MLE for $Sigma$ based on sequences of observations
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Given $bf{x}$$_1$,...,$bf{x}$$_n$ be a sequence of random vectors that are independent and identically distributed from $N_p(mu_0,Sigma)$ where $mu_0$ is known.
(i) Show that the MLE for $Sigma$ is;
$$
widehat{Sigma}=frac{1}{n}sumlimits_{i=1}^n(mathbf{x}_i-mu_0)(mathbf{x}_i-mu_0)'
$$
(ii) Now let $bf{y}$$_1$,...,$bf{y}$$_m$ be another sequence of random vectors that are independent and identically distributed from $N_p(mu_1,Sigma)$ where $mu_1$ is known. Calculate the MLE for $Sigma$ based on the sequences of observations {$bf{x}$$_1$,...,$bf{x}$$_n$} and {$bf{y}$$_1$,...,$bf{y}$$_m$}. What happens to $Sigma$ if both $mu_i$ for $i=0,1$ are assumed to be unknown?
I do not have any questions on part (i), I have already shown that the MLE for $Sigma$ is indeed $widehat{Sigma}$. My concerns are towards part (ii). I don't quite understand in what ways $widehat{Sigma}$ will change under this scenario and how to represent it notation-wise. Also, when $mu_i$ for $i=0,1$ are assumed to be unknown; do we simply replace $mu_0$ and $mu_1$ in the new MLE for $Sigma$ by $hat{mathbf{mu}}_0$ and $hat{mathbf{mu}}_1$?
Any form of help is much appreciated.
statistics normal-distribution maximum-likelihood
asked Nov 26 at 2:28
Nelly
74110
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show 1 more comment
up vote
0
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Given $bf{x}$$_1$,...,$bf{x}$$_n$ be a sequence of random vectors that are independent and identically distributed from $N_p(mu_0,Sigma)$ where $mu_0$ is known.
(i) Show that the MLE for $Sigma$ is;
$$
widehat{Sigma}=frac{1}{n}sumlimits_{i=1}^n(mathbf{x}_i-mu_0)(mathbf{x}_i-mu_0)'
$$
(ii) Now let $bf{y}$$_1$,...,$bf{y}$$_m$ be another sequence of random vectors that are independent and identically distributed from $N_p(mu_1,Sigma)$ where $mu_1$ is known. Calculate the MLE for $Sigma$ based on the sequences of observations {$bf{x}$$_1$,...,$bf{x}$$_n$} and {$bf{y}$$_1$,...,$bf{y}$$_m$}. What happens to $Sigma$ if both $mu_i$ for $i=0,1$ are assumed to be unknown?
I do not have any questions on part (i), I have already shown that the MLE for $Sigma$ is indeed $widehat{Sigma}$. My concerns are towards part (ii). I don't quite understand in what ways $widehat{Sigma}$ will change under this scenario and how to represent it notation-wise. Also, when $mu_i$ for $i=0,1$ are assumed to be unknown; do we simply replace $mu_0$ and $mu_1$ in the new MLE for $Sigma$ by $hat{mathbf{mu}}_0$ and $hat{mathbf{mu}}_1$?
Any form of help is much appreciated.
statistics normal-distribution maximum-likelihood
asked Nov 26 at 2:28
Nelly
74110
I think you need to find the joint probability $$p(mathbf{x}_1,ldots,mathbf{x}_n,mathbf{y}_1,ldots,mathbf{y}_m)$$ and extract the covariance matrix from that.
– BlackMath
Nov 26 at 3:31
Thus, would claiming $(mathbf{x}_1,...,mathbf{x}_n,mathbf{y}_1,...,mathbf{y}_n) sim N_p(nmu_0+mmu_1,(n+m)Sigma)$ and then considering the MLE for $(n+m)Sigma$ be the correct approach?
