Eigenfunctions heuristics for self-conjugate priors
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I am looking for a citable reference (books, research papers, PhD theses, not websites, internal reports, etc.) about the heuristic interpretation of self-conjugate priors as "eigenfunctions for the conditioning operator".
This is best explained (though without reference) in the Wikipedia page Conjugate prior#Interpretations.
In a nutshell: A statistical model is any family ${P_theta}_theta$ of probability distributions, parametrized by $thetain T$ (e.g., $Tsubset mathbb R$ or $Tsubset mathbb R^k$).
Some data $X:=(X_1,dotsc, X_k)$ is sampled from the model in two stages:
- firstly, we let $Thetain T$ be a $Q$-distributed random variable;
- secondly, we draw $X_1,dotsc, X_k$ in such a way that $X_1,dotsc, X_kmid Thetasim P_Theta$ conditionally i.i.d..
The posterior distribution of $Q$ given $X$ is then defined as $Q^X:=Q[Thetain cdotmid X]$.
A family of distributions ${Q_alpha}_{alphain A}$ (again $Asubset mathbb R$ or $mathbb R^k$) is self-conjugate (for a fixed model) if the posterior distribution $Q_alpha^X$ satisfies $Q_alpha^X=Q_{alpha'}$ for some $alpha'in A$.
Heuristically, $Q_alpha$ is (sort of) an "eigenfunction" for the conditioning operator $mathbf E[cdot mid X]$.
statistics reference-request bayesian
asked Jan 10 at 14:23
AlephBethAlephBeth
233
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add a comment |
$begingroup$
I am looking for a citable reference (books, research papers, PhD theses, not websites, internal reports, etc.) about the heuristic interpretation of self-conjugate priors as "eigenfunctions for the conditioning operator".
This is best explained (though without reference) in the Wikipedia page Conjugate prior#Interpretations.
In a nutshell: A statistical model is any family ${P_theta}_theta$ of probability distributions, parametrized by $thetain T$ (e.g., $Tsubset mathbb R$ or $Tsubset mathbb R^k$).
Some data $X:=(X_1,dotsc, X_k)$ is sampled from the model in two stages:
- firstly, we let $Thetain T$ be a $Q$-distributed random variable;
- secondly, we draw $X_1,dotsc, X_k$ in such a way that $X_1,dotsc, X_kmid Thetasim P_Theta$ conditionally i.i.d..
The posterior distribution of $Q$ given $X$ is then defined as $Q^X:=Q[Thetain cdotmid X]$.
A family of distributions ${Q_alpha}_{alphain A}$ (again $Asubset mathbb R$ or $mathbb R^k$) is self-conjugate (for a fixed model) if the posterior distribution $Q_alpha^X$ satisfies $Q_alpha^X=Q_{alpha'}$ for some $alpha'in A$.
Heuristically, $Q_alpha$ is (sort of) an "eigenfunction" for the conditioning operator $mathbf E[cdot mid X]$.
statistics reference-request bayesian
asked Jan 10 at 14:23
AlephBethAlephBeth
233
$endgroup$
add a comment |
$begingroup$
I am looking for a citable reference (books, research papers, PhD theses, not websites, internal reports, etc.) about the heuristic interpretation of self-conjugate priors as "eigenfunctions for the conditioning operator".
This is best explained (though without reference) in the Wikipedia page Conjugate prior#Interpretations.
In a nutshell: A statistical model is any family ${P_theta}_theta$ of probability distributions, parametrized by $thetain T$ (e.g., $Tsubset mathbb R$ or $Tsubset mathbb R^k$).
Some data $X:=(X_1,dotsc, X_k)$ is sampled from the model in two stages:
- firstly, we let $Thetain T$ be a $Q$-distributed random variable;
- secondly, we draw $X_1,dotsc, X_k$ in such a way that $X_1,dotsc, X_kmid Thetasim P_Theta$ conditionally i.i.d..
The posterior distribution of $Q$ given $X$ is then defined as $Q^X:=Q[Thetain cdotmid X]$.
A family of distributions ${Q_alpha}_{alphain A}$ (again $Asubset mathbb R$ or $mathbb R^k$) is self-conjugate (for a fixed model) if the posterior distribution $Q_alpha^X$ satisfies $Q_alpha^X=Q_{alpha'}$ for some $alpha'in A$.
Heuristically, $Q_alpha$ is (sort of) an "eigenfunction" for the conditioning operator $mathbf E[cdot mid X]$.
statistics reference-request bayesian
asked Jan 10 at 14:23
AlephBethAlephBeth
233
$endgroup$
I am looking for a citable reference (books, research papers, PhD theses, not websites, internal reports, etc.) about the heuristic interpretation of self-conjugate priors as "eigenfunctions for the conditioning operator".
This is best explained (though without reference) in the Wikipedia page Conjugate prior#Interpretations.
In a nutshell: A statistical model is any family ${P_theta}_theta$ of probability distributions, parametrized by $thetain T$ (e.g., $Tsubset mathbb R$ or $Tsubset mathbb R^k$).
Some data $X:=(X_1,dotsc, X_k)$ is sampled from the model in two stages:
- firstly, we let $Thetain T$ be a $Q$-distributed random variable;
- secondly, we draw $X_1,dotsc, X_k$ in such a way that $X_1,dotsc, X_kmid Thetasim P_Theta$ conditionally i.i.d..
The posterior distribution of $Q$ given $X$ is then defined as $Q^X:=Q[Thetain cdotmid X]$.
A family of distributions ${Q_alpha}_{alphain A}$ (again $Asubset mathbb R$ or $mathbb R^k$) is self-conjugate (for a fixed model) if the posterior distribution $Q_alpha^X$ satisfies $Q_alpha^X=Q_{alpha'}$ for some $alpha'in A$.
Heuristically, $Q_alpha$ is (sort of) an "eigenfunction" for the conditioning operator $mathbf E[cdot mid X]$.
statistics reference-request bayesian
statistics reference-request bayesian
asked Jan 10 at 14:23
AlephBethAlephBeth
233
asked Jan 10 at 14:23
AlephBethAlephBeth
233
asked Jan 10 at 14:23
AlephBethAlephBeth
233
asked Jan 10 at 14:23
AlephBethAlephBeth
233
asked Jan 10 at 14:23
AlephBethAlephBeth
233
233
add a comment |
add a comment |
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