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Is uniform distribution in exponential family?
Comments: (1) For an exponential family, the support of the distribution (i.e., ) cannot depend on . Thus, iid Uniform is not an exponential family model. (2) For an exponential family model, is a sufficient statistic by the factorization theorem.
Is uniform distribution part of exponential family?
Uniform distribution U([0, θ]), θ ∈ R+ does not belong to the exponential fam- ily, since its support depends on θ If the probability distribution of X1 belongs to an exponential family, the prob- ability distribution of (X1, ··· ,Xn) also belongs to the same exponential family, where Xi are iid with distribution same …
How to describe the exponential family of distributions?
Exponential Family of distributions The exponential family of distribution is the set of distributions parametrized by θ ∈ RD that can be described in the form: p(x ∣ θ) = h(x)exp(η(θ)TT(x) − A(θ)) or in a more extensive notation:
Why are exponential families important for Bayesian statistics?
Finally, the exponential families have conjugate priors (i.e. same distributions for prior and posterior distributions), and the posterior predictive distribution has always a closed-form solution (provided that the normalizing factor can also be stated in closed-form), both important properties for Bayesian statistics.
How to calculate the exponential family of Bernoulli distributions?
Similarly, to compute the exponential family parameters in the Bernoulli distribution we follow as: A(η) = log(1 + eη). We now compute the mean of T(x) as: which is the mean of a Bernoulli variable. Taking a second derivative yields:
How to reparametrize exponential families to the natural parameter?
Natural Exponential Families It is often convenient to reparametrize exponential families to the natural parameter η= η(θ) ∈ Rq, leading (with A(η(θ)) ≡ B(θ)) to f(x| η) = eη·t(x)−A(η)h(x) (2) Since any pdf integrates to unity we have eA(η) = Z.