mirror of
https://github.com/ION606/COGMOD-HWI.git
synced 2026-05-14 22:16:57 +00:00
276 lines
34 KiB
Plaintext
276 lines
34 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"import stan\n",
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"import arviz as az\n",
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"\n",
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"# stan problems\n",
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"import nest_asyncio\n",
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"nest_asyncio.apply()\n",
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"\n",
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"np.random.seed(42)\n",
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"N = 100\n",
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"alpha = 2.3\n",
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"beta = 4.0\n",
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"sigma = 2.0\n",
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"x = np.random.normal(size=N)\n",
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"y = alpha + beta * x + sigma * np.random.normal(size=N)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"mu_0 = 3\n",
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"sigma_0 = 3\n",
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"sample_mean = np.mean(y)\n",
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"n = N\n",
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"\n",
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"# posterior\n",
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"sigma_sq = sigma ** 2\n",
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"sigma_0_sq = sigma_0 ** 2\n",
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"\n",
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"mu_post = ((n / sigma_sq) * sample_mean + (1 / sigma_0_sq) * mu_0) / (n / sigma_sq + 1 / sigma_0_sq)\n",
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"sigma_post_sq = 1 / (n / sigma_sq + 1 / sigma_0_sq)\n",
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"sigma_post = np.sqrt(sigma_post_sq)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": "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"text/plain": [
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"<Figure size 640x480 with 1 Axes>"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"prior_samples = np.random.normal(mu_0, sigma_0, 10000)\n",
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"posterior_samples = np.random.normal(mu_post, sigma_post, 10000)\n",
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"\n",
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"plt.hist(prior_samples, bins=50, density=True, alpha=0.5, label='Prior')\n",
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"plt.hist(posterior_samples, bins=50, density=True, alpha=0.5, label='Posterior')\n",
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"\n",
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"plt.legend()\n",
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"plt.xlabel('Average Height (meters)')\n",
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"plt.ylabel('Density')\n",
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"plt.title('Prior vs Posterior')\n",
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"plt.savefig('Q5.png')\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"stanCode = \"\"\"\n",
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"data {\n",
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" int<lower=0> N; // observations\n",
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" vector[N] x; // predictor\n",
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" vector[N] y; // response\n",
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"}\n",
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"parameters {\n",
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" real alpha; // intercept\n",
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" real beta; // slope\n",
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" real<lower=0> sigma; // noise standard deviation\n",
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"}\n",
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"model {\n",
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" sigma ~ inv_gamma(1, 1); // noise\n",
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" alpha ~ normal(0, 10); // intercept\n",
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" beta ~ normal(0, 10); // slope\n",
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" y ~ normal(alpha + beta * x, sigma);\n",
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"}\n",
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"\"\"\"\n",
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"\n",
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"data = {\n",
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" 'N': N,\n",
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" 'x': x,\n",
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" 'y': y\n",
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"}"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Building...\n"
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]
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},
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"\n",
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"Building: found in cache, done.Sampling: 0%\n",
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"Sampling: 25% (3000/12000)\n",
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"Sampling: 50% (6000/12000)\n",
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"Sampling: 75% (9000/12000)\n",
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"Sampling: 100% (12000/12000)\n",
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"Sampling: 100% (12000/12000), done.\n",
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"Messages received during sampling:\n",
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" Gradient evaluation took 4.2e-05 seconds\n",
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" 1000 transitions using 10 leapfrog steps per transition would take 0.42 seconds.\n",
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" Adjust your expectations accordingly!\n",
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" Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:\n",
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" Exception: normal_lpdf: Scale parameter is 0, but must be positive! (in '/tmp/httpstan_lipc1ba5/model_oj7ef663.stan', line 16, column 2 to column 38)\n",
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" If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,\n",
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" but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.\n",
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" Gradient evaluation took 2.8e-05 seconds\n",
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" 1000 transitions using 10 leapfrog steps per transition would take 0.28 seconds.\n",
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" Adjust your expectations accordingly!\n",
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" Gradient evaluation took 2.2e-05 seconds\n",
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" 1000 transitions using 10 leapfrog steps per transition would take 0.22 seconds.\n",
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" Adjust your expectations accordingly!\n",
