Measure-theoretic definition of conditional probability
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I am trying to reconcile the formal definition of conditional probability via measure theory (as in this question's answer and the Wikipedia definition given here) with the more elementary definitions of conditional probability given in early probability courses.
By elementary definitions, I mean the following: for two continuous real-valued r.v. $X, Y$ which have densities $f_X, f_Y$ (with respect to the Lebesgue measure), we define the conditional density of $X$ given $Y$ to be $$f_{X|Y}(x|y) = frac{f_{X,Y}(x,y)}{f_Y(y)} text{ when } f_Y(y) > 0$$ where $f_{X,Y}$ is the joint density of $X,Y$ (with respect to the product measure on $mathbb{R}^2$). We have similar equations in the mixture (discrete-continuous) and discrete-discrete cases, as given on Wikipedia here.
Question: How can we derive these "definitions" from the more general notions of conditional probability distributions defined via measure theory?
I am also trying to make sense of things like $f_{X,Y|Z}(x,y|z)$, i.e. the conditional density/distribution of $X,Y$ given $Z$ (which could be any combination of discrete and continuous r.v.'s or even taking values in a general space). It seems cumbersome to have to define all these cases in the elementary way. I hope answering my question will also clarify this.
Thank you.
probability-theory measure-theory conditional-probability
asked Dec 12 '18 at 15:52
dstivddstivd
616
$endgroup$
add a comment |
$begingroup$
I am trying to reconcile the formal definition of conditional probability via measure theory (as in this question's answer and the Wikipedia definition given here) with the more elementary definitions of conditional probability given in early probability courses.
By elementary definitions, I mean the following: for two continuous real-valued r.v. $X, Y$ which have densities $f_X, f_Y$ (with respect to the Lebesgue measure), we define the conditional density of $X$ given $Y$ to be $$f_{X|Y}(x|y) = frac{f_{X,Y}(x,y)}{f_Y(y)} text{ when } f_Y(y) > 0$$ where $f_{X,Y}$ is the joint density of $X,Y$ (with respect to the product measure on $mathbb{R}^2$). We have similar equations in the mixture (discrete-continuous) and discrete-discrete cases, as given on Wikipedia here.
Question: How can we derive these "definitions" from the more general notions of conditional probability distributions defined via measure theory?
I am also trying to make sense of things like $f_{X,Y|Z}(x,y|z)$, i.e. the conditional density/distribution of $X,Y$ given $Z$ (which could be any combination of discrete and continuous r.v.'s or even taking values in a general space). It seems cumbersome to have to define all these cases in the elementary way. I hope answering my question will also clarify this.
Thank you.
probability-theory measure-theory conditional-probability
asked Dec 12 '18 at 15:52
dstivddstivd
616
$endgroup$
$begingroup$
Are you asking how the "elementary" definition of conditional density fits in the abstract measure-theoretical one?
$endgroup$
– user587192
Dec 12 '18 at 16:01
$begingroup$
@user587192: Yes, that's right. More generally, there is also the mixture case which involves probability mass and density functions (as in this link given in the question). How do all of these fit in with the abstract measure-theoretic definition?
$endgroup$
– dstivd
Dec 12 '18 at 16:21
add a comment |
$begingroup$
I am trying to reconcile the formal definition of conditional probability via measure theory (as in this question's answer and the Wikipedia definition given here) with the more elementary definitions of conditional probability given in early probability courses.
By elementary definitions, I mean the following: for two continuous real-valued r.v. $X, Y$ which have densities $f_X, f_Y$ (with respect to the Lebesgue measure), we define the conditional density of $X$ given $Y$ to be $$f_{X|Y}(x|y) = frac{f_{X,Y}(x,y)}{f_Y(y)} text{ when } f_Y(y) > 0$$ where $f_{X,Y}$ is the joint density of $X,Y$ (with respect to the product measure on $mathbb{R}^2$). We have similar equations in the mixture (discrete-continuous) and discrete-discrete cases, as given on Wikipedia here.
Question: How can we derive these "definitions" from the more general notions of conditional probability distributions defined via measure theory?
