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constructor-s/bayesian_regression.ipynb

Created August 27, 2020 00:18
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Bayesian Linear Regression
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bayesian_regression.ipynb
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"cells": [
{
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"metadata": {},
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"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from scipy.stats import multivariate_normal\n",
"from sklearn.preprocessing import StandardScaler"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Generate some data"
]
},
{
"cell_type": "code",
"execution_count": 448,
"metadata": {},
"outputs": [
{
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"data": {
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"metadata": {},
"execution_count": 448
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}
}
],
"source": [
"np.random.seed(2)\n",
"xdata = np.linspace(70, 75, 11)\n",
"slope = 0\n",
"ydata = slope * xdata + 80-slope*xdata[0] + 2*np.random.normal(loc=0, scale=1, size=xdata.shape)\n",
"\n",
"xdata = xdata[::]\n",
"ydata = ydata[::]\n",
"\n",
"plt.plot(xdata, ydata, 'x')\n",
"plt.ylim([0, 100])\n",
"slope"
]
},
{
"cell_type": "code",
"execution_count": 449,
"metadata": {},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "((11,),\n (11,),\n array([72.5]),\n None,\n array([78.95235539]),\n None,\n array([-2.5, -2. , -1.5, -1. , -0.5, 0. , 0.5, 1. , 1.5, 2. , 2.5]),\n array([ 0.21412892, 0.93511096, -3.22474758, 4.32818623, -2.53922656,\n -0.63585012, 2.05340745, -1.44293156, -1.06825982, -0.77037062,\n 2.1505527 ]))"
},
"metadata": {},
"execution_count": 449
},
{
"output_type": "display_data",
"data": {
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},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"xscalar = StandardScaler(with_mean=True, with_std=False).fit(xdata.reshape(-1, 1))\n",
"x = xscalar.transform(xdata.reshape(-1, 1)).ravel()\n",
"yscalar = StandardScaler(with_mean=True, with_std=False).fit(ydata.reshape(-1, 1))\n",
"y = yscalar.transform(ydata.reshape(-1, 1)).ravel()\n",
"plt.plot(x, y, 'x')\n",
"xdata.shape, ydata.shape, xscalar.mean_, xscalar.var_, yscalar.mean_, yscalar.var_, x, y"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Bayesian Linear Regression\n",
"https://www.cs.toronto.edu/~rgrosse/courses/csc411_f18/slides/lec19-slides.pdf"
]
},
{
"cell_type": "code",
"execution_count": 450,
"metadata": {},
"outputs": [
{
"output_type": "execute_result",
"data": {
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},
"metadata": {},
"execution_count": 450
},
{
"output_type": "display_data",
"data": {
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},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"sig = 2 # Data measurement standard deviation\n",
"S = np.array([[2*2, 0], [0, np.var(y)]]) # Bayesian prior of covariance of weights\n",
"S_inv = np.linalg.pinv(S)\n",
"\n",
"Phi = x.reshape(-1, 1); Phi = np.hstack([Phi, np.ones_like(Phi)]) # Feature vector\n",
"\n",
"Sigma_inv = sig ** (-2) * Phi.T @ Phi + S_inv\n",
"Sigma = np.linalg.pinv(Sigma_inv)\n",
"\n",
"Mu = sig ** (-2) * Sigma @ Phi.T @ y\n",
"\n",
"plt.plot(x, y, 'x', x, Mu @ Phi.T, '--')\n",
"Mu, Sigma, np.sqrt(np.diag(Sigma))"
]
},
{
"cell_type": "code",
"execution_count": 451,
"metadata": {},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "array([4.28838613e-02, 1.14352972e-14])"
},
"metadata": {},
"execution_count": 451
}
],
"source": [
"lambd = 1 / S[0, 0]\n",
"w = np.linalg.pinv(Phi.T @ Phi + lambd * np.eye(Phi.shape[1])) @ Phi.T @ y\n",
"w"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Visualization"
]
},
{
"cell_type": "code",
"execution_count": 452,
"metadata": {
"tags": []
},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "Text(0.5, 0, 'w[0]')"
},
"metadata": {},
"execution_count": 452
},
{
"output_type": "display_data",
"data": {
"text/plain": "<Figure size 648x216 with 2 Axes>",
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\n"
},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"xgrid, ygrid = np.mgrid[-2:2:.1, -2:2:.1]\n",
"pos = np.dstack((xgrid, ygrid))\n",
"fig, ax = plt.subplots(1, 2, figsize=(9,3), sharey=True)\n",
"ax[0].contourf(xgrid, ygrid, multivariate_normal(mean=[0, 0], cov=S).pdf(pos), vmin=0, vmax=1)\n",
"ax[0].axis('equal')\n",
"ax[0].set_xlabel('w[0]')\n",
"ax[0].set_ylabel('w[1]')\n",
"ax[1].contourf(xgrid, ygrid, multivariate_normal(mean=Mu, cov=Sigma).pdf(pos), vmin=0, vmax=1)\n",
"ax[1].axis('equal')\n",
"ax[1].set_xlabel('w[0]')"
]
},
{
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{
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\n"
},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"sigma2_pred = np.diag(Phi @ Sigma @ Phi.T + sig * sig)\n",
"plt.plot(x, y, 'x', x, Mu @ Phi.T, '-', x, Mu @ Phi.T + 1.96 * np.sqrt(sigma2_pred), 'b--', x, Mu @ Phi.T - 1.96 * np.sqrt(sigma2_pred), 'b--')"
]
},
{
"cell_type": "code",
"execution_count": 454,
"metadata": {},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "Text(0.5, 1.0, 'Slope = $0.042 \\\\pm $0.734')"
},
"metadata": {},
"execution_count": 454
},
{
"output_type": "display_data",
"data": {
"text/plain": "<Figure size 432x288 with 1 Axes>",
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\n"
},
"metadata": {
"needs_background": "light"
}
}
],
"source": [
"plt.plot(xscalar.inverse_transform(x), yscalar.inverse_transform(y), 'x', \n",
" xscalar.inverse_transform(x), yscalar.inverse_transform(Mu @ Phi.T), '-', \n",
" xscalar.inverse_transform(x), yscalar.inverse_transform(Mu @ Phi.T + 1.96 * np.sqrt(sigma2_pred)), 'b--', \n",
" xscalar.inverse_transform(x), yscalar.inverse_transform(Mu @ Phi.T - 1.96 * np.sqrt(sigma2_pred)), 'b--')\n",
"plt.xlabel(\"Age\")\n",
"plt.ylabel(\"Index\")\n",
"plt.ylim([0, 100])\n",
"plt.grid()\n",
"plt.title(fr\"Slope = ${Mu[0]:.3f} \\pm ${np.sqrt(Sigma[0,0])*1.96:.3f}\")"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
]
}
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