Clarification on mixture of pdf and mixture of cdf
I have a question on how to define mixture distributions for continuous random variables. In short, what I'm confused about is whether they can be equivalently written using the cdf (cumulative distribution function) or the pdf (probability density function).
Let me explain my doubt with an example.
Let $Y,X,W$ be real-valued random variables, respectively with supports denoted by $mathcal{Y},mathcal{X},mathcal{W}$.
(A1) Assume that $mathcal{X},mathcal{W}$ are finite.
(A2) Assume that $Y$ is a continuous random variable, with well defined pdf.
Consider now the pdf $f(cdot)$ of $Y$ evaluated at $yin mathcal{Y}$:
$$f(y)=sum_{xin mathcal{X},win mathcal{W}} f(y,x,w)=overbrace{sum_{xin mathcal{X},win mathcal{W}} overbrace{p(x,w)}^{text{Mixing weight}}times overbrace{f(y| X=x, W=w)}^{text{Mixing density}}}^{text{Finite mixture}}
$$
where $p(cdot,cdot)$ is the probability mass function of $(X,W)$.
It seems to me that the pdf of $Y$ can be written as a finite mixture (assuming that $f(y|X=x, W=w)$ is well-defined at each $xin mathcal{X}, win mathcal{W}$).
I don't think though that the same relation holds for the cdf of $Y$. In other words, let $F(cdot)$ denote the cdf of $Y$; I think that
$$
F(y)neq sum_{xin mathcal{X},win mathcal{W}} p(x,w)times F(y| X=x, W=w)
$$
Therefore, I would like you help to understand the following:
Can we write $F(cdot)$ as a mixture?
If yes, using which weights and what is the relation of those weights with the weights ${p(x,w)}_{x,w}$ used above?
If not, why do I have the impression that in the stat literature mixtures are flexibly defined using cdf or pdf, as in this Wikipedia article?
probability probability-theory probability-distributions random-variables conditional-probability
asked Dec 28 '18 at 15:12
STFSTF
801420
add a comment |
I have a question on how to define mixture distributions for continuous random variables. In short, what I'm confused about is whether they can be equivalently written using the cdf (cumulative distribution function) or the pdf (probability density function).
Let me explain my doubt with an example.
Let $Y,X,W$ be real-valued random variables, respectively with supports denoted by $mathcal{Y},mathcal{X},mathcal{W}$.
(A1) Assume that $mathcal{X},mathcal{W}$ are finite.
(A2) Assume that $Y$ is a continuous random variable, with well defined pdf.
Consider now the pdf $f(cdot)$ of $Y$ evaluated at $yin mathcal{Y}$:
$$f(y)=sum_{xin mathcal{X},win mathcal{W}} f(y,x,w)=overbrace{sum_{xin mathcal{X},win mathcal{W}} overbrace{p(x,w)}^{text{Mixing weight}}times overbrace{f(y| X=x, W=w)}^{text{Mixing density}}}^{text{Finite mixture}}
$$
where $p(cdot,cdot)$ is the probability mass function of $(X,W)$.
It seems to me that the pdf of $Y$ can be written as a finite mixture (assuming that $f(y|X=x, W=w)$ is well-defined at each $xin mathcal{X}, win mathcal{W}$).
I don't think though that the same relation holds for the cdf of $Y$. In other words, let $F(cdot)$ denote the cdf of $Y$; I think that
$$
F(y)neq sum_{xin mathcal{X},win mathcal{W}} p(x,w)times F(y| X=x, W=w)
$$
Therefore, I would like you help to understand the following:
Can we write $F(cdot)$ as a mixture?
If yes, using which weights and what is the relation of those weights with the weights ${p(x,w)}_{x,w}$ used above?
If not, why do I have the impression that in the stat literature mixtures are flexibly defined using cdf or pdf, as in this Wikipedia article?
probability probability-theory probability-distributions random-variables conditional-probability
asked Dec 28 '18 at 15:12
STFSTF
801420
add a comment |
I have a question on how to define mixture distributions for continuous random variables. In short, what I'm confused about is whether they can be equivalently written using the cdf (cumulative distribution function) or the pdf (probability density function).
Let me explain my doubt with an example.
Let $Y,X,W$ be real-valued random variables, respectively with supports denoted by $mathcal{Y},mathcal{X},mathcal{W}$.
(A1) Assume that $mathcal{X},mathcal{W}$ are finite.
(A2) Assume that $Y$ is a continuous random variable, with well defined pdf.
Consider now the pdf $f(cdot)$ of $Y$ evaluated at $yin mathcal{Y}$:
$$f(y)=sum_{xin mathcal{X},win mathcal{W}} f(y,x,w)=overbrace{sum_{xin mathcal{X},win mathcal{W}} overbrace{p(x,w)}^{text{Mixing weight}}times overbrace{f(y| X=x, W=w)}^{text{Mixing density}}}^{text{Finite mixture}}
$$
where $p(cdot,cdot)$ is the probability mass function of $(X,W)$.
