Shifted Dirichlet Distributions as Second-Order Probability Distributions that Factors into Marginals

• In classic decision theory it is assumed that a decisionmaker can assign precise numerical values corresponding to the true value of each consequence, as well as precise numerical probabilities for their occurrences. In attempting to address real-life problems, where uncertainty in the input data prevails, some kind of representation of imprecise information is important. Second-order distributions, probability distributions over probabilities, is one way to achieve such a representation. However, it is hard to intuitively understand statements in a multi-dimensional space and user statements must be provided more locally. But the information-theoretic interplay between joint and marginal distributions may give rise to unwanted effects on the global level. We consider this problem in a setting of second-order probability distributions and find a family of distributions that normalised over the probability simplex equals its own product of marginals. For such distributions, there is no flow of information between the joint distributions and the marginal distributions other than the trivial fact that the variables belong to the probability simplex.

Subject headings

NATURVETENSKAP  -- Data- och informationsvetenskap -- Systemvetenskap, informationssystem och informatik (hsv//swe)

NATURAL SCIENCES  -- Computer and Information Sciences -- Information Systems (hsv//eng)

NATURVETENSKAP  -- Matematik -- Sannolikhetsteori och statistik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Probability Theory and Statistics (hsv//eng)

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Keyword

Second-order probability distribution

Dirichlet distribution

Beta distribution

Kullback-Leibler divergence

relative entropy

product of marginal distributions

Computer and systems science

Data- och systemvetenskap

data- och systemvetenskap

Computer and Systems Sciences

Publication and Content Type

ref (subject category)

kon (subject category)