Uniform distribution on simplex

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SUMMARY

The Dirichlet distribution with parameters α = (1, 1, ..., 1) is uniformly distributed on a K-dimensional unit simplex. This is established by recognizing that when all parameters are equal to one, the distribution effectively assigns equal probability to all points within the simplex. The uniformity arises from the symmetry of the Dirichlet distribution under these conditions, confirming that each vertex and point within the simplex has an equal likelihood of being sampled.

PREREQUISITES
  • Understanding of Dirichlet distribution
  • Familiarity with K-dimensional unit simplex
  • Knowledge of probability theory
  • Basic concepts of statistical distributions
NEXT STEPS
  • Study the properties of the Dirichlet distribution in detail
  • Explore applications of the Dirichlet distribution in Bayesian statistics
  • Learn about the geometric interpretation of K-dimensional simplices
  • Investigate the relationship between Dirichlet and multinomial distributions
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Statisticians, data scientists, and researchers in probability theory looking to deepen their understanding of the Dirichlet distribution and its applications in statistical modeling.

chowpy
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How can we show that Dirichlet distribution with parameters α = (α1, ..., αK) all equal to one is uniformly distributed on a K-dimensional unit simplex?
 
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chowpy said:
How can we show that Dirichlet distribution with parameters α = (α1, ..., αK) all equal to one is uniformly distributed on a K-dimensional unit simplex?

Isn't it true by definition?
 

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