Radon-Nikodym derivative when distributions are involved.

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I have two cases I'm interested in looking at (I'm computing two KL divergences specifically).



The domain is $Omega in mathbbR^2n$. Let's say $(x, y) in mathbbR^2n$. Assume the intersection $Omega cap x=y$ has Lebesgue measure (of the corresponding dimension) greater than zero.



I have two measures, $P ll Q$, which written as "functions" (I mean, there are Delta distributions too...) would be $int dP = int_x int_y delta(x-y)f(x)dxdy$ and $int dQ = int_x int_y delta(x-y)g(x) dx dy$. I also have $tildeQ$ such that $int dtildeQ = int_x int_y tildeg(x,y) dx dy$.



OK now, are the following things correct?



  1. $int_A fracdPdQ = int_Acapx=y fracf(x)g(x)dx$.

  2. $D(P || Q) = int_x f(x) logfracf(x)g(x) dx$.

  3. $P$ is not absolutely continuous under $tildeQ$ and therefore the KL divergence does not exist.

Edit: I previously had some assertions about $D(P || tildeQ)$ that were false. $P$ is not absolutely continuous under $Q$. Is there something that can be done about that? It would be very convenient to have a way of measuring the KL divergence (for example if $Q$ is not a delta but a very narrow gaussian around the hyperplane)



Intuitively 1,2 make sense but I am struggling with how to prove it.



Any recommendation on where to look at these kind of manipulations with deltas and distributions would be very appreciated.



Thanks!




probability-theory measure-theory distribution-theory




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  • 1




    This might be helpful: stats.stackexchange.com/questions/328123/…
    – E-A
    Jul 26 at 19:33














up vote
0
down vote

favorite












I have two cases I'm interested in looking at (I'm computing two KL divergences specifically).



The domain is $Omega in mathbbR^2n$. Let's say $(x, y) in mathbbR^2n$. Assume the intersection $Omega cap x=y$ has Lebesgue measure (of the corresponding dimension) greater than zero.



I have two measures, $P ll Q$, which written as "functions" (I mean, there are Delta distributions too...) would be $int dP = int_x int_y delta(x-y)f(x)dxdy$ and $int dQ = int_x int_y delta(x-y)g(x) dx dy$. I also have $tildeQ$ such that $int dtildeQ = int_x int_y tildeg(x,y) dx dy$.



OK now, are the following things correct?



  1. $int_A fracdPdQ = int_Acapx=y fracf(x)g(x)dx$.

  2. $D(P || Q) = int_x f(x) logfracf(x)g(x) dx$.

  3. $P$ is not absolutely continuous under $tildeQ$ and therefore the KL divergence does not exist.

Edit: I previously had some assertions about $D(P || tildeQ)$ that were false. $P$ is not absolutely continuous under $Q$. Is there something that can be done about that? It would be very convenient to have a way of measuring the KL divergence (for example if $Q$ is not a delta but a very narrow gaussian around the hyperplane)



Intuitively 1,2 make sense but I am struggling with how to prove it.



Any recommendation on where to look at these kind of manipulations with deltas and distributions would be very appreciated.



Thanks!




probability-theory measure-theory distribution-theory




share|cite|improve this question

















  • 1




    This might be helpful: stats.stackexchange.com/questions/328123/…
    – E-A
    Jul 26 at 19:33












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I have two cases I'm interested in looking at (I'm computing two KL divergences specifically).



The domain is $Omega in mathbbR^2n$. Let's say $(x, y) in mathbbR^2n$. Assume the intersection $Omega cap x=y$ has Lebesgue measure (of the corresponding dimension) greater than zero.



I have two measures, $P ll Q$, which written as "functions" (I mean, there are Delta distributions too...) would be $int dP = int_x int_y delta(x-y)f(x)dxdy$ and $int dQ = int_x int_y delta(x-y)g(x) dx dy$. I also have $tildeQ$ such that $int dtildeQ = int_x int_y tildeg(x,y) dx dy$.



OK now, are the following things correct?



  1. $int_A fracdPdQ = int_Acapx=y fracf(x)g(x)dx$.

  2. $D(P || Q) = int_x f(x) logfracf(x)g(x) dx$.

  3. $P$ is not absolutely continuous under $tildeQ$ and therefore the KL divergence does not exist.

Edit: I previously had some assertions about $D(P || tildeQ)$ that were false. $P$ is not absolutely continuous under $Q$. Is there something that can be done about that? It would be very convenient to have a way of measuring the KL divergence (for example if $Q$ is not a delta but a very narrow gaussian around the hyperplane)



Intuitively 1,2 make sense but I am struggling with how to prove it.



Any recommendation on where to look at these kind of manipulations with deltas and distributions would be very appreciated.



Thanks!




probability-theory measure-theory distribution-theory




share|cite|improve this question













I have two cases I'm interested in looking at (I'm computing two KL divergences specifically).



The domain is $Omega in mathbbR^2n$. Let's say $(x, y) in mathbbR^2n$. Assume the intersection $Omega cap x=y$ has Lebesgue measure (of the corresponding dimension) greater than zero.



I have two measures, $P ll Q$, which written as "functions" (I mean, there are Delta distributions too...) would be $int dP = int_x int_y delta(x-y)f(x)dxdy$ and $int dQ = int_x int_y delta(x-y)g(x) dx dy$. I also have $tildeQ$ such that $int dtildeQ = int_x int_y tildeg(x,y) dx dy$.



OK now, are the following things correct?



  1. $int_A fracdPdQ = int_Acapx=y fracf(x)g(x)dx$.

  2. $D(P || Q) = int_x f(x) logfracf(x)g(x) dx$.

  3. $P$ is not absolutely continuous under $tildeQ$ and therefore the KL divergence does not exist.

Edit: I previously had some assertions about $D(P || tildeQ)$ that were false. $P$ is not absolutely continuous under $Q$. Is there something that can be done about that? It would be very convenient to have a way of measuring the KL divergence (for example if $Q$ is not a delta but a very narrow gaussian around the hyperplane)



Intuitively 1,2 make sense but I am struggling with how to prove it.



Any recommendation on where to look at these kind of manipulations with deltas and distributions would be very appreciated.



Thanks!





probability-theory measure-theory distribution-theory





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 26 at 9:44
























asked Jul 26 at 9:30









bernatguillen

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  • 1




    This might be helpful: stats.stackexchange.com/questions/328123/…
    – E-A
    Jul 26 at 19:33












  • 1




    This might be helpful: stats.stackexchange.com/questions/328123/…
    – E-A
    Jul 26 at 19:33







1




1




This might be helpful: stats.stackexchange.com/questions/328123/…
– E-A
Jul 26 at 19:33




This might be helpful: stats.stackexchange.com/questions/328123/…
– E-A
Jul 26 at 19:33















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