page_title: Dirichlet process
section_title: Use in Dirichlet mixture models
hierarchical_section_title: Dirichlet process / Use in Dirichlet mixture models
caption_attribution_description: English: Simulation of 1000 observations drawn from a Dirichlet mixture model. Each observation within a cluster is drawn iid from N ( m u k , 1 / 4 ) {\displaystyle N(mu_{k},1/4)} where the cluster means &#x03BC; k {\displaystyle \mu _{k}} are drawn from a distribution G drawn from a Dirichlet Process with concentration parameter &#x03B1; = 0.5 {\displaystyle \alpha =0.5} and base distribution H = N ( 2 , 16 ) {\displaystyle H=N(2,16)} .
context_page_description: In probability theory, Dirichlet processes are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a probability distribution whose range is itself a set of probability distributions. It is often used in Bayesian inference to describe the prior knowledge about the distribution of random variables—how likely it is that the random variables are distributed according to one or another particular distribution.
  The Dirichlet process is specified by a base distribution and a positive real number called the concentration parameter. The base distribution is the expected value of the process, i.e., the Dirichlet process draws distributions "around" the base distribution the way a normal distribution draws real numbers around its mean. However, even if the base distribution is continuous, the distributions drawn from the Dirichlet process are almost surely discrete. The scaling parameter specifies how strong this discretization is: in the limit of, the realizations are all concentrated at a single value, while in the limit of the realizations become continuous.
context_section_description: To understand what Dirichlet processes are and the problem they solve we consider the example of data clustering. It is a common situation that data points are assumed to be distributed in a hierarchical fashion where each data point belongs to a (randomly chosen) cluster and the members of a cluster are further distributed randomly within that cluster.
caption: Simulation of 1000 observations drawn from a Dirichlet mixture model. Each observation within a cluster is drawn independently from the multivariate normal distribution 
    
      
        
          N
          (
          
            μ
            
              k
            
          
          ,
          1
          
            /
          
          4
          )
        
      
      {\displaystyle N(\mu _{k},1/4)}
    
  . The cluster means 
    
      
        
          
            μ
            
              k
            
          
        
      
      {\displaystyle \mu _{k}}
    
   are drawn from a distribution G which itself is drawn from a Dirichlet process with concentration parameter 
    
      
        
          α
          =
          0.5
        
      
      {\displaystyle \alpha =0.5}
    
   and base distribution 
    
      
        
          H
          =
          N
          (
          2
          ,
          16
          )
        
      
      {\displaystyle H=N(2,16)}
    
  . Each row is a new simulation.
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