Unsurprisingly, the DM outclasses the multinomial by a mile, assigning a weight of nearly 100% to the over-dispersed model. set_title ( "Dirichlet-multinomial" ) axs. legend ( loc = "upper center", fontsize = 10 ) axs. We’ll parameterize this distribution with three things: - \(\mathrm ", xy = ( 0.96, 0.9 ), xycoords = "axes fraction", ha = "right", va = "top", color = c, ) axs. Our simulation will produce a two-dimensional matrix of integers (counts) where each row, (zero-)indexed by \(i \in (0.n-1)\), is an observation (different forest), and each column \(j \in (0.k-1)\) is a category (tree species). Here we will discuss a community ecology example, pretending that we have observed counts of \(k=5\) different tree species in \(n=10\) different forests. Here we are simulating from the DM distribution itself, so it is perhaps tautological to fit that model, but rest assured that data like these really do appear in the counts of different: (1) words in text corpuses, (2) types of RNA molecules in a cell, (3) items purchased by shoppers. Let us simulate some over-dispersed, categorical count data for this example. use ( "arviz-darkgrid" ) Simulation data ¶ This notebook will demonstrate the performance benefits that come from taking that approach. The DM is also an example of marginalizing a mixture distribution over its latent parameters. Other examples of over-dispersed count distributions are the Beta-binomial (which can be thought of as a special case of the DM) or the Negative binomial distributions. To accommodate more variable (a.k.a, over-dispersed) count data than the Multinomial. This contrasts with the Multinomial distribution, which assumes that all observations arise from a single fixed probability vector. The Dirichlet-multinomial can be understood as draws from a Multinomial distribution where each sample has a slightly different probability vector, which is itself drawn from a common Dirichlet distribution. Models like this one are important in a variety of areas, including natural language processing, ecology, bioinformatics, and more. This example notebook demonstrates the use of a Dirichlet mixture of multinomials (a.k.a Dirichlet-multinomial or DM) to model categorical count data.
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