– Nelly
Nov 26 at 5:39
How did you find the covariance in part (i)? I would assume that you found the joint PDF $p(mathbf{x}_1,ldots,mathbf{x}_n)$, which is a multivariate Gaussian distribution, and from there you found the covariance matrix. Here I think it's the same thing, but you need to find the joint PDF $p(mathbf{x}_1,ldots,mathbf{x}_n,mathbf{y}_1,ldots,mathbf{y}_m)$ instead. Since these are independent, this can be written as $$prod_ip(mathbf{x}_i)prod_jp(mathbf{y}_j)$$
– BlackMath
Nov 26 at 6:00
In part (i) I derived $widehat{Sigma}$ by using the maximum likelihood method of finding the log-likelihood, taking the partial derivative with respect to $Sigma^{-1}$ and setting the equation equal to 0. I didn't necessarily find $Sigma$ as you're suggesting. Maybe I'm just not following what you're saying?
– Nelly
Nov 26 at 7:02
1
I'm describing the same thing, I think. You need first to find the joint PDF, find the log-likelihood of that function, and then do the derivation. The joint PDF in the first case is $$prod_ip(mathbf{x}_i)$$ while in the second case it will be $$prod_ip(mathbf{x}_i)prod_jp(mathbf{y}_j)$$ where $$p(mathbf{x}_i)=text{det}left(2piSigmaright)^{-1/2}expleft(-frac{1}{2}(mathbf{x}_i-mu_0)^TSigma^{-1}(mathbf{x}_i-mu_0)right)$$ and $$p(mathbf{y}_j)=text{det}left(2piSigmaright)^{-1/2}expleft(-frac{1}{2}(mathbf{y}_j-mu_1)^TSigma^{-1}(mathbf{y}_j-mu_1)right)$$
– BlackMath
Nov 26 at 7:13
|
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given $bf{x}$$_1$,...,$bf{x}$$_n$ be a sequence of random vectors that are independent and identically distributed from $N_p(mu_0,Sigma)$ where $mu_0$ is known.
(i) Show that the MLE for $Sigma$ is;
$$
widehat{Sigma}=frac{1}{n}sumlimits_{i=1}^n(mathbf{x}_i-mu_0)(mathbf{x}_i-mu_0)'
$$
(ii) Now let $bf{y}$$_1$,...,$bf{y}$$_m$ be another sequence of random vectors that are independent and identically distributed from $N_p(mu_1,Sigma)$ where $mu_1$ is known. Calculate the MLE for $Sigma$ based on the sequences of observations {$bf{x}$$_1$,...,$bf{x}$$_n$} and {$bf{y}$$_1$,...,$bf{y}$$_m$}. What happens to $Sigma$ if both $mu_i$ for $i=0,1$ are assumed to be unknown?
I do not have any questions on part (i), I have already shown that the MLE for $Sigma$ is indeed $widehat{Sigma}$. My concerns are towards part (ii). I don't quite understand in what ways $widehat{Sigma}$ will change under this scenario and how to represent it notation-wise. Also, when $mu_i$ for $i=0,1$ are assumed to be unknown; do we simply replace $mu_0$ and $mu_1$ in the new MLE for $Sigma$ by $hat{mathbf{mu}}_0$ and $hat{mathbf{mu}}_1$?
Any form of help is much appreciated.
statistics normal-distribution maximum-likelihood
asked Nov 26 at 2:28
Nelly
74110
Given $bf{x}$$_1$,...,$bf{x}$$_n$ be a sequence of random vectors that are independent and identically distributed from $N_p(mu_0,Sigma)$ where $mu_0$ is known.
(i) Show that the MLE for $Sigma$ is;
$$
widehat{Sigma}=frac{1}{n}sumlimits_{i=1}^n(mathbf{x}_i-mu_0)(mathbf{x}_i-mu_0)'
$$
(ii) Now let $bf{y}$$_1$,...,$bf{y}$$_m$ be another sequence of random vectors that are independent and identically distributed from $N_p(mu_1,Sigma)$ where $mu_1$ is known. Calculate the MLE for $Sigma$ based on the sequences of observations {$bf{x}$$_1$,...,$bf{x}$$_n$} and {$bf{y}$$_1$,...,$bf{y}$$_m$}. What happens to $Sigma$ if both $mu_i$ for $i=0,1$ are assumed to be unknown?
