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Trying to understand some concepts of Measure Theory for Statistical Inference
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I have the next set of measures $P_theta,thetain Theta$ for a parameter space $Theta$. I found the next statement in a book:
$$boxedtextWe assume that the measures P_theta;textare all dominated by some sigmatext-finite textmeasure Q,;textso that the density function f_X(cdot;theta);textunder P_theta;textof X;textwith respect to Q;textexists. Formally, f_X(cdot;theta);textis the Radon-Nikodym derivate of P_theta;textwith respect to Q.$$
I have never been in a Measure Theory class, so for me most of the concepts are new. I have try to study myself but I end up with the same questions.
Is there a way to understand the statement without going to the details? I just want to move on to study statistical inference.
Will I have some problems understanding some concepts of statistics if I don't understand the concepts related to the statement presentend?
probability-theory measure-theory statistical-inference
asked Jul 16 at 16:46
Roberto Cabal
1137
 |Â
show 1 more comment
up vote
1
down vote
favorite
I have the next set of measures $P_theta,thetain Theta$ for a parameter space $Theta$. I found the next statement in a book:
$$boxedtextWe assume that the measures P_theta;textare all dominated by some sigmatext-finite textmeasure Q,;textso that the density function f_X(cdot;theta);textunder P_theta;textof X;textwith respect to Q;textexists. Formally, f_X(cdot;theta);textis the Radon-Nikodym derivate of P_theta;textwith respect to Q.$$
I have never been in a Measure Theory class, so for me most of the concepts are new. I have try to study myself but I end up with the same questions.
Is there a way to understand the statement without going to the details? I just want to move on to study statistical inference.
Will I have some problems understanding some concepts of statistics if I don't understand the concepts related to the statement presentend?
probability-theory measure-theory statistical-inference
asked Jul 16 at 16:46
Roberto Cabal
1137
1
This is a fairly deep result (existence of Radon-Nikodym derivative). Just read the statement as you have a bunch of probability densities and move on (assuming you learned about densities in your elementary probability class, how you compute stuff with them and their properties ).
– Calculon
Jul 16 at 16:56
Yes, I have a solid background in probability distributions, both univariate and multivariate. But, will I have some problems understanding some concepts of statistics if I don't understand the concepts related to the statement presentend?
– Roberto Cabal
Jul 16 at 17:02
I doubt it. I would suggest you move forward and if you encounter anything you don't understand, just ask here.
– Calculon
Jul 16 at 17:04
Thanks a lot! I'll move forward.
– Roberto Cabal
Jul 16 at 17:06
You are welcome. Good luck.
– Calculon
Jul 16 at 17:12
 |Â
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have the next set of measures $P_theta,thetain Theta$ for a parameter space $Theta$. I found the next statement in a book:
$$boxedtextWe assume that the measures P_theta;textare all dominated by some sigmatext-finite textmeasure Q,;textso that the density function f_X(cdot;theta);textunder P_theta;textof X;textwith respect to Q;textexists. Formally, f_X(cdot;theta);textis the Radon-Nikodym derivate of P_theta;textwith respect to Q.$$
I have never been in a Measure Theory class, so for me most of the concepts are new. I have try to study myself but I end up with the same questions.
Is there a way to understand the statement without going to the details? I just want to move on to study statistical inference.
Will I have some problems understanding some concepts of statistics if I don't understand the concepts related to the statement presentend?
probability-theory measure-theory statistical-inference
asked Jul 16 at 16:46
Roberto Cabal
1137
I have the next set of measures $P_theta,thetain Theta$ for a parameter space $Theta$. I found the next statement in a book:
$$boxedtextWe assume that the measures P_theta;textare all dominated by some sigmatext-finite textmeasure Q,;textso that the density function f_X(cdot;theta);textunder P_theta;textof X;textwith respect to Q;textexists. Formally, f_X(cdot;theta);textis the Radon-Nikodym derivate of P_theta;textwith respect to Q.$$
I have never been in a Measure Theory class, so for me most of the concepts are new. I have try to study myself but I end up with the same questions.
Is there a way to understand the statement without going to the details? I just want to move on to study statistical inference.
