edits

Author: bmeyers

_posts/2024-11-18-Conditioning_joint_gaussian_on_sum.md

@@ -14,7 +14,7 @@ Let $X$ be a jointly Gaussian (j-g) random vector with mean $\mu\in\mathbf{R}^n$
 
 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/#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-nsdb7c7pEdSklSgHtfrkfHDqLAmr0T0ZCgqhEq09Q5HDajE9jBKJJLgSirHG6YRmO0ZHEyUSsgkSR4CkaTAXAYDqDAGIENYwpXNY6EcHQeCkEgYAzqsMSEeYyGINYSB9FcyEwJmtCmKoLDQMAODAAQJzarqCgGkan6mjR+R0Z4MSGAWkq-sSBCQSycQwRyaSAfGqx0WhGE5FccaYB0cinlum4AeukrYdYuH4TAhEzjpeBUCRZEsBRVG9LGIkwIx+YsLoYDdGQ5DcXqwB8Y6AlCXkIk4GJElXFJ-7WR0yyhBgzLRPJ7JwZyxHxjZdl3qpKGYcJKFPGQIDckmQU4DOwSfLs5isexnHqGFSHFeCeBlZ65CVfFtk1fIUBuMQhggDA5gAMK8IJ2wcXBo0TbwAgCH8OzfN5vkUC1bkecx3WIDggQnNYcAQCkOoiEF+qGiFkZxL8rnFaYcpBg99GReJP6EtJsmpWysHwVyiHaQleWtTAhV5JgAKrFDeAxAOoxNTgHB4GxdDVZKIntWVUMnLV3wNRxJwtbj6gwyIcMI0T6jI6jODo8DfWY21pUrtDA1DVAI1jSwk3TXAiNzTzk1LStXx7F5PmHuQW3FVo1htkxFgQDgsv0e5MBKwAomxhAAmcx2nQFvHXcaKB3WrHBPYQL3M29UWfX+MnJVBaX-ZlykCeBZIMsQr68mD9IUkcEPSnAHE5nD8DrdLCD4QQHDh-A1h0Ho6j+dYuggHA5QnU4eZQBTGcWkMunMJidzEBw+uOlWU6lvpy4ohS1cnGclZ3oORAfpe0aFi2CBTCI4KQtCMKepnZUoh0-XT7ZuJ2yVHXpOSldQK3gId2+3ejo2174xLhNNZb2MrzlB-mDHfl-Egg3DfNvNTSts3cwtouvVbeDPdFRXqzthgsBAPHf4UdoDWAxFnE6KQr4UDjgaOQABvCeWcz4zyyBwRYhgAC+lt3o-3xF9OKc87JJQiK7P6ikuQ8nyqhf4odWx4ShDnOgpB-jh2OsqawoBkJ4FNKsWoeBE4OAIAIIgOBSDAApkqUkk9jq514IvU+IgP4ayYgAl81ws6MCQdESez5sgYOwbgh2klCHOzIXJChGUlI0PUqHUy5kCLSFUMpUi0hyKaGcpbVRnlXSXgAsYj6pinY-WgulAGLj-Y0NDnTdmsTKbME9IjE4tM0YuI-qfWJJwVE7RYDrOUNdDYpCUIE-BBQzFwCyWTSpbEKYxFCW7ShBZNL5G4dAeAt9UBKmOhkSUbTYDVizmALQ3S7x9PAO064+EXD-kotcdC4zIADMtMhRwownAcGGas+Acy6LBOkoPKJgd5nWDCMAAGnEgHYCYMwFg2AkAcWwCwFpeQoLwgkkAA)) 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!
+There is a Marimo notebook ([app](https://marimo.io/p/@gismo/notebook-mf27rb), [editable code](https://marimo.app/#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-nsdb7c7pEdSklSgHtfrkfHDqLAmr0T0ZCgqhEq09Q5HDajE9jBKJJLgSirHG6YRmO0ZHEyUSsgkSR4CkaTAXAYDqDAGIENYwpXNY6EcHQeCkEgYAzqsMSEeYyGINYSB9FcyEwJmtCmKoLDQMAODAAQJzarqCgGkan6mjR+R0Z4MSGAWkq-sSBCQSycQwRyaSAfGqx0WhGE5FccaYB0cinlum4AeukrYdYuH4TAhEzjpeBUCRZEsBRVG9LGIkwIx+YsLoYDdGQ5DcXqwB8Y6AlCXkIk4GJElXFJ-7WR0yyhBgzLRPJ7JwZyxHxjZdl3qpKGYcJKFPGQIDckmQU4DOwSfLs5isexnHqGFSHFeCeBlZ65CVfFtk1fIUBuMQhggDA5gAMK8IJ2wcXBo0TbwAgCH8OzfN5vkUC1bkecx3WIDggQnNYcAQCkOoiEF+qGiFkZxL8rnFaYcpBg99GReJP6EtJsmpWysHwVyiHaQleWtTAhV5JgAKrFDeAxAOoxNTgHB4GxdDVZKIntWVUMnLV3wNRxJwtbj6gwyIcMI0T6jI6jODo8DfWY21pUrtDA1DVAI1jSwk3TXAiNzTzk1LStXx7F5PmHuQW3FVo1htkxFgQDgsv0e5MBKwAomxhAAmcx2nQFvHXcaKB3WrHBPYQL3M29UWfX+MnJVBaX-ZlykCeBZIMsQr68mD9IUkcEPSnAHE5nD8DrdLCD4QQHDh-A1h0Ho6j+dYuggHA5QnU4eZQBTGcWkMunMJidzEBw+uOlWU6lvpy4ohS1cnGclZ3oORAfpe0aFi2CBTCI4KQtCMKepnZUoh0-XT7ZuJ2yVHXpOSldQK3gId2+3ejo2174xLhNNZb2MrzlB-mDHfl-Egg3DfNvNTSts3cwtouvVbeDPdFRXqzthgsBAPHf4UdoDWAxFnE6KQr4UDjgaOQABvCeWcz4zyyBwRYhgAC+lt3o-3xF9OKc87JJQiK7P6ikuQ8nyqhf4odWx4ShDnOgpB-jh2OsqawoBkJ4FNKsWoeBE4OAIAIIgOBSDAApkqUkk9jq514IvU+IgP4ayYgAl81ws6MCQdESez5sgYOwbgh2klCHOzIXJChGUlI0PUqHUy5kCLSFUMpUi0hyKaGcpbVRnlXSXgAsYj6pinY-WgulAGLj-Y0NDnTdmsTKbME9IjE4tM0YuI-qfWJJwVE7RYDrOUNdDYpCUIE-BBQzFwCyWTSpbEKYxFCW7Shu8vzhjvKHT03pfT+jzBQVQAAWRYP5Ax+mGIgcApgBydL9AGH0TQACsbBQJ72bDFCpqgpkgFGeMvAiUGlWIiZKYChzz4WlIMMAAjrAZYpywJNmuVcYh9yBirzRE8vI1DJLDM2dYMZPDrmhzACgaAmBurgmnJKB5ZzrCXJTHGIsc4NlbJ4cZK4VxSxnxbjXLe94d5e2-DCLI-sOAdEQKaYA1hsSSmCCi4SEBwQgvJfAZgo4LmwF6lQeFBQvlItMOuEsPsOgEqJSSslFLqVrmCdJJCdKDCgtoYC4AwK5V7IUtYqhAc1JISBVABl4IAUs2XsgsqCqlWMtWKS8ljBeD91osVHxzE1SQKNoQK0w4oFwAWdcX5pgilasVRMOVKB6iKqCB-PBjtvou0saqiJ1DpX0rle06ISBTTlSeCg3MsqzVKlFVawscBbyehABxNNejM26toRa6weaI0khVeEzKcbNXlsTZpX+S9Oq6IzTKit5rc3WpyZrTyjrs7OoIK631nqfnbN9Zm4g-quiMqDfoENJMdQ+l4OC1F+Qi2ktLd2hN2aq01oLZ3YtTQjWqBbUe-tNrt1wAADzMF4HgAQfSqW9nXeQRYA4gF7QvdEFoI5L3XvBH2y1A6ixPrLAADjmZKYA8Ev2bsDKkL9iw23bpEloUYZUBBbDgNaxe4bJV1qjb9GNmVQ7cOgPAW+qAlTHQyAh8AtHqxZzAFoRjd4WOQFgNcfCLh-yUWuOhXjbHLTIUcKMJwHBONSfgCJuipHSQZCiYHUT1h63uzSDyUOHJLSspTJaUOVwZTjT9GAFdBAcCLv2odUYdoJESnvZUwpqxRCgKgMdRJ0Rkk0xRmkkzEL8gyi1uhYkHFbPQt0MgR0UBXB0f8bwxlWnXOUQLvAVQog25uoAHzMAuBwE46gylYRhZgASchPMxHKN57FTwYXZcKZypJ1NkYcREM1rzGEQsEKdtcCrAkVNNyhTC3ZGBEOck4kA7ATBmAsGwEgDi2AWCYaVCyeEEkgA)) 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!
 
