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proper horizontal spacing between f(x) and dx
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edit approved May 1, 2022 at 23:33
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edit approved May 1, 2022 at 23:33
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I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2}\textbf{x}^TH\textbf{x})d\textbf{x}$$$$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$How can I compute probability densities within the hyperplane $\textbf{x}\cdot \textbf{g}=C$ using the standard Lebesgue measure? ${}$

The closest question & answer I have found on this site was for a particular instance of Gaussian and hyperplane here.

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2}\textbf{x}^TH\textbf{x})d\textbf{x}$$How can I compute probability densities within the hyperplane $\textbf{x}\cdot \textbf{g}=C$ using the standard Lebesgue measure?

The closest question & answer I have found on this site was for a particular instance of Gaussian and hyperplane here.

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$How can I compute probability densities within the hyperplane $\textbf{x}\cdot \textbf{g}=C$ using the standard Lebesgue measure? ${}$

The closest question & answer I have found on this site was for a particular instance of Gaussian and hyperplane here.

clarified question and removed non-answer
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user44143
user44143

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $\mathcal{N}(\textbf{0}, H^{-1}) = det(2\pi H^{-1})^{-1/2}exp(-\frac{1}{2}\textbf{x}^TH\textbf{x})$. $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2}\textbf{x}^TH\textbf{x})d\textbf{x}$$How can I want to compute the probability density ofdensities within the hyperplane $\textbf{x}\cdot \textbf{g}=C$ using the standard Lebesgue measure.

Let $\textbf{x}^* = arg\min_{\textbf{x}\cdot\textbf{g}=C} \textbf{x}^TH\textbf{x}$. My current approach tries to decompose the integral into the probability density of a one-dimensional gaussian involving $\textbf{x}^*$ and an integral over one less dimension evaluating to 1. However, I haven't been able to formalize this or obtain the exact expression.?

The closest question & answer I have found on this site was for a particular instance of Gaussian and hyperplane here.

I have a multivariate Gaussian distribution $\mathcal{N}(\textbf{0}, H^{-1}) = det(2\pi H^{-1})^{-1/2}exp(-\frac{1}{2}\textbf{x}^TH\textbf{x})$. I want to compute the probability density of the hyperplane $\textbf{x}\cdot \textbf{g}=C$ using the standard Lebesgue measure.

Let $\textbf{x}^* = arg\min_{\textbf{x}\cdot\textbf{g}=C} \textbf{x}^TH\textbf{x}$. My current approach tries to decompose the integral into the probability density of a one-dimensional gaussian involving $\textbf{x}^*$ and an integral over one less dimension evaluating to 1. However, I haven't been able to formalize this or obtain the exact expression.

The closest question & answer I have found on this site was for a particular instance of Gaussian and hyperplane here.

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2}\textbf{x}^TH\textbf{x})d\textbf{x}$$How can I compute probability densities within the hyperplane $\textbf{x}\cdot \textbf{g}=C$ using the standard Lebesgue measure?

The closest question & answer I have found on this site was for a particular instance of Gaussian and hyperplane here.

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etal
etal
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Probability density of a hyperplane for a Gaussian distribution

I have a multivariate Gaussian distribution $\mathcal{N}(\textbf{0}, H^{-1}) = det(2\pi H^{-1})^{-1/2}exp(-\frac{1}{2}\textbf{x}^TH\textbf{x})$. I want to compute the probability density of the hyperplane $\textbf{x}\cdot \textbf{g}=C$ using the standard Lebesgue measure.

Let $\textbf{x}^* = arg\min_{\textbf{x}\cdot\textbf{g}=C} \textbf{x}^TH\textbf{x}$. My current approach tries to decompose the integral into the probability density of a one-dimensional gaussian involving $\textbf{x}^*$ and an integral over one less dimension evaluating to 1. However, I haven't been able to formalize this or obtain the exact expression.

The closest question & answer I have found on this site was for a particular instance of Gaussian and hyperplane here.