I do not have any questions on part (i), I have already shown that the MLE for $Sigma$ is indeed $widehat{Sigma}$. My concerns are towards part (ii). I don't quite understand in what ways $widehat{Sigma}$ will change under this scenario and how to represent it notation-wise. Also, when $mu_i$ for $i=0,1$ are assumed to be unknown; do we simply replace $mu_0$ and $mu_1$ in the new MLE for $Sigma$ by $hat{mathbf{mu}}_0$ and $hat{mathbf{mu}}_1$?
Any form of help is much appreciated.
statistics normal-distribution maximum-likelihood
statistics normal-distribution maximum-likelihood
asked Nov 26 at 2:28
Nelly
74110
asked Nov 26 at 2:28
Nelly
74110
asked Nov 26 at 2:28
Nelly
74110
asked Nov 26 at 2:28
Nelly
74110
asked Nov 26 at 2:28
Nelly
74110
74110
I think you need to find the joint probability $$p(mathbf{x}_1,ldots,mathbf{x}_n,mathbf{y}_1,ldots,mathbf{y}_m)$$ and extract the covariance matrix from that.
– BlackMath
Nov 26 at 3:31
Thus, would claiming $(mathbf{x}_1,...,mathbf{x}_n,mathbf{y}_1,...,mathbf{y}_n) sim N_p(nmu_0+mmu_1,(n+m)Sigma)$ and then considering the MLE for $(n+m)Sigma$ be the correct approach?
– Nelly
Nov 26 at 5:39
How did you find the covariance in part (i)? I would assume that you found the joint PDF $p(mathbf{x}_1,ldots,mathbf{x}_n)$, which is a multivariate Gaussian distribution, and from there you found the covariance matrix. Here I think it's the same thing, but you need to find the joint PDF $p(mathbf{x}_1,ldots,mathbf{x}_n,mathbf{y}_1,ldots,mathbf{y}_m)$ instead. Since these are independent, this can be written as $$prod_ip(mathbf{x}_i)prod_jp(mathbf{y}_j)$$
– BlackMath
Nov 26 at 6:00
In part (i) I derived $widehat{Sigma}$ by using the maximum likelihood method of finding the log-likelihood, taking the partial derivative with respect to $Sigma^{-1}$ and setting the equation equal to 0. I didn't necessarily find $Sigma$ as you're suggesting. Maybe I'm just not following what you're saying?
– Nelly
Nov 26 at 7:02
1
I'm describing the same thing, I think. You need first to find the joint PDF, find the log-likelihood of that function, and then do the derivation. The joint PDF in the first case is $$prod_ip(mathbf{x}_i)$$ while in the second case it will be $$prod_ip(mathbf{x}_i)prod_jp(mathbf{y}_j)$$ where $$p(mathbf{x}_i)=text{det}left(2piSigmaright)^{-1/2}expleft(-frac{1}{2}(mathbf{x}_i-mu_0)^TSigma^{-1}(mathbf{x}_i-mu_0)right)$$ and $$p(mathbf{y}_j)=text{det}left(2piSigmaright)^{-1/2}expleft(-frac{1}{2}(mathbf{y}_j-mu_1)^TSigma^{-1}(mathbf{y}_j-mu_1)right)$$
– BlackMath
Nov 26 at 7:13
|
show 1 more comment
I think you need to find the joint probability $$p(mathbf{x}_1,ldots,mathbf{x}_n,mathbf{y}_1,ldots,mathbf{y}_m)$$ and extract the covariance matrix from that.
– BlackMath
Nov 26 at 3:31
Thus, would claiming $(mathbf{x}_1,...,mathbf{x}_n,mathbf{y}_1,...,mathbf{y}_n) sim N_p(nmu_0+mmu_1,(n+m)Sigma)$ and then considering the MLE for $(n+m)Sigma$ be the correct approach?
– Nelly
Nov 26 at 5:39
How did you find the covariance in part (i)? I would assume that you found the joint PDF $p(mathbf{x}_1,ldots,mathbf{x}_n)$, which is a multivariate Gaussian distribution, and from there you found the covariance matrix. Here I think it's the same thing, but you need to find the joint PDF $p(mathbf{x}_1,ldots,mathbf{x}_n,mathbf{y}_1,ldots,mathbf{y}_m)$ instead. Since these are independent, this can be written as $$prod_ip(mathbf{x}_i)prod_jp(mathbf{y}_j)$$
– BlackMath
Nov 26 at 6:00
In part (i) I derived $widehat{Sigma}$ by using the maximum likelihood method of finding the log-likelihood, taking the partial derivative with respect to $Sigma^{-1}$ and setting the equation equal to 0. I didn't necessarily find $Sigma$ as you're suggesting. Maybe I'm just not following what you're saying?
