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DOC/BUG: fix covariance matrix notation in Nevo tutorial
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jeffgortmaker committed Mar 22, 2023
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10 changes: 4 additions & 6 deletions docs/notebooks/tutorial/nevo.ipynb
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"It is common to assume that $f(\\beta_i \\mid \\theta)$ follows a multivariate normal distribution, and to break it up into three parts:\n",
"\n",
"1. A mean $K_1 \\times 1$ taste which all individuals agree on, $\\beta$.\n",
"2. A $K_2 \\times K_2$ covariance matrix, $\\Sigma$ .\n",
"2. A $K_2 \\times K_2$ covariance matrix, $V$. As is common with multivariate normal distributions, $V$ is not estimated directly. Rather, its matrix square (Cholesky) root $\\Sigma$ is estimated where $\\Sigma\\Sigma' = V$.\n",
"3. Any $K_2 \\times D$ interactions, $\\Pi$, with observed $D \\times 1$ demographic data, $d_i$.\n",
"\n",
"Together this gives us that\n",
"\n",
"$$\\beta_i \\sim N(\\beta + \\Pi d_i, \\Sigma).$$\n",
"$$\\beta_i \\sim N(\\beta + \\Pi d_i, \\Sigma\\Sigma').$$\n",
"\n",
":meth:`Problem.solve` takes an initial guess $\\Sigma_0$ of $\\Sigma$. It guarantees that $\\hat{\\Sigma}$ (the estimated parameters) will have the same sparsity structure as $\\Sigma_0$. So any zero element of $\\Sigma$ is restricted to be zero in the solution $\\hat{\\Sigma}$. For example, a popular restriction is that $\\Sigma$ is diagonal, this can be achieved by passing a diagonal matrix as $\\Sigma_0$.\n",
"\n",
"As is common with multivariate normal distributions, $\\Sigma$ is not estimated directly. Rather, its matrix square (Cholesky) root $L$ is estimated where $LL' = \\Sigma$."
":meth:`Problem.solve` takes an initial guess $\\Sigma_0$ of $\\Sigma$. It guarantees that $\\hat{\\Sigma}$ (the estimated parameters) will have the same sparsity structure as $\\Sigma_0$. So any zero element of $\\Sigma$ is restricted to be zero in the solution $\\hat{\\Sigma}$. For example, a popular restriction is that $\\Sigma$ is diagonal, this can be achieved by passing a diagonal matrix as $\\Sigma_0$."
]
},
{
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