| \n", " | School ID | \n", "score | \n", "
|---|---|---|
| Student ID | \n", "\n", " | \n", " |
| 1697 | \n", "33 | \n", "64.510645 | \n", "
| 314 | \n", "6 | \n", "62.005591 | \n", "
| 1786 | \n", "35 | \n", "67.453794 | \n", "
| 1730 | \n", "35 | \n", "73.349892 | \n", "
| 1224 | \n", "23 | \n", "56.454934 | \n", "
\n", " \n", "% Assuming a normal model for the data $y_j$ for group j with a normal prior for the mean $\\alpha_j$.\n", "\n", "% Let $y_j = N(\\alpha_j, \\sigma^2_y)$ and $\\alpha_j = N(\\mu, \\sigma^2_{\\alpha})$\n", "\n", "% The likelihood is given by the normal distribution:\n", "$$\n", "p(y_j | \\alpha_j) = \\frac{1}{\\sqrt{2\\pi\\sigma^2_y}} \\exp\\left(-\\frac{(y_j - \\alpha_j)^2}{2\\sigma^2_y}\\right)\n", "$$\n", "\n", "% The log likelihood is the logarithm of the above:\n", "$$\n", "\\log p(y_j | \\alpha_j) = \\log\\left(\\frac{1}{\\sqrt{2\\pi\\sigma^2_y}}\\right) -\\frac{(y_j - \\alpha_j)^2}{2\\sigma^2_y}\n", "$$\n", "\n", "% Simplifying the constant and assuming a sample size of $n_j$ for group j, i.e. to add log prob terms of all samples as they are i.i.d:\n", "$$\n", "\\log p(y_j | \\alpha_j) = -\\frac{n_j}{2}\\log(2\\pi\\sigma^2_y) -\\frac{1}{2\\sigma^2_y}\\sum_{i=1}^{n_j}(y_{ji} - \\alpha_j)^2\n", "$$\n", "\n", "% For the prior:\n", "$$\n", "p(\\alpha_j) = \\frac{1}{\\sqrt{2\\pi\\sigma^2_\\alpha}} \\exp\\left(-\\frac{(\\alpha_j - \\mu)^2}{2\\sigma^2_\\alpha}\\right)\n", "$$\n", "\n", "% The log prior is the logarithm of the above:\n", "$$\n", "\\log p(\\alpha_j) = \\log\\left(\\frac{1}{\\sqrt{2\\pi\\sigma^2_\\alpha}}\\right) -\\frac{(\\alpha_j - \\mu)^2}{2\\sigma^2_\\alpha}\n", "$$\n", "\n", "% Simplifying the constant:\n", "$$\n", "\\log p(\\alpha_j) = -\\frac{1}{2}\\log(2\\pi\\sigma^2_\\alpha) -\\frac{1}{2\\sigma^2_\\alpha}(\\alpha_j - \\mu)^2\n", "$$\n", "\n", "% Now, combining the log likelihood and the log prior for the log posterior:\n", "$$\n", "\\log p(\\alpha_j | y_j) \\propto -\\frac{n_j}{2}\\log(2\\pi\\sigma^2_y) -\\frac{1}{2\\sigma^2_y}\\sum_{i=1}^{n_j}(y_{ji} - \\alpha_j)^2 -\\frac{1}{2}\\log(2\\pi\\sigma^2_\\alpha) -\\frac{1}{2\\sigma^2_\\alpha}(\\alpha_j - \\mu)^2\n", "$$\n", "\n", "% Taking the derivative of the log posterior with respect to $\\alpha_j$, setting it to zero, and solving for $\\alpha_j$ will yield the MAP estimate.\n", "\n", "% Continuing from the log posterior, we have:\n", "$$\n", "\\log p(\\alpha_j | y_j) \\propto -\\frac{1}{2\\sigma^2_y}\\sum_{i=1}^{n_j}(y_{ji} - \\alpha_j)^2 -\\frac{1}{2\\sigma^2_\\alpha}(\\alpha_j - \\mu)^2\n", "$$\n", "\n", "% The derivative of the log posterior with respect to alpha_j is:\n", "$$\n", "\\frac{d}{d\\alpha_j}\\log p(\\alpha_j | y_j) = \\frac{1}{\\sigma^2_y}\\sum_{i=1}^{n_j}(y_{ji} - \\alpha_j) - \\frac{1}{\\sigma^2_\\alpha}(\\alpha_j - \\mu)\n", "$$\n", "\n", "% Setting the derivative to zero for the MAP estimate:\n", "$$\n", "\\frac{1}{\\sigma^2_y}\\sum_{i=1}^{n_j}(y_{ji} - \\alpha_j) - \\frac{1}{\\sigma^2_\\alpha}(\\alpha_j - \\mu) = 0\n", "$$\n", "\n", "% Solving for alpha_j gives us:\n", "$$\n", "\\frac{1}{\\sigma^2_y}\\sum_{i=1}^{n_j}y_{ji} - \\frac{n_j}{\\sigma^2_y}\\alpha_j - \\frac{1}{\\sigma^2_\\alpha}\\alpha_j + \\frac{\\mu}{\\sigma^2_\\alpha} = 0\n", "$$\n", "\n", "% Collect terms involving alpha_j:\n", "$$\n", "\\left(\\frac{n_j}{\\sigma^2_y} + \\frac{1}{\\sigma^2_\\alpha}\\right)\\alpha_j = \\frac{1}{\\sigma^2_y}\\sum_{i=1}^{n_j}y_{ji} + \\frac{\\mu}{\\sigma^2_\\alpha}\n", "$$\n", "\n", "% Since $\\sum_{i=1}^{n_j}y_{ji}$ is just $n_j$ times the sample mean $\\bar{y}_j$:\n", "$$\n", "\\left(\\frac{n_j}{\\sigma^2_y} + \\frac{1}{\\sigma^2_\\alpha}\\right)\\alpha_j = \\frac{n_j}{\\sigma^2_y}\\bar{y}_j + \\frac{\\mu}{\\sigma^2_\\alpha}\n", "$$\n", "\n", "% Divide through by the coefficient of $\\alpha_j$ to solve for it:\n", "$$\n", "\\alpha_j = \\frac{\\frac{n_j}{\\sigma^2_y}\\bar{y}_j + \\frac{\\mu}{\\sigma^2_\\alpha}}{\\frac{n_j}{\\sigma^2_y} + \\frac{1}{\\sigma^2_\\alpha}}\n", "$$\n", "\n", "% If we let $\\mu = \\bar{y}_{\\text{all}}$, we get the original equation provided:\n", "$$\n", "\\hat{\\alpha}_{j}^{\\text{partial-pooled}} \\approx \\frac{\\frac{n_j}{\\sigma^2_y}\\bar{y}_j + \\frac{1}{\\sigma^2_\\alpha}\\bar{y}_{\\text{all}}}{\\frac{n_j}{\\sigma^2_y} + \\frac{1}{\\sigma^2_\\alpha}}\n", "$$\n", "\n", "
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