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" Informational Message: The current Metropolis proposal is about to be rejected because of the following issue:\n",
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" Exception: normal_lpdf: Scale parameter is 0, but must be positive! (in '/tmp/httpstan_lipc1ba5/model_oj7ef663.stan', line 16, column 2 to column 38)\n",
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" If this warning occurs sporadically, such as for highly constrained variable types like covariance matrices, then the sampler is fine,\n",
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" but if this warning occurs often then your model may be either severely ill-conditioned or misspecified.\n",
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" Gradient evaluation took 3.8e-05 seconds\n",
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" 1000 transitions using 10 leapfrog steps per transition would take 0.38 seconds.\n",
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" Adjust your expectations accordingly!\n"
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]
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},
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{
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"data": {
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"text/html": [
|
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|
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"<div>\n",
|
||
|
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"<style scoped>\n",
|
||
|
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" .dataframe tbody tr th:only-of-type {\n",
|
||
|
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" vertical-align: middle;\n",
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||
|
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" }\n",
|
||
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"\n",
|
||
|
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" .dataframe tbody tr th {\n",
|
||
|
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" vertical-align: top;\n",
|
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|
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" }\n",
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"\n",
|
||
|
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" .dataframe thead th {\n",
|
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|
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" text-align: right;\n",
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" }\n",
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"</style>\n",
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"<table border=\"1\" class=\"dataframe\">\n",
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|
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" <thead>\n",
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|
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" <tr style=\"text-align: right;\">\n",
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" <th></th>\n",
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" <th>mean</th>\n",
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" <th>sd</th>\n",
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" <th>hdi_3%</th>\n",
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" <th>hdi_97%</th>\n",
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" <th>mcse_mean</th>\n",
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" <th>mcse_sd</th>\n",
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" <th>ess_bulk</th>\n",
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" <th>ess_tail</th>\n",
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" <th>r_hat</th>\n",
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" </tr>\n",
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|
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" </thead>\n",
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" <tbody>\n",
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" <tr>\n",
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" <th>alpha</th>\n",
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" <td>2.315</td>\n",
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" <td>0.193</td>\n",
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" <td>1.946</td>\n",
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" <td>2.672</td>\n",
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" <td>0.002</td>\n",
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" <td>0.002</td>\n",
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" <td>7672.0</td>\n",
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" <td>5912.0</td>\n",
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" <td>1.0</td>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>beta</th>\n",
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" <td>3.716</td>\n",
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" <td>0.214</td>\n",
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" <td>3.305</td>\n",
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" <td>4.108</td>\n",
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" <td>0.003</td>\n",
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" <td>0.002</td>\n",
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" <td>6876.0</td>\n",
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" <td>5564.0</td>\n",
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" <td>1.0</td>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>sigma</th>\n",
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" <td>1.910</td>\n",
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" <td>0.140</td>\n",
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" <td>1.662</td>\n",
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" <td>2.175</td>\n",
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" <td>0.002</td>\n",
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" <td>0.002</td>\n",
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||
|
|
" <td>7423.0</td>\n",
|
||
|
|
" <td>5615.0</td>\n",
|
||
|
|
" <td>1.0</td>\n",
|
||
|
|
" </tr>\n",
|
||
|
|
" </tbody>\n",
|
||
|
|
"</table>\n",
|
||
|
|
"</div>"
|
||
|
|
],
|
||
|
|
"text/plain": [
|
||
|
|
" mean sd hdi_3% hdi_97% mcse_mean mcse_sd ess_bulk ess_tail \\\n",
|
||
|
|
"alpha 2.315 0.193 1.946 2.672 0.002 0.002 7672.0 5912.0 \n",
|
||
|
|
"beta 3.716 0.214 3.305 4.108 0.003 0.002 6876.0 5564.0 \n",
|
||
|
|
"sigma 1.910 0.140 1.662 2.175 0.002 0.002 7423.0 5615.0 \n",
|
||
|
|
"\n",
|
||
|
|
" r_hat \n",
|
||
|
|
"alpha 1.0 \n",
|
||
|
|
"beta 1.0 \n",
|
||
|
|
"sigma 1.0 "
|
||
|
|
]
|
||
|
|
},
|
||
|
|
"execution_count": 6,
|
||
|
|
"metadata": {},
|
||
|
|
"output_type": "execute_result"
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"model = stan.build(stanCode, data=data)\n",
|
||
|
|
"fit = model.sample(num_samples=2000, num_chains=4)\n",
|
||
|
|
"\n",
|
||
|
|
"az.summary(az.from_pystan(fit))"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"metadata": {
|
||
|
|
"kernelspec": {
|
||
|
|
"display_name": ".venv",
|
||
|
|
"language": "python",
|
||
|
|
"name": "python3"
|
||
|
|
},
|
||
|
|
"language_info": {
|
||
|
|
"codemirror_mode": {
|
||
|
|
"name": "ipython",
|
||
|
|
"version": 3
|
||
|
|
},
|
||
|
|
"file_extension": ".py",
|
||
|
|
"mimetype": "text/x-python",
|
||
|
|
"name": "python",
|
||
|
|
"nbconvert_exporter": "python",
|
||
|
|
"pygments_lexer": "ipython3",
|
||
|
|
"version": "3.12.9"
|
||
|
|
}
|
||
|
|
},
|
||
|
|
"nbformat": 4,
|
||
|
|
"nbformat_minor": 2
|
||
|
|
}
|