It seems to me that the pdf of $Y$ can be written as a finite mixture (assuming that $f(y|X=x, W=w)$ is well-defined at each $xin mathcal{X}, win mathcal{W}$).
I don't think though that the same relation holds for the cdf of $Y$. In other words, let $F(cdot)$ denote the cdf of $Y$; I think that
$$
F(y)neq sum_{xin mathcal{X},win mathcal{W}} p(x,w)times F(y| X=x, W=w)
$$
Therefore, I would like you help to understand the following:
Can we write $F(cdot)$ as a mixture?
If yes, using which weights and what is the relation of those weights with the weights ${p(x,w)}_{x,w}$ used above?
If not, why do I have the impression that in the stat literature mixtures are flexibly defined using cdf or pdf, as in this Wikipedia article?
probability probability-theory probability-distributions random-variables conditional-probability
asked Dec 28 '18 at 15:12
STFSTF
801420
I have a question on how to define mixture distributions for continuous random variables. In short, what I'm confused about is whether they can be equivalently written using the cdf (cumulative distribution function) or the pdf (probability density function).
Let me explain my doubt with an example.
Let $Y,X,W$ be real-valued random variables, respectively with supports denoted by $mathcal{Y},mathcal{X},mathcal{W}$.
(A1) Assume that $mathcal{X},mathcal{W}$ are finite.
(A2) Assume that $Y$ is a continuous random variable, with well defined pdf.
Consider now the pdf $f(cdot)$ of $Y$ evaluated at $yin mathcal{Y}$:
$$f(y)=sum_{xin mathcal{X},win mathcal{W}} f(y,x,w)=overbrace{sum_{xin mathcal{X},win mathcal{W}} overbrace{p(x,w)}^{text{Mixing weight}}times overbrace{f(y| X=x, W=w)}^{text{Mixing density}}}^{text{Finite mixture}}
$$
where $p(cdot,cdot)$ is the probability mass function of $(X,W)$.
It seems to me that the pdf of $Y$ can be written as a finite mixture (assuming that $f(y|X=x, W=w)$ is well-defined at each $xin mathcal{X}, win mathcal{W}$).
I don't think though that the same relation holds for the cdf of $Y$. In other words, let $F(cdot)$ denote the cdf of $Y$; I think that
$$
F(y)neq sum_{xin mathcal{X},win mathcal{W}} p(x,w)times F(y| X=x, W=w)
$$
Therefore, I would like you help to understand the following:
Can we write $F(cdot)$ as a mixture?
If yes, using which weights and what is the relation of those weights with the weights ${p(x,w)}_{x,w}$ used above?
If not, why do I have the impression that in the stat literature mixtures are flexibly defined using cdf or pdf, as in this Wikipedia article?
probability probability-theory probability-distributions random-variables conditional-probability
probability probability-theory probability-distributions random-variables conditional-probability
asked Dec 28 '18 at 15:12
STFSTF
801420
asked Dec 28 '18 at 15:12
STFSTF
801420
edited Dec 28 '18 at 15:30
STF
asked Dec 28 '18 at 15:12
STFSTF
801420
asked Dec 28 '18 at 15:12
STFSTF
801420
asked Dec 28 '18 at 15:12
STFSTF
801420
801420
add a comment |
add a comment |
1 Answer
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For any Borel set $B$ we can write:$$P(Yin B)=sum_{xin mathcal{X},win mathcal{W}}P(X=x,W=w)P(Yin Bmid X=x,W=w)=sum_{xin mathcal{X},win mathcal{W}}p(x,w)P_{x,w}(B)$$
Doing so for $B=(-infty,y]$ we find something like:
$$F_Y(y)=sum_{xin mathcal{X},win mathcal{W}}p(x,w)F_{x,w}(y)$$where every $F_{x,w}$ can be recognized as a CDF.
answered Dec 28 '18 at 15:41
drhabdrhab
98.4k544129
Thanks. From your claim, I understand that I can write BOTH (1) $f(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) f(y|X=x, W=w)$ (i.e., the mixture with the pdf) and (2) $F(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) F(y|X=x, W=w)$ (i.e., the mixture with the cdf). Notice that (1) and (2) use the same mixing weights. Correct?
– STF
Dec 28 '18 at 15:43
1
Yes, that is correct.
– drhab
Dec 28 '18 at 15:44
add a comment |
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For any Borel set $B$ we can write:$$P(Yin B)=sum_{xin mathcal{X},win mathcal{W}}P(X=x,W=w)P(Yin Bmid X=x,W=w)=sum_{xin mathcal{X},win mathcal{W}}p(x,w)P_{x,w}(B)$$
Doing so for $B=(-infty,y]$ we find something like:
$$F_Y(y)=sum_{xin mathcal{X},win mathcal{W}}p(x,w)F_{x,w}(y)$$where every $F_{x,w}$ can be recognized as a CDF.
answered Dec 28 '18 at 15:41
drhabdrhab
98.4k544129
Thanks. From your claim, I understand that I can write BOTH (1) $f(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) f(y|X=x, W=w)$ (i.e., the mixture with the pdf) and (2) $F(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) F(y|X=x, W=w)$ (i.e., the mixture with the cdf). Notice that (1) and (2) use the same mixing weights. Correct?
