How do you incorporate prior knowledge or domain expertise into a mixture model?

1 What are mixture models?

Mixture models are statistical models that assume that the data are generated by a combination of two or more underlying distributions, each with its own parameters and proportions. For example, you can use a mixture model to describe the heights of men and women in a population, by assuming that they follow two different normal distributions with different means and variances. Mixture models can also handle more complex and diverse data, such as images, text, or gene expression.

2 Why use mixture models?

Mixture models can offer many advantages and applications in various domains. These models allow you to identify and separate hidden subgroups or patterns in your data, such as customer segments, disease subtypes, or topic clusters. They also help you estimate the parameters and proportions of each component, which can provide insights into the characteristics and behavior of each subgroup or source. Additionally, mixture models account for heterogeneity and uncertainty in your data by allowing for different sources of variation and noise. Furthermore, they enable you to incorporate prior knowledge or domain expertise into your model by choosing appropriate component distributions and priors.

3 How to choose the component distributions?

When using a mixture model, one of the key decisions to make is what kind of distribution to use for each component. This depends on the data's nature and shape, as well as domain knowledge and assumptions. Normal distributions are suitable for continuous, symmetric, bell-shaped data like heights, weights, or test scores. Poisson distributions are suitable for discrete data that count the number of events or occurrences such as words, accidents, or cells. Multinomial distributions are suitable for categorical data representing proportions of different outcomes or classes such as colors, topics, or labels. Lastly, Dirichlet distributions are suitable for compositional data representing fractions of different components or features such as proportions of metals, genes, or words.

4 How to choose the number of components?

When using a mixture model, it is important to decide how many components to include. This decision depends on how well the model fits your data, as well as your domain knowledge and goals. Common methods for determining the number of components are using a fixed number based on prior knowledge or expectation, such as the number of known subgroups or sources in the data, and using a criterion or test to compare different models with different numbers of components based on their likelihood, complexity, or information content. Additionally, one can use a flexible number of components based on a prior distribution that allows for uncertainty and adaptation, such as Dirichlet process (DP), which is a nonparametric Bayesian method that can infer the number of components from the data.

5 How to incorporate prior knowledge or domain expertise?

Using a mixture model offers the advantage of incorporating prior knowledge or domain expertise into the model by selecting suitable priors for the parameters and proportions of each component. Priors are distributions that reflect your beliefs or assumptions about the values of the parameters and proportions before seeing the data. These can help express uncertainty or confidence in the parameters and proportions, as well as incorporate domain knowledge or expertise. Additionally, they can regularize or constrain the model, favoring simpler or more plausible solutions. Examples of priors include uniform, normal, beta distributions, conjugate, hierarchical, empirical Bayes priors, sparsity, smoothness, and shrinkage priors.

6 How to fit and evaluate a mixture model?

Once you have chosen your component distributions, number of components, and priors, you need to fit your mixture model to your data and evaluate its performance and validity. Common methods include using an optimization or sampling algorithm such as the EM algorithm, MCMC algorithm, or VI algorithm to estimate the parameters and proportions of each component, as well as the component membership of each data point. Additionally, you can use diagnostic or visualization tools like PPC, trace plot, or density plot to check the quality and adequacy of your model. Lastly, metrics or measures such as log-likelihood, BIC, AIC, or cross-validation score can be used to compare and select the best model among different alternatives based on their fit, complexity, or predictive power.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?