Mixing
How does the CLT justify statistical models which are not modeling our data as a sum of random variables?
$begingroup$
Wikipedia says (emphasis mine):
The Central Limit Theorem states "that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.
Now the CLT says something about the sum of independent random variables:
$$
X = X_1 + X_2 + dots X_n
$$
where $X_i$ is the $i$th draw of a random variable. But the statistical models I am familiar with do not model the sum of random variables; they model the random variable directly.
For example, factor analysis models a random variable $textbf{x}$ as
$$
textbf{x} sim mathcal{N}(textbf{0}, Lambda Lambda^{top} + Psi)
$$
(It does not matter what $Lambda$ and $Psi$ are; only that $textbf{x}$ is modeled as Gaussian.) And this modeling assumption is justified using the CLT (see A Unifying Review of Linear Gaussian Models, footnote on page 2). But aren't we modeling $textbf{x}$, not the sum of $textbf{x}$?
In summary: How does the CLT justify statistical models which are not modeling our data as a sum of random variables?
probability statistical-inference central-limit-theorem
asked Jan 24 at 23:19
gwggwg
1,00711023
$endgroup$
add a comment |
$begingroup$
Wikipedia says (emphasis mine):
The Central Limit Theorem states "that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.
Now the CLT says something about the sum of independent random variables:
$$
X = X_1 + X_2 + dots X_n
$$
where $X_i$ is the $i$th draw of a random variable. But the statistical models I am familiar with do not model the sum of random variables; they model the random variable directly.
For example, factor analysis models a random variable $textbf{x}$ as
$$
textbf{x} sim mathcal{N}(textbf{0}, Lambda Lambda^{top} + Psi)
$$
(It does not matter what $Lambda$ and $Psi$ are; only that $textbf{x}$ is modeled as Gaussian.) And this modeling assumption is justified using the CLT (see A Unifying Review of Linear Gaussian Models, footnote on page 2). But aren't we modeling $textbf{x}$, not the sum of $textbf{x}$?
In summary: How does the CLT justify statistical models which are not modeling our data as a sum of random variables?
probability statistical-inference central-limit-theorem
asked Jan 24 at 23:19
gwggwg
1,00711023
$endgroup$
$begingroup$
It doesn't, but many processes turn out to be a sum of other ones at a deeper level. In the case of your footnote, the variable of interest is noise, and noise is formed when there is interference (read: a sum) of many different external sources.
$endgroup$
– nathan.j.mcdougall
Jan 24 at 23:36
$begingroup$
I see. This makes a lot of sense. It's too lengthy for a comment, but it explains a few other things in that paper as well. Basically, the latent variable in factor analysis—which is typically denoted $textbf{z} sim mathcal{N}(0, 1)$—is actually a constant with additive Gaussian noise as well.
$endgroup$
– gwg
Jan 25 at 0:45
add a comment |
$begingroup$
Wikipedia says (emphasis mine):
The Central Limit Theorem states "that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.
Now the CLT says something about the sum of independent random variables:
$$
X = X_1 + X_2 + dots X_n
$$
where $X_i$ is the $i$th draw of a random variable. But the statistical models I am familiar with do not model the sum of random variables; they model the random variable directly.
For example, factor analysis models a random variable $textbf{x}$ as
$$
textbf{x} sim mathcal{N}(textbf{0}, Lambda Lambda^{top} + Psi)
$$
(It does not matter what $Lambda$ and $Psi$ are; only that $textbf{x}$ is modeled as Gaussian.) And this modeling assumption is justified using the CLT (see A Unifying Review of Linear Gaussian Models, footnote on page 2). But aren't we modeling $textbf{x}$, not the sum of $textbf{x}$?
In summary: How does the CLT justify statistical models which are not modeling our data as a sum of random variables?
probability statistical-inference central-limit-theorem
asked Jan 24 at 23:19
gwggwg
1,00711023
$endgroup$
Wikipedia says (emphasis mine):
The Central Limit Theorem states "that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.
Now the CLT says something about the sum of independent random variables:
$$
X = X_1 + X_2 + dots X_n
$$
where $X_i$ is the $i$th draw of a random variable. But the statistical models I am familiar with do not model the sum of random variables; they model the random variable directly.
