Mixture Transition Distribution Model

TL;DR

• Models an observed variable’s distribution as a combination of multiple component distributions.
• Lets you infer the number and parameters of those sub-distributions using estimation techniques.
• Supports prediction of future behavior from the estimated mixture; used in sequence models such as HMMs for tasks like speech recognition.

Definition

A mixture transition distribution model is a type of statistical model that represents the distribution of a random variable as a mixture of other distributions. It is a probabilistic model that assumes the underlying distribution of the random variable is composed of multiple sub-distributions, each with its own unique characteristics.

Explanation

The model represents an observed variable’s overall distribution as a weighted combination of several component distributions. Each component captures a subset of the data’s behavior (for example, with its own mean and variance in the Gaussian case). By fitting a mixture model, one can infer the underlying structure of the data — including how many sub-distributions exist and the parameters of each — using various estimation techniques. Once estimated, the mixture can be used to make predictions about future values of the random variable.

Examples

Mixture of Gaussians

This model assumes the underlying distribution is a mixture of multiple Gaussian distributions, each with its own mean and variance. For example, fitting a mixture of Gaussians to a dataset of heights might reveal two Gaussian components, one corresponding to the heights of men and the other to the heights of women.

Hidden Markov Model (HMM)

Hidden Markov models represent sequences of observations generated by a system with multiple latent states. In this context, the underlying distribution of observations is a mixture of distributions associated with different states. For instance, in a speech recognition system, the states might represent different phonemes or words, and the observations might be the acoustic features of the speech signal.

Use cases

• Modeling complex distributions that are not well represented by a single distribution.
• Inferring underlying structure of data (number of sub-distributions and their parameters) via estimation techniques.
• Making predictions about future behavior based on the estimated mixture distribution.
• Sequence modeling applications such as natural language processing and speech recognition (via HMMs).

Related terms

• Mixture of Gaussians
• Hidden Markov Model (HMM)