– Nelly
Nov 26 at 7:02
1
I'm describing the same thing, I think. You need first to find the joint PDF, find the log-likelihood of that function, and then do the derivation. The joint PDF in the first case is $$prod_ip(mathbf{x}_i)$$ while in the second case it will be $$prod_ip(mathbf{x}_i)prod_jp(mathbf{y}_j)$$ where $$p(mathbf{x}_i)=text{det}left(2piSigmaright)^{-1/2}expleft(-frac{1}{2}(mathbf{x}_i-mu_0)^TSigma^{-1}(mathbf{x}_i-mu_0)right)$$ and $$p(mathbf{y}_j)=text{det}left(2piSigmaright)^{-1/2}expleft(-frac{1}{2}(mathbf{y}_j-mu_1)^TSigma^{-1}(mathbf{y}_j-mu_1)right)$$
– BlackMath
Nov 26 at 7:13
I think you need to find the joint probability $$p(mathbf{x}_1,ldots,mathbf{x}_n,mathbf{y}_1,ldots,mathbf{y}_m)$$ and extract the covariance matrix from that.
– BlackMath
Nov 26 at 3:31
I think you need to find the joint probability $$p(mathbf{x}_1,ldots,mathbf{x}_n,mathbf{y}_1,ldots,mathbf{y}_m)$$ and extract the covariance matrix from that.
– BlackMath
Nov 26 at 3:31
Thus, would claiming $(mathbf{x}_1,...,mathbf{x}_n,mathbf{y}_1,...,mathbf{y}_n) sim N_p(nmu_0+mmu_1,(n+m)Sigma)$ and then considering the MLE for $(n+m)Sigma$ be the correct approach?
– Nelly
Nov 26 at 5:39
Thus, would claiming $(mathbf{x}_1,...,mathbf{x}_n,mathbf{y}_1,...,mathbf{y}_n) sim N_p(nmu_0+mmu_1,(n+m)Sigma)$ and then considering the MLE for $(n+m)Sigma$ be the correct approach?
– Nelly
Nov 26 at 5:39
How did you find the covariance in part (i)? I would assume that you found the joint PDF $p(mathbf{x}_1,ldots,mathbf{x}_n)$, which is a multivariate Gaussian distribution, and from there you found the covariance matrix. Here I think it's the same thing, but you need to find the joint PDF $p(mathbf{x}_1,ldots,mathbf{x}_n,mathbf{y}_1,ldots,mathbf{y}_m)$ instead. Since these are independent, this can be written as $$prod_ip(mathbf{x}_i)prod_jp(mathbf{y}_j)$$
– BlackMath
Nov 26 at 6:00
How did you find the covariance in part (i)? I would assume that you found the joint PDF $p(mathbf{x}_1,ldots,mathbf{x}_n)$, which is a multivariate Gaussian distribution, and from there you found the covariance matrix. Here I think it's the same thing, but you need to find the joint PDF $p(mathbf{x}_1,ldots,mathbf{x}_n,mathbf{y}_1,ldots,mathbf{y}_m)$ instead. Since these are independent, this can be written as $$prod_ip(mathbf{x}_i)prod_jp(mathbf{y}_j)$$
– BlackMath
Nov 26 at 6:00
In part (i) I derived $widehat{Sigma}$ by using the maximum likelihood method of finding the log-likelihood, taking the partial derivative with respect to $Sigma^{-1}$ and setting the equation equal to 0. I didn't necessarily find $Sigma$ as you're suggesting. Maybe I'm just not following what you're saying?
– Nelly
Nov 26 at 7:02
In part (i) I derived $widehat{Sigma}$ by using the maximum likelihood method of finding the log-likelihood, taking the partial derivative with respect to $Sigma^{-1}$ and setting the equation equal to 0. I didn't necessarily find $Sigma$ as you're suggesting. Maybe I'm just not following what you're saying?