Will I have some problems understanding some concepts of statistics if I don't understand the concepts related to the statement presentend?
probability-theory measure-theory statistical-inference
asked Jul 16 at 16:46
Roberto Cabal
1137
edited Jul 16 at 17:02
asked Jul 16 at 16:46
Roberto Cabal
1137
asked Jul 16 at 16:46
Roberto Cabal
1137
asked Jul 16 at 16:46
Roberto Cabal
1137
1137
1
This is a fairly deep result (existence of Radon-Nikodym derivative). Just read the statement as you have a bunch of probability densities and move on (assuming you learned about densities in your elementary probability class, how you compute stuff with them and their properties ).
– Calculon
Jul 16 at 16:56
Yes, I have a solid background in probability distributions, both univariate and multivariate. But, will I have some problems understanding some concepts of statistics if I don't understand the concepts related to the statement presentend?
– Roberto Cabal
Jul 16 at 17:02
I doubt it. I would suggest you move forward and if you encounter anything you don't understand, just ask here.
– Calculon
Jul 16 at 17:04
Thanks a lot! I'll move forward.
– Roberto Cabal
Jul 16 at 17:06
You are welcome. Good luck.
– Calculon
Jul 16 at 17:12
 |Â
show 1 more comment
1
This is a fairly deep result (existence of Radon-Nikodym derivative). Just read the statement as you have a bunch of probability densities and move on (assuming you learned about densities in your elementary probability class, how you compute stuff with them and their properties ).
– Calculon
Jul 16 at 16:56
Yes, I have a solid background in probability distributions, both univariate and multivariate. But, will I have some problems understanding some concepts of statistics if I don't understand the concepts related to the statement presentend?
– Roberto Cabal
Jul 16 at 17:02
I doubt it. I would suggest you move forward and if you encounter anything you don't understand, just ask here.
– Calculon
Jul 16 at 17:04
Thanks a lot! I'll move forward.
– Roberto Cabal
Jul 16 at 17:06
You are welcome. Good luck.
– Calculon
Jul 16 at 17:12
1
1
This is a fairly deep result (existence of Radon-Nikodym derivative). Just read the statement as you have a bunch of probability densities and move on (assuming you learned about densities in your elementary probability class, how you compute stuff with them and their properties ).
– Calculon
Jul 16 at 16:56
This is a fairly deep result (existence of Radon-Nikodym derivative). Just read the statement as you have a bunch of probability densities and move on (assuming you learned about densities in your elementary probability class, how you compute stuff with them and their properties ).
– Calculon
Jul 16 at 16:56
Yes, I have a solid background in probability distributions, both univariate and multivariate. But, will I have some problems understanding some concepts of statistics if I don't understand the concepts related to the statement presentend?
– Roberto Cabal
Jul 16 at 17:02
Yes, I have a solid background in probability distributions, both univariate and multivariate. But, will I have some problems understanding some concepts of statistics if I don't understand the concepts related to the statement presentend?
– Roberto Cabal
Jul 16 at 17:02
I doubt it. I would suggest you move forward and if you encounter anything you don't understand, just ask here.
– Calculon
Jul 16 at 17:04
I doubt it. I would suggest you move forward and if you encounter anything you don't understand, just ask here.
– Calculon
Jul 16 at 17:04
Thanks a lot! I'll move forward.
– Roberto Cabal
Jul 16 at 17:06
Thanks a lot! I'll move forward.
– Roberto Cabal
Jul 16 at 17:06
You are welcome. Good luck.
– Calculon
Jul 16 at 17:12
You are welcome. Good luck.
– Calculon
Jul 16 at 17:12
 |Â
show 1 more comment
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1
This is a fairly deep result (existence of Radon-Nikodym derivative). Just read the statement as you have a bunch of probability densities and move on (assuming you learned about densities in your elementary probability class, how you compute stuff with them and their properties ).
– Calculon
Jul 16 at 16:56
Yes, I have a solid background in probability distributions, both univariate and multivariate. But, will I have some problems understanding some concepts of statistics if I don't understand the concepts related to the statement presentend?
– Roberto Cabal
Jul 16 at 17:02
I doubt it. I would suggest you move forward and if you encounter anything you don't understand, just ask here.
– Calculon
Jul 16 at 17:04
Thanks a lot! I'll move forward.
– Roberto Cabal
Jul 16 at 17:06
You are welcome. Good luck.
– Calculon
Jul 16 at 17:12