– STF
Dec 28 '18 at 15:43
1
Yes, that is correct.
– drhab
Dec 28 '18 at 15:44
add a comment |
For any Borel set $B$ we can write:$$P(Yin B)=sum_{xin mathcal{X},win mathcal{W}}P(X=x,W=w)P(Yin Bmid X=x,W=w)=sum_{xin mathcal{X},win mathcal{W}}p(x,w)P_{x,w}(B)$$
Doing so for $B=(-infty,y]$ we find something like:
$$F_Y(y)=sum_{xin mathcal{X},win mathcal{W}}p(x,w)F_{x,w}(y)$$where every $F_{x,w}$ can be recognized as a CDF.
answered Dec 28 '18 at 15:41
drhabdrhab
98.4k544129
Thanks. From your claim, I understand that I can write BOTH (1) $f(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) f(y|X=x, W=w)$ (i.e., the mixture with the pdf) and (2) $F(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) F(y|X=x, W=w)$ (i.e., the mixture with the cdf). Notice that (1) and (2) use the same mixing weights. Correct?
– STF
Dec 28 '18 at 15:43
1
Yes, that is correct.
– drhab
Dec 28 '18 at 15:44
add a comment |
For any Borel set $B$ we can write:$$P(Yin B)=sum_{xin mathcal{X},win mathcal{W}}P(X=x,W=w)P(Yin Bmid X=x,W=w)=sum_{xin mathcal{X},win mathcal{W}}p(x,w)P_{x,w}(B)$$
Doing so for $B=(-infty,y]$ we find something like:
$$F_Y(y)=sum_{xin mathcal{X},win mathcal{W}}p(x,w)F_{x,w}(y)$$where every $F_{x,w}$ can be recognized as a CDF.
answered Dec 28 '18 at 15:41
drhabdrhab
98.4k544129
For any Borel set $B$ we can write:$$P(Yin B)=sum_{xin mathcal{X},win mathcal{W}}P(X=x,W=w)P(Yin Bmid X=x,W=w)=sum_{xin mathcal{X},win mathcal{W}}p(x,w)P_{x,w}(B)$$
Doing so for $B=(-infty,y]$ we find something like:
$$F_Y(y)=sum_{xin mathcal{X},win mathcal{W}}p(x,w)F_{x,w}(y)$$where every $F_{x,w}$ can be recognized as a CDF.
answered Dec 28 '18 at 15:41
drhabdrhab
98.4k544129
answered Dec 28 '18 at 15:41
drhabdrhab
98.4k544129
answered Dec 28 '18 at 15:41
drhabdrhab
98.4k544129
answered Dec 28 '18 at 15:41
drhabdrhab
98.4k544129
98.4k544129
Thanks. From your claim, I understand that I can write BOTH (1) $f(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) f(y|X=x, W=w)$ (i.e., the mixture with the pdf) and (2) $F(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) F(y|X=x, W=w)$ (i.e., the mixture with the cdf). Notice that (1) and (2) use the same mixing weights. Correct?
– STF
Dec 28 '18 at 15:43
1
Yes, that is correct.
– drhab
Dec 28 '18 at 15:44
add a comment |
Thanks. From your claim, I understand that I can write BOTH (1) $f(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) f(y|X=x, W=w)$ (i.e., the mixture with the pdf) and (2) $F(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) F(y|X=x, W=w)$ (i.e., the mixture with the cdf). Notice that (1) and (2) use the same mixing weights. Correct?
– STF
Dec 28 '18 at 15:43
1
Yes, that is correct.
– drhab
Dec 28 '18 at 15:44
Thanks. From your claim, I understand that I can write BOTH (1) $f(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) f(y|X=x, W=w)$ (i.e., the mixture with the pdf) and (2) $F(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) F(y|X=x, W=w)$ (i.e., the mixture with the cdf). Notice that (1) and (2) use the same mixing weights. Correct?
– STF
Dec 28 '18 at 15:43
Thanks. From your claim, I understand that I can write BOTH (1) $f(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) f(y|X=x, W=w)$ (i.e., the mixture with the pdf) and (2) $F(y)=sum_{xin mathcal{X}, win mathcal{W}} p(x,w) F(y|X=x, W=w)$ (i.e., the mixture with the cdf). Notice that (1) and (2) use the same mixing weights. Correct?
– STF
Dec 28 '18 at 15:43
1
1
Yes, that is correct.
– drhab
Dec 28 '18 at 15:44
Yes, that is correct.
– drhab
Dec 28 '18 at 15:44
add a comment |
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