I am also trying to make sense of things like $f_{X,Y|Z}(x,y|z)$, i.e. the conditional density/distribution of $X,Y$ given $Z$ (which could be any combination of discrete and continuous r.v.'s or even taking values in a general space). It seems cumbersome to have to define all these cases in the elementary way. I hope answering my question will also clarify this.
Thank you.
probability-theory measure-theory conditional-probability
asked Dec 12 '18 at 15:52
dstivddstivd
616
$endgroup$
I am trying to reconcile the formal definition of conditional probability via measure theory (as in this question's answer and the Wikipedia definition given here) with the more elementary definitions of conditional probability given in early probability courses.
By elementary definitions, I mean the following: for two continuous real-valued r.v. $X, Y$ which have densities $f_X, f_Y$ (with respect to the Lebesgue measure), we define the conditional density of $X$ given $Y$ to be $$f_{X|Y}(x|y) = frac{f_{X,Y}(x,y)}{f_Y(y)} text{ when } f_Y(y) > 0$$ where $f_{X,Y}$ is the joint density of $X,Y$ (with respect to the product measure on $mathbb{R}^2$). We have similar equations in the mixture (discrete-continuous) and discrete-discrete cases, as given on Wikipedia here.
Question: How can we derive these "definitions" from the more general notions of conditional probability distributions defined via measure theory?
I am also trying to make sense of things like $f_{X,Y|Z}(x,y|z)$, i.e. the conditional density/distribution of $X,Y$ given $Z$ (which could be any combination of discrete and continuous r.v.'s or even taking values in a general space). It seems cumbersome to have to define all these cases in the elementary way. I hope answering my question will also clarify this.
Thank you.
probability-theory measure-theory conditional-probability
probability-theory measure-theory conditional-probability
asked Dec 12 '18 at 15:52
dstivddstivd
616
asked Dec 12 '18 at 15:52
dstivddstivd
616
asked Dec 12 '18 at 15:52
dstivddstivd
616
asked Dec 12 '18 at 15:52
dstivddstivd
616
asked Dec 12 '18 at 15:52
dstivddstivd
616
616
$begingroup$
Are you asking how the "elementary" definition of conditional density fits in the abstract measure-theoretical one?
$endgroup$
– user587192
Dec 12 '18 at 16:01
$begingroup$
@user587192: Yes, that's right. More generally, there is also the mixture case which involves probability mass and density functions (as in this link given in the question). How do all of these fit in with the abstract measure-theoretic definition?
$endgroup$
– dstivd
Dec 12 '18 at 16:21
add a comment |
$begingroup$
Are you asking how the "elementary" definition of conditional density fits in the abstract measure-theoretical one?
$endgroup$
– user587192
Dec 12 '18 at 16:01
$begingroup$
@user587192: Yes, that's right. More generally, there is also the mixture case which involves probability mass and density functions (as in this link given in the question). How do all of these fit in with the abstract measure-theoretic definition?
$endgroup$
– dstivd
Dec 12 '18 at 16:21
$begingroup$
Are you asking how the "elementary" definition of conditional density fits in the abstract measure-theoretical one?
$endgroup$
– user587192
Dec 12 '18 at 16:01
$begingroup$
Are you asking how the "elementary" definition of conditional density fits in the abstract measure-theoretical one?
$endgroup$
– user587192
Dec 12 '18 at 16:01
$begingroup$
@user587192: Yes, that's right. More generally, there is also the mixture case which involves probability mass and density functions (as in this link given in the question). How do all of these fit in with the abstract measure-theoretic definition?
$endgroup$
– dstivd
Dec 12 '18 at 16:21
$begingroup$
@user587192: Yes, that's right. More generally, there is also the mixture case which involves probability mass and density functions (as in this link given in the question). How do all of these fit in with the abstract measure-theoretic definition?
$endgroup$
– dstivd
Dec 12 '18 at 16:21
add a comment |
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$begingroup$
Are you asking how the "elementary" definition of conditional density fits in the abstract measure-theoretical one?
$endgroup$
– user587192
Dec 12 '18 at 16:01
$begingroup$
@user587192: Yes, that's right. More generally, there is also the mixture case which involves probability mass and density functions (as in this link given in the question). How do all of these fit in with the abstract measure-theoretic definition?
$endgroup$
– dstivd
Dec 12 '18 at 16:21