For example, factor analysis models a random variable $textbf{x}$ as
$$
textbf{x} sim mathcal{N}(textbf{0}, Lambda Lambda^{top} + Psi)
$$
(It does not matter what $Lambda$ and $Psi$ are; only that $textbf{x}$ is modeled as Gaussian.) And this modeling assumption is justified using the CLT (see A Unifying Review of Linear Gaussian Models, footnote on page 2). But aren't we modeling $textbf{x}$, not the sum of $textbf{x}$?
In summary: How does the CLT justify statistical models which are not modeling our data as a sum of random variables?
probability statistical-inference central-limit-theorem
probability statistical-inference central-limit-theorem
asked Jan 24 at 23:19
gwggwg
1,00711023
asked Jan 24 at 23:19
gwggwg
1,00711023
asked Jan 24 at 23:19
gwggwg
1,00711023
asked Jan 24 at 23:19
gwggwg
1,00711023
asked Jan 24 at 23:19
gwggwg
1,00711023
1,00711023
$begingroup$
It doesn't, but many processes turn out to be a sum of other ones at a deeper level. In the case of your footnote, the variable of interest is noise, and noise is formed when there is interference (read: a sum) of many different external sources.
$endgroup$
– nathan.j.mcdougall
Jan 24 at 23:36
$begingroup$
I see. This makes a lot of sense. It's too lengthy for a comment, but it explains a few other things in that paper as well. Basically, the latent variable in factor analysis—which is typically denoted $textbf{z} sim mathcal{N}(0, 1)$—is actually a constant with additive Gaussian noise as well.
$endgroup$
– gwg
Jan 25 at 0:45
add a comment |
$begingroup$
It doesn't, but many processes turn out to be a sum of other ones at a deeper level. In the case of your footnote, the variable of interest is noise, and noise is formed when there is interference (read: a sum) of many different external sources.
$endgroup$
– nathan.j.mcdougall
Jan 24 at 23:36
$begingroup$
I see. This makes a lot of sense. It's too lengthy for a comment, but it explains a few other things in that paper as well. Basically, the latent variable in factor analysis—which is typically denoted $textbf{z} sim mathcal{N}(0, 1)$—is actually a constant with additive Gaussian noise as well.
$endgroup$
– gwg
Jan 25 at 0:45
$begingroup$
It doesn't, but many processes turn out to be a sum of other ones at a deeper level. In the case of your footnote, the variable of interest is noise, and noise is formed when there is interference (read: a sum) of many different external sources.
$endgroup$
– nathan.j.mcdougall
Jan 24 at 23:36
$begingroup$
It doesn't, but many processes turn out to be a sum of other ones at a deeper level. In the case of your footnote, the variable of interest is noise, and noise is formed when there is interference (read: a sum) of many different external sources.
$endgroup$
– nathan.j.mcdougall
Jan 24 at 23:36
$begingroup$
I see. This makes a lot of sense. It's too lengthy for a comment, but it explains a few other things in that paper as well. Basically, the latent variable in factor analysis—which is typically denoted $textbf{z} sim mathcal{N}(0, 1)$—is actually a constant with additive Gaussian noise as well.
$endgroup$
– gwg
Jan 25 at 0:45
$begingroup$
I see. This makes a lot of sense. It's too lengthy for a comment, but it explains a few other things in that paper as well. Basically, the latent variable in factor analysis—which is typically denoted $textbf{z} sim mathcal{N}(0, 1)$—is actually a constant with additive Gaussian noise as well.
$endgroup$
– gwg
Jan 25 at 0:45
add a comment |
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$begingroup$
It doesn't, but many processes turn out to be a sum of other ones at a deeper level. In the case of your footnote, the variable of interest is noise, and noise is formed when there is interference (read: a sum) of many different external sources.
$endgroup$
– nathan.j.mcdougall
Jan 24 at 23:36
$begingroup$
I see. This makes a lot of sense. It's too lengthy for a comment, but it explains a few other things in that paper as well. Basically, the latent variable in factor analysis—which is typically denoted $textbf{z} sim mathcal{N}(0, 1)$—is actually a constant with additive Gaussian noise as well.
$endgroup$
– gwg
Jan 25 at 0:45