Page 161 - Elements of Distribution Theory

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052184472Xc05 CUNY148/Severini May 24, 2005 17:53

5.4 Hierarchical Models 147

Note that
2
H 0 ={(η 1 ,η 2 ) ∈ R :exp(η 1 ) + exp(η 2 ) = 1}
+ 2
={(z 1 , z 2 ) ∈ (R ) : z 1 + z 2 = 1}.
2
It follows that H 0 is a one-dimensional subset of R and, hence, it does not contain an open
2
subset of R .

5.4 Hierarchical Models

Let X denote a random variable with a distribution depending on a parameter λ. Suppose
that λ is not a constant, but is instead itself a random variable with a distribution depending
on a parameter θ. This description yields a model for X with parameter θ.
More specifically, consider random variables X and λ, each of which may be a vector.
The random variable λ is assumed to be unobserved and we are interested in the model for
X. This model is specified by giving the conditional distribution of X given λ, along with
the marginal distribution of λ. Both of these distributions may depend on the parameter
m
θ, taking values in a set ⊂ R , for some m. Hence, probabilities regarding X may be
calculated by first conditioning on λ and then averaging with respect to the distribution of
λ.For instance,
Pr(X ≤ x; θ) = E[Pr(X ≤ x|λ; θ); θ]

where, in this expression, the expectation is with respect to the distribution of λ. The result
is a parametric model for X.
If the conditional distribution of X given λ is an absolutely continuous distribution, then
the marginal distribution of X is also absolutely continuous. Similarly, if the conditional
distribution of X given λ is discrete, the marginal distribution of X is discrete as well.

Theorem 5.5. Let X and λ denote random variables such that the conditional distribution
of X given λ is absolutely continuous with density function p(x|λ; θ) where θ ∈ is a
parameter with parameter space . Then the marginal distribution of X is absolutely
continuous with density function

p X (x; θ) = E[p(x|λ; θ); θ],θ ∈ .
Let X and λ denote random variables such that the conditional distribution of X given
λ is discrete and that there exists a countable set X, not depending on λ, and a conditional
frequency function p(x|λ; θ) such that

p(x|λ; θ) = 1,θ ∈
x∈X
for all λ. Then the marginal distribution of X is discrete with frequency function

p X (x; θ) = E[p(x|λ; θ); θ], x ∈ X,θ ∈ .

Proof. First suppose that the conditonal distribution of X given λ is absolutely continuous.
d
Let g denote a bounded continuous function on the range of X, which we take to be R ,