– Nelly
Nov 26 at 7:02
1
1
I'm describing the same thing, I think. You need first to find the joint PDF, find the log-likelihood of that function, and then do the derivation. The joint PDF in the first case is $$prod_ip(mathbf{x}_i)$$ while in the second case it will be $$prod_ip(mathbf{x}_i)prod_jp(mathbf{y}_j)$$ where $$p(mathbf{x}_i)=text{det}left(2piSigmaright)^{-1/2}expleft(-frac{1}{2}(mathbf{x}_i-mu_0)^TSigma^{-1}(mathbf{x}_i-mu_0)right)$$ and $$p(mathbf{y}_j)=text{det}left(2piSigmaright)^{-1/2}expleft(-frac{1}{2}(mathbf{y}_j-mu_1)^TSigma^{-1}(mathbf{y}_j-mu_1)right)$$
– BlackMath
Nov 26 at 7:13
I'm describing the same thing, I think. You need first to find the joint PDF, find the log-likelihood of that function, and then do the derivation. The joint PDF in the first case is $$prod_ip(mathbf{x}_i)$$ while in the second case it will be $$prod_ip(mathbf{x}_i)prod_jp(mathbf{y}_j)$$ where $$p(mathbf{x}_i)=text{det}left(2piSigmaright)^{-1/2}expleft(-frac{1}{2}(mathbf{x}_i-mu_0)^TSigma^{-1}(mathbf{x}_i-mu_0)right)$$ and $$p(mathbf{y}_j)=text{det}left(2piSigmaright)^{-1/2}expleft(-frac{1}{2}(mathbf{y}_j-mu_1)^TSigma^{-1}(mathbf{y}_j-mu_1)right)$$
– BlackMath
Nov 26 at 7:13
|
show 1 more comment
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I think you need to find the joint probability $$p(mathbf{x}_1,ldots,mathbf{x}_n,mathbf{y}_1,ldots,mathbf{y}_m)$$ and extract the covariance matrix from that.
– BlackMath
Nov 26 at 3:31
Thus, would claiming $(mathbf{x}_1,...,mathbf{x}_n,mathbf{y}_1,...,mathbf{y}_n) sim N_p(nmu_0+mmu_1,(n+m)Sigma)$ and then considering the MLE for $(n+m)Sigma$ be the correct approach?
– Nelly
Nov 26 at 5:39
How did you find the covariance in part (i)? I would assume that you found the joint PDF $p(mathbf{x}_1,ldots,mathbf{x}_n)$, which is a multivariate Gaussian distribution, and from there you found the covariance matrix. Here I think it's the same thing, but you need to find the joint PDF $p(mathbf{x}_1,ldots,mathbf{x}_n,mathbf{y}_1,ldots,mathbf{y}_m)$ instead. Since these are independent, this can be written as $$prod_ip(mathbf{x}_i)prod_jp(mathbf{y}_j)$$
– BlackMath
Nov 26 at 6:00
In part (i) I derived $widehat{Sigma}$ by using the maximum likelihood method of finding the log-likelihood, taking the partial derivative with respect to $Sigma^{-1}$ and setting the equation equal to 0. I didn't necessarily find $Sigma$ as you're suggesting. Maybe I'm just not following what you're saying?
– Nelly
Nov 26 at 7:02
1
I'm describing the same thing, I think. You need first to find the joint PDF, find the log-likelihood of that function, and then do the derivation. The joint PDF in the first case is $$prod_ip(mathbf{x}_i)$$ while in the second case it will be $$prod_ip(mathbf{x}_i)prod_jp(mathbf{y}_j)$$ where $$p(mathbf{x}_i)=text{det}left(2piSigmaright)^{-1/2}expleft(-frac{1}{2}(mathbf{x}_i-mu_0)^TSigma^{-1}(mathbf{x}_i-mu_0)right)$$ and $$p(mathbf{y}_j)=text{det}left(2piSigmaright)^{-1/2}expleft(-frac{1}{2}(mathbf{y}_j-mu_1)^TSigma^{-1}(mathbf{y}_j-mu_1)right)$$
– BlackMath
Nov 26 at 7:13