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Exponential Family of Distributions

  1. Aug 13, 2014 #1
    I'm reading about the exponential family of distributions. In my book, I have the expression

    [itex] f(x|\theta) = h(x)c(\theta)exp(\sum_{i=1}^{k} w_i(\theta) t_i(x)) [/itex]

    where [itex]h(x) \geq 0 [/itex], [itex]t_1(x), t_2(x),...,t_k(x) [/itex] are real valued functions of the observation [itex] x[/itex], [itex]c(\theta) \geq 0 [/itex], and [itex]w_1(\theta),w_2(\theta),...,w_k(\theta) [/itex] are real-valued functions of the possibly vector-valued parameter [itex] \theta[/itex].

    What's being stressed in the few examples available (in the book), is the indicator function. Here's the indicator function:

    [itex]I_A(x) = 1[/itex], if [itex] x \in A[/itex], and [itex]I_A(x) = 0 [/itex], if [itex]x \notin A [/itex], where [itex]A[/itex] is the set values the observation or parameter may take.

    What I'm seeing, is that the indicator function is inserted with [itex]h(x) [/itex] or [itex]c(\theta) [/itex] whenever these functions are constants.

    Do you guys know of any examples where an indicator function is used and [itex]h(x)[/itex] or [itex]c(\theta) [/itex] are not constants?

    I'm thinking the whole point of using indicator functions is to make the expression exactly like the probability distribution function.
  2. jcsd
  3. Aug 21, 2014 #2
    I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
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