adding marimo notebook

Author: bmeyers

_posts/2024-11-18-Conditioning_joint_gaussian_on_sum.md

@@ -12,6 +12,8 @@ Let $X$ be a jointly Gaussian (j-g) random vector with mean $\mu\in\mathbf{R}^n$
 
 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](https://marimo.app/#code/JYWwDg9gTgLgBCAhlUEBQaD6mDmBTAOzykRjwBNMB3YGACzgF44AiABgDoBODgRi4C05PADcAzCzSIwYJgmSoOAQRkAKAJQY0AAWlgOAYzwAbY2mEAzONlUgI6gFxo4LuFACuBTACN3MGBAEcnYc7sAcvv6BqsaI3iaMLERUcOSkiCyargjumADOxsDCUMEQoeEFRcSqAgBMADRwDXCcvI2x8caJIO4sjSKIxu54jGxZrnn5hcWl5RyVxTXNza3tcQkseX1wA0MjY86uIXR5MIgGANaqANoeXpEBBI09U1VQjZMLxAC64y5QeBg7igQReX3ebk8Pj8jw+r2KWl0MkMJjMlmsmFUBDAjTu0KiBEch3+UIegWJcAAxHB8EQSGQ4Ig4AAFADKABE4AYIAMUIgCEZ5DAUAAPClIYXAEWs4AALzwcl4bApAFk5NiOCQCOQICBNfzyLZSKKZfLnsapaa8H84AAhdX6HUwVQqxoqjjC-l5SB5PAadRwADULT4cAAVHANXgAJ5+iUmuXWinUgyDAzuWIMkQUzAQB0cQJ4PJGyXSxM2kRyVT27TWexwAD0cFUuY4ABU4LWa3WbSm0xnSAqlBSlPmY3GLWX5QGBJH9BA-NURI0NYXi-HLeWbQCgSC4EpGra3ebS1b+oi9CjTOY8FYbEe4GBjDAPgQ8kTsnk3xw6HhSEgwGrRoDAAxIn0QPIkEyCknxgD1aGMP0WGgYAcGAAhBi5HkFH5QUNxFaDslgjgcAMCwNApHdgQIC9kSMa90RsA87UaWDX3fJxsltTBkjkUcu07fd2wpL88h-P8YAA6seLwKhgNAlhwMgjIbWImAEKQ9wwDSMhyCw3lgFwhV8MI1xiNI8jt0Bajm245J6k0DAkX0ei0VvDEgLtGS5LnVjnw-Mzn3mMgQCxR1DJwat1DWTpEhQtCMOMUyXGI048FCjVyAi6Tkmi+QoAuYgLBAGBEgImKNi0nSKGSx8gvUmBENUFgssQHBAjyOAICsbkREMvkBWMi0jC2VSgsQ2lDTGuCLIo7IqJBWiXNRG870xZj8LPBAIGeXJwRXHE4E+aZiH6AKXGpdDvTwAx4Cqwc9JAP8CA4OBaCOugF2MPS8ncEA4ACLrvF9KARAoI7xVyXjmGOt4OF2YZwx2IN90E2w9pOqB4cGRGI1XIh10nK11BtEIRFOc4rmuDVftCl5ct27zfko6y93R7yHKWq83LWsK-JfHZzqpLlf0uAG6FIZHgE6pk8lAJ88BFRpwjwV6CECAQiBwUhgDBxkglpzrAd4GCgrS0KRGm+DGr9CwWGXI6-sYABvGm-tUS2HA4WoLAAX1q8yyLm1wFpopzL1c1aPOYtijrfIXRPE-9pFUZiQOkMDYmUgOgtmqzdzDnQI5Wxj1oO-mhbwVDdkaKucDBgx80KRKcA4Ou6FTq3zdUOvditvOWYLuBe8GWvq5urnI9LjRONceXoHgJAUDsRlOrsCl59gSM-rAaNV7nDfwAXo6-28aAggguO8kPyAt4lJ8IEa4BvA4XeH-gS-YMHmy7HLuqBdEloYAa0MJPWwEwZgLBsBIHQtgFgs8XCXjuBRIAA) 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
 
 The distribution of $X$ given the observation that the sum $S=s$ is given by
@@ -71,4 +73,4 @@ 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.
 
-$A$ has rank $n-1$ with exactly one negative eigenvalue, and so the covariance is always shrunk in a way that depends on the original covariance matrix. This has the effect in which blocks of more highly correlated variable will have their variance (diagonal entries) shrunk more than independent variables. 
+$A$ has rank $n-1$ with exactly one "near-zero" eigenvalue, and so the covariance is always shrunk in a way that depends on the original covariance matrix. This has the effect in which blocks of more highly correlated variable will have their variance (diagonal entries) shrunk more than independent variables.