Mixing
Eigenfunctions heuristics for self-conjugate priors
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Also asked on CrossValidated.
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
235
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add a comment |
$begingroup$
Also asked on CrossValidated.
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
235
$endgroup$
1
$begingroup$
You have asked in a flag to cross-post this to CrossValidated (and not migrated). I think you are under the impression that this is something that the mods (or the community) does for you --- but in fact, all you need to do is go to CrossValidated and ask your question. If you decide to do this, be sure to include links to each post, so that if one gets answered then no one will waste their time duplicating the effort. Good luck
$endgroup$
– davidlowryduda♦
Jan 23 at 22:17
$begingroup$
@davidlowryduda thank you for your kind the clarification. I was in fact under the impression that cross-posting is a kind of duplication, thus requiring moderators' approval. (I will read the guidelines more carefully...)
$endgroup$
– AlephBeth
Jan 25 at 10:29
add a comment |
$begingroup$
Also asked on CrossValidated.
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
235
$endgroup$
Also asked on CrossValidated.
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
235
asked Jan 10 at 14:23
AlephBethAlephBeth
235
edited Jan 25 at 10:45
AlephBeth
asked Jan 10 at 14:23
AlephBethAlephBeth
235
asked Jan 10 at 14:23
AlephBethAlephBeth
235
asked Jan 10 at 14:23
AlephBethAlephBeth
235
235
1
$begingroup$
You have asked in a flag to cross-post this to CrossValidated (and not migrated). I think you are under the impression that this is something that the mods (or the community) does for you --- but in fact, all you need to do is go to CrossValidated and ask your question. If you decide to do this, be sure to include links to each post, so that if one gets answered then no one will waste their time duplicating the effort. Good luck
$endgroup$
– davidlowryduda♦
Jan 23 at 22:17
$begingroup$
@davidlowryduda thank you for your kind the clarification. I was in fact under the impression that cross-posting is a kind of duplication, thus requiring moderators' approval. (I will read the guidelines more carefully...)
$endgroup$
– AlephBeth
Jan 25 at 10:29
add a comment |
1
$begingroup$
You have asked in a flag to cross-post this to CrossValidated (and not migrated). I think you are under the impression that this is something that the mods (or the community) does for you --- but in fact, all you need to do is go to CrossValidated and ask your question. If you decide to do this, be sure to include links to each post, so that if one gets answered then no one will waste their time duplicating the effort. Good luck
$endgroup$
– davidlowryduda♦
Jan 23 at 22:17
$begingroup$
@davidlowryduda thank you for your kind the clarification. I was in fact under the impression that cross-posting is a kind of duplication, thus requiring moderators' approval. (I will read the guidelines more carefully...)
$endgroup$
– AlephBeth
Jan 25 at 10:29
1
1
$begingroup$
You have asked in a flag to cross-post this to CrossValidated (and not migrated). I think you are under the impression that this is something that the mods (or the community) does for you --- but in fact, all you need to do is go to CrossValidated and ask your question. If you decide to do this, be sure to include links to each post, so that if one gets answered then no one will waste their time duplicating the effort. Good luck
$endgroup$
– davidlowryduda♦
Jan 23 at 22:17
$begingroup$
You have asked in a flag to cross-post this to CrossValidated (and not migrated). I think you are under the impression that this is something that the mods (or the community) does for you --- but in fact, all you need to do is go to CrossValidated and ask your question. If you decide to do this, be sure to include links to each post, so that if one gets answered then no one will waste their time duplicating the effort. Good luck
$endgroup$
– davidlowryduda♦
Jan 23 at 22:17
$begingroup$
@davidlowryduda thank you for your kind the clarification. I was in fact under the impression that cross-posting is a kind of duplication, thus requiring moderators' approval. (I will read the guidelines more carefully...)
$endgroup$
– AlephBeth
Jan 25 at 10:29
$begingroup$
@davidlowryduda thank you for your kind the clarification. I was in fact under the impression that cross-posting is a kind of duplication, thus requiring moderators' approval. (I will read the guidelines more carefully...)
$endgroup$
– AlephBeth
Jan 25 at 10:29
add a comment |
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$begingroup$
You have asked in a flag to cross-post this to CrossValidated (and not migrated). I think you are under the impression that this is something that the mods (or the community) does for you --- but in fact, all you need to do is go to CrossValidated and ask your question. If you decide to do this, be sure to include links to each post, so that if one gets answered then no one will waste their time duplicating the effort. Good luck
$endgroup$
– davidlowryduda♦
Jan 23 at 22:17
$begingroup$
@davidlowryduda thank you for your kind the clarification. I was in fact under the impression that cross-posting is a kind of duplication, thus requiring moderators' approval. (I will read the guidelines more carefully...)
$endgroup$
– AlephBeth
Jan 25 at 10:29