edits

Author: bmeyers

_posts/2024-11-18-Conditioning_joint_gaussian_on_sum.md

@@ -8,9 +8,9 @@ tags: [probability, statistics, gaussian, ee278]
 _This is a follow-up to the [previous post]({{ "/conditional_distribution_for_jointly_gaussian_random_vectors/" | absolute_url}}) on conditioning a jointly Gaussian probability distribution on partial observations_
 ### Setting
 
-Let $X$ be a jointly Gaussian (j-g) random vector with mean $\mu\in\mathbf{R}^n$ and covariance matrix $\Sigma\in\mathbf{R}^{n\times n}$, such that $X\sim\mathcal{N}(\mu, \Sigma)$, and let $S$ be the sum of the entries of $X$. Because $S$ is a linear transform of j-g randon variables, $S$ is also itself Gaussian. We are interested in the setting where we wish to update our belief about the distribuition of $X$ given our observation that $S=s$.
+Let $X$ be a jointly Gaussian (j-g) random vector with mean $\mu\in\mathbf{R}^n$ and covariance matrix $\Sigma\in\mathbf{R}^{n\times n}$, such that $X\sim\mathcal{N}(\mu, \Sigma)$, and let $S$ be the sum of the entries of $X$. We are interested in the setting where we wish to update our belief about the distribuition of $X$ given our observation that the sum equals a specific value, $S=s$.
 
-As in the [prior post]({{ "/conditional_distribution_for_jointly_gaussian_random_vectors/" | absolute_url}}), we will exploit the unique property of Gaussians that uncorrelated variables are also independent.
+Because $S$ is a linear transform of j-g randon variables, $S$ is also itself Gaussian. As in the [prior post]({{ "/conditional_distribution_for_jointly_gaussian_random_vectors/" | absolute_url}}), we will exploit the unique property of Gaussians that uncorrelated variables are also independent.
 
 There is a Marimo notebook ([app](https://marimo.io/p/@gismo/notebook-mf27rb), [editable code](https://marimo.app/#code/JYWwDg9gTgLgBCAhlUEBQaD6mDmBTAOzykRjwBNMB3YGACzgF44AiABgDoBODgRi4C05PADcAzCzSIwYJgmSoOAQRkAKAJQY0AAWlgOAYzwAbY2mEAzONlUgI6gFxo4LuFACuBTACN3MGBAEcnYc7sAcvv6BqsaI3iaMLERUcOSkiCyargjumADOxsDCUMEQoeEFRcSqAgBMADRwDXCcvI2x8caJIO4sjSKIxu54jGxZrnn5hcWl5RyVxTXNza3tcQkseX1wA0MjY86uIXR5MIgGANaqANoeXpEBBI09U1VQjZMLxAC64y5QeBg7igQReX3ebk8Pj8jw+r2KWl0MkMJjMlmsmFUBDAjTu0KiBEch3+UIegWJcAAxHB8EQSGQ4Ig4AAFADKABE4AYIAMUIgCEZ5DAUAAPClIYXAEWs4AALzwcl4bApAFk5NiOCQCOQICBNfzyLZSKKZfLnsapaa8H84AAhdX6HUwVQqxoqjjC-l5SB5PAadRwADULT4cAAVHANXgAJ5+iUmuXWinUgyDAzuWIMkQUzAQB0cQJ4PJGyXSxM2kRyVT27TWexwAD0cFUuY4ABU4LWa3WbSm0xnSAqlBSlPmY3GLWX5QGBJH9BA-NURI0NYXi-HLeWbQCgSC4EpGra3ebS1b+oi9CjTOY8FYbEe4GBjDAPgQ8kTsnk3xw6HhSEgwGrRoDAAxIn0QPIkEyCknxgD1aGMP0WGgYAcGAAhBi5HkFH5QUNxFaDslgjgcAMCwNApHdgQIC9kSMa90RsA87UaWDX3fJxsltTBkjkUcu07fd2wpL88h-P8YAA6seLwKhgNAlhwMgjIbWImAEKQ9wwDSMhyCw3lgFwhV8MI1xiNI8jt0Bajm245J6k0DAkX0ei0VvDEgLtGS5LnVjnw-Mzn3mMgQCxR1DJwat1DWTpEhQtCMOMUyXGI048FCjVyAi6Tkmi+QoAuYgLBAGBEgImKNi0nSKGSx8gvUmBENUFgssQHBAkGPI4AgKxuREQy+QFYyLSMLZVKCxDaUNca4IsijsiokFaJc1EbzvTFmPws8EAgZ5cnBFccTgT5pmIfoApcal0O9PADHgKrBz0kA-wIDg4FoY66AXYw9LydwQDgAJuu8X0oBEChjvFXJeOYE63g4XZhnDHYg33QTbH206oARwYkYjVciHXScrXUG0QhEU5ziua4NT+0KXlyvbvN+SjrL3DHvIc5arzc9awr8l8dguqkuV-S5AboUgUeALqmTyUAnzwEVGnCPA3oIQIBCIHBSGAcHGSCOmuqB3gYKCtLQpEGb4Mav0LBYZdjv+xgAG9af+1QrYcDhagsABfWrzLI+bXEWminMvVy1o85i2OOt9hdE8T-2kVRmJA6QwNiZTA6CuarN3cOdEj1bGI2w6BeFvBUN2Rpq5wcGDHzQpEpwDh67oNPrYt1R692a389Zwu4D7wY65r27uajsuNE41wFegeAkBQOxGS6uwKQX2BI3+sBozXudN-ARfjr-bxoCCCD47yI-IG3iUnwgRrgG8Dg98f+Ar9goebLsCu6sFqJLQwB1oYWetgJgzAWDYCQOhbALA54uEvHcCiQA)) accompanying this post that allows you to play with some relevent numerical experiments. It runs in your browser, so feel free to edit and play around!