 ### The Formula
 
@@ -74,7 +74,7 @@ thus proving the first result.
 
 Note that both $v$ and $A$ do not depend at all on $s$ and can be pre-calculated. The entries of $v$ are all on the interval $[0,1]$ and, in fact, form a simplex (their values sum to $1$). The posterior mean is always updated to be exactly consistent with the observed sum.
 
-The updated covariance matrix is always "shrunk," _i.e._, $\Sigma - A\Sigma A^T \succeq 0$, so that the uncertainty is reduced in the posterior distribution. Assuming $\Sigma$ is rank $n$, $A$ has rank $n-1$, and the updated covariance becomes degenerate (singular), also with rank $n-1$. This degeneracy is important! It establishes a subspace in $\mathbf{R}^n$ (the nullspace of the updated covariance matrix) along which our posterior distribution has no variance. This subspace is exactly $\mathbf{1}\in\mathbf{R}^n$. That means the posterior distribution has no variability in the sum of entries. Any sample of this posterior distribution will have the exact same sum!
+The updated covariance matrix is always "shrunk," _i.e._, $\Sigma - A\Sigma A^T \succeq 0$, so that the uncertainty is reduced in the posterior distribution. Assuming $\Sigma$ is rank $n$, $A$ has rank $n-1$, and the updated covariance becomes degenerate (singular), also with rank $n-1$. This degeneracy is important! It establishes a subspace in $\mathbf{R}^n$ (the nullspace of the updated covariance matrix) along which our posterior distribution has no variance. This subspace is one dimensional with the basis matrix, $\mathbf{1}\in\mathbf{R}^{n\times 1}$. That means the posterior distribution has no variability in the sum of entries. Any sample of this posterior distribution will have the exact same sum!
 
 #### See also