Mixing
Function which behave having other face
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I encountered a formula in density of exponential family of distributions,
begin{eqnarray*}
f(y;theta) = expBig[a(y)b(theta)+d(y)+c(theta)Big],
end{eqnarray*}
and it seemed to me that there is other formula expression of argument of exponential function.
To be more precise,
begin{eqnarray*}
left(
begin{array}{c}
alpha(y) \
gamma(theta) \
epsilon_1(y,theta)
end{array}
right) otimes left(
begin{array}{c}
delta(y) \
beta(theta) \
epsilon_2(y,theta)
end{array}
right)^{mathrm{T}} = left(
begin{array}{ccc}
alpha(y)delta(y) & alpha(y)beta(theta) & alpha(y)epsilon_2 \
gamma(theta)delta(y) & gamma(theta)beta(theta) & gamma(theta)epsilon_2 \
epsilon_1delta(y) & epsilon_1beta(theta) & epsilon_1epsilon_2
end{array}
right),
end{eqnarray*}
so if there is convenience function $epsilon_1(y,theta)$ such that,
begin{eqnarray*}
left{
begin{array}{l}
epsilon_1(y,theta)delta(y) approx Cdelta(y) (C>>1) \
epsilon_1(y,theta)beta(theta) approx 0
end{array}
right.,
end{eqnarray*}
and $epsilon_2$ which suppress $alpha$ and emphasize $gamma$,
(1,1),(1,2),(2,2),(2,3),(3,1),(3,3)-element of the matrix remain where $beta(theta)gamma(y) << alpha(y)gamma(theta)$.
This calculation may not make sense in first plobrem of formula exchange because $epsilon_1epsilon_2$ remain.
But existence of such function $epsilon_1$ interested me.
Do you have any idea of construction of $epsilon_1(y,theta)?$
probability statistics
asked yesterday
quickybrown
12
add a comment |
up vote
0
down vote
favorite
I encountered a formula in density of exponential family of distributions,
begin{eqnarray*}
f(y;theta) = expBig[a(y)b(theta)+d(y)+c(theta)Big],
end{eqnarray*}
and it seemed to me that there is other formula expression of argument of exponential function.
To be more precise,
begin{eqnarray*}
left(
begin{array}{c}
alpha(y) \
gamma(theta) \
epsilon_1(y,theta)
end{array}
right) otimes left(
begin{array}{c}
delta(y) \
beta(theta) \
epsilon_2(y,theta)
end{array}
right)^{mathrm{T}} = left(
begin{array}{ccc}
alpha(y)delta(y) & alpha(y)beta(theta) & alpha(y)epsilon_2 \
gamma(theta)delta(y) & gamma(theta)beta(theta) & gamma(theta)epsilon_2 \
epsilon_1delta(y) & epsilon_1beta(theta) & epsilon_1epsilon_2
end{array}
right),
end{eqnarray*}
so if there is convenience function $epsilon_1(y,theta)$ such that,
begin{eqnarray*}
left{
begin{array}{l}
epsilon_1(y,theta)delta(y) approx Cdelta(y) (C>>1) \
epsilon_1(y,theta)beta(theta) approx 0
end{array}
right.,
end{eqnarray*}
and $epsilon_2$ which suppress $alpha$ and emphasize $gamma$,
(1,1),(1,2),(2,2),(2,3),(3,1),(3,3)-element of the matrix remain where $beta(theta)gamma(y) << alpha(y)gamma(theta)$.
This calculation may not make sense in first plobrem of formula exchange because $epsilon_1epsilon_2$ remain.
But existence of such function $epsilon_1$ interested me.
Do you have any idea of construction of $epsilon_1(y,theta)?$
probability statistics
asked yesterday
quickybrown
12
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I encountered a formula in density of exponential family of distributions,
begin{eqnarray*}
f(y;theta) = expBig[a(y)b(theta)+d(y)+c(theta)Big],
end{eqnarray*}
and it seemed to me that there is other formula expression of argument of exponential function.
To be more precise,
begin{eqnarray*}
left(
begin{array}{c}
alpha(y) \
gamma(theta) \
epsilon_1(y,theta)
end{array}
right) otimes left(
begin{array}{c}
delta(y) \
beta(theta) \
epsilon_2(y,theta)
end{array}
right)^{mathrm{T}} = left(
begin{array}{ccc}
alpha(y)delta(y) & alpha(y)beta(theta) & alpha(y)epsilon_2 \
gamma(theta)delta(y) & gamma(theta)beta(theta) & gamma(theta)epsilon_2 \
epsilon_1delta(y) & epsilon_1beta(theta) & epsilon_1epsilon_2
end{array}
right),
end{eqnarray*}
so if there is convenience function $epsilon_1(y,theta)$ such that,
begin{eqnarray*}
left{
begin{array}{l}
epsilon_1(y,theta)delta(y) approx Cdelta(y) (C>>1) \
epsilon_1(y,theta)beta(theta) approx 0
end{array}
right.,
end{eqnarray*}
and $epsilon_2$ which suppress $alpha$ and emphasize $gamma$,
(1,1),(1,2),(2,2),(2,3),(3,1),(3,3)-element of the matrix remain where $beta(theta)gamma(y) << alpha(y)gamma(theta)$.
This calculation may not make sense in first plobrem of formula exchange because $epsilon_1epsilon_2$ remain.
But existence of such function $epsilon_1$ interested me.
Do you have any idea of construction of $epsilon_1(y,theta)?$
probability statistics
asked yesterday
quickybrown
12
I encountered a formula in density of exponential family of distributions,
begin{eqnarray*}
f(y;theta) = expBig[a(y)b(theta)+d(y)+c(theta)Big],
end{eqnarray*}
and it seemed to me that there is other formula expression of argument of exponential function.
To be more precise,
begin{eqnarray*}
left(
begin{array}{c}
alpha(y) \
gamma(theta) \
epsilon_1(y,theta)
end{array}
right) otimes left(
begin{array}{c}
delta(y) \
beta(theta) \
epsilon_2(y,theta)
end{array}
right)^{mathrm{T}} = left(
begin{array}{ccc}
alpha(y)delta(y) & alpha(y)beta(theta) & alpha(y)epsilon_2 \
gamma(theta)delta(y) & gamma(theta)beta(theta) & gamma(theta)epsilon_2 \
epsilon_1delta(y) & epsilon_1beta(theta) & epsilon_1epsilon_2
end{array}
right),
end{eqnarray*}
so if there is convenience function $epsilon_1(y,theta)$ such that,
begin{eqnarray*}
left{
begin{array}{l}
epsilon_1(y,theta)delta(y) approx Cdelta(y) (C>>1) \
epsilon_1(y,theta)beta(theta) approx 0
end{array}
right.,
end{eqnarray*}
and $epsilon_2$ which suppress $alpha$ and emphasize $gamma$,
(1,1),(1,2),(2,2),(2,3),(3,1),(3,3)-element of the matrix remain where $beta(theta)gamma(y) << alpha(y)gamma(theta)$.
This calculation may not make sense in first plobrem of formula exchange because $epsilon_1epsilon_2$ remain.
But existence of such function $epsilon_1$ interested me.
Do you have any idea of construction of $epsilon_1(y,theta)?$
probability statistics
probability statistics
asked yesterday
quickybrown
12
asked yesterday
quickybrown
12
edited 25 mins ago
asked yesterday
quickybrown
12
asked yesterday
quickybrown
12
asked yesterday
quickybrown
12
12
add a comment |
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