will be a mean and variance for each component. If the mixture components are categorical distributions (e.g., when each observation is a token from a finite alphabet of size V), there will be a vector of V probabilities summing to 1.
In addition, in a Bayesian setting, the mixture weights and parameters will themselves be random variables, and prior distributions will be placed over the variables. In such a case, the weights are typically viewed as a K-dimensional random vector drawn from a Dirichlet distribution (the conjugate prior of the categorical distribution), and the parameters will be distributed according to their respective conjugate priors.
Mathematically, a basic parametric mixture model can be described as follows:
In a Bayesian setting, all parameters are associated with random variables, as follows:
This characterization uses F and H to describe arbitrary distributions over observations and parameters, respectively. Typically H will be the conjugate prior of F. The two most common choices of F are Gaussian aka "normal" (for real-valued observations) and categorical (for discrete observations). Other common possibilities for the distribution of the mixture components are: Binomial distribution, for the number of "positive occurrences" (e.g., successes, yes votes, etc.) given a fixed number of total occurrences Multinomial distribution, similar to the binomial distribution, but for counts of multi-way occurrences (e.g., yes/no/maybe in a survey) Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs Poisson distribution, for the number of occurrences of an event in a given period of time, for an event that is characterized by a fixed rate of occurrence Exponential distribution, for the time before the next event occurs, for an event that is characterized by a fixed rate of occurrence Log-normal distribution, for positive real numbers that are assumed to grow exponentially, such as incomes or prices Multivariate normal distribution (aka multivariate Gaussian distribution), for vectors of correlated outcomes that are individually Gaussian-distributed Multivariate Student's t-distribution, for vectors of heavy-tailed correlated outcomes A vector of Bernoulli-distributed values, corresponding, e.g., to a black-and-white image, with each value representing a pixel; see the handwriting-recognition example below
Specific examples
Gaussian mixture model
A typical non-Bayesian Gaussian mixture model looks like this:
A Bayesian version of a Gaussian mixture model is as follows:
Multivariate Gaussian mixture model A Bayesian Gaussian mixture model is commonly extended to fit a vector of unknown parameters (denoted in bold), or multivariate normal distributions. In a multivariate distribution (i.e. one modelling a vector with N random variables) one may model a vector of parameters (such as several observations of a signal or patches within an image) using a Gaussian mixture model prior distribution on the vector of estimates given by
where the ith vector component is characterized by normal distributions with weights , means and covariance matrices . To incorporate this prior into a Bayesian estimation, the prior is multiplied with the known distribution of the data conditioned on the parameters to be estimated. With this formulation, the posterior distribution is also a 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 8×8 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 modelDinov, ID. "Expectation Maximization and Mixture Modeling Tutorial". California Digital Library, Statistics Online Computational Resource, Paper EM_MM, http://repositories.cdlib.org/socr/EM_MM, December 9, 2008 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 particular combination of house type/neighborhood. Fitting this model to observed prices, e.g., using the expectation-maximization algorithm, would tend to cluster the prices according to house type/neighborhood and reveal the spread of prices in each type/neighborhood. (Note that for values such as prices or incomes that are guaranteed to be positive and which tend to grow exponentially, a log-normal distribution might actually be a better model than a normal distribution.)
Topics in a document Assume that a document is composed of N different words from a total vocabulary of size V, where each word corresponds to one of K possible topics. The distribution of such words could be modelled as a mixture of K different V-dimensional categorical distributions. A model of this sort is commonly termed a topic model. Note that expectation maximization applied to such a model will typically fail to produce realistic results, due (among other things) to the excessive number of parameters. Some sorts of additional assumptions are typically necessary to get good results. Typically two sorts of additional components are added to the model: A prior distribution is placed over the parameters describing the topic distributions, using a Dirichlet distribution with a concentration parameter that is set significantly below 1, so as to encourage sparse distributions (where only a small number of words have significantly non-zero probabilities). Some sort of additional constraint is placed over the topic identities of words, to take advantage of natural clustering. For example, a Markov chain could be placed on the topic identities (i.e., the latent variables specifying the mixture component of each observation), corresponding to the fact that nearby words belong to similar topics. (This results in a hidden Markov model, specifically one where a prior distribution is placed over state transitions that favors transitions that stay in the same state.) Another possibility is the latent Dirichlet allocation model, which divides up the words into D different documents and assumes that in each document only a small number of topics occur with any frequency.
Handwriting recognition The following example is based on an example in Christopher M. Bishop, Pattern Recognition and Machine Learning.
Imagine that we are given an N×N black-and-white image that is known to be a scan of a hand-written digit between 0 and 9, but we don't know which digit is written. We can create a mixture model with different components, where each component is a vector of size of Bernoulli distributions (one per pixel). Such a model can be trained with the expectation-maximization algorithm on an unlabeled set of hand-written digits, and will effectively cluster the images according to the digit being written. The same model could then be used to recognize the digit of another image simply by holding the parameters constant, computing the probability of the new image for each possible digit (a trivial calculation), and returning the digit that generated the highest probability.
Assessing projectile accuracy (a.k.a. circular error probable, CEP)