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‎_notebooks/aiyagari.ipynb‎

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‎_notebooks/ak2.ipynb‎

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‎_notebooks/ar1_bayes.ipynb‎

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"cells": [
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{
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"cell_type": "markdown",
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"id": "18ced5a7",
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"id": "4480886e",
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"metadata": {},
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"source": [
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"# Posterior Distributions for AR(1) Parameters"
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]
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},
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{
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"cell_type": "markdown",
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"id": "4550bc58",
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"metadata": {},
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"source": [
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"# GPU\n",
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"\n",
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"This lecture was built using a machine with the latest CUDA and CUDANN frameworks installed with access to a GPU.\n",
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"This lecture was built using a machine with access to a GPU.\n",
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"\n",
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"To run this lecture on [Google Colab](https://colab.research.google.com/), click on the “play” icon top right, select Colab, and set the runtime environment to include a GPU.\n",
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"[Google Colab](https://colab.research.google.com/) has a free tier with GPUs\n",
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"that you can access as follows:\n",
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"\n",
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"To run this lecture on your own machine, you need to install the software listed following this notice."
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"1. Click on the “play” icon top right \n",
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"1. Select Colab \n",
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"1. Set the runtime environment to include a GPU "
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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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"id": "bc11defe",
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"metadata": {
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"hide-output": false
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},
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},
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{
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"cell_type": "markdown",
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"id": "4d1620ca",
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"metadata": {},
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"source": [
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"In addition to what’s included in base Anaconda, we need to install the following packages"
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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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"hide-output": false
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},
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},
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{
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"cell_type": "markdown",
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"id": "9492ed94",
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"metadata": {},
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"source": [
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"We’ll begin with some Python imports."
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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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},
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{
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"cell_type": "markdown",
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"id": "806e7cad",
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"metadata": {},
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"source": [
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"This lecture uses Bayesian methods offered by [pymc](https://www.pymc.io/projects/docs/en/stable/) and [numpyro](https://num.pyro.ai/en/stable/) to make statistical inferences about two parameters of a univariate first-order autoregression.\n",
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "d4d988b3",
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"metadata": {
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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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"id": "270e075d",
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"metadata": {
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"hide-output": false
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},
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{
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"cell_type": "markdown",
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"id": "0eb5e83e",
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"metadata": {},
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"source": [
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"Now we shall use Bayes’ law to construct a posterior distribution, conditioning on the initial value of $ y_0 $.\n",
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"## PyMC Implementation\n",
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{
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"cell_type": "code",
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},
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{
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"cell_type": "markdown",
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"source": [
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"[pmc.sample](https://www.pymc.io/projects/docs/en/v5.10.0/api/generated/pymc.sample.html#pymc-sample) by default uses the NUTS samplers to generate samples as shown in the below cell:"
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{
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"execution_count": null,
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{
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"cell_type": "code",
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"metadata": {
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Evidently, the posteriors aren’t centered on the true values of $ .5, 1 $ that we used to generate the data.\n",
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{
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"cell_type": "code",
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"execution_count": null,
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},
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{
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"cell_type": "markdown",
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"id": "f42ea1fd",
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"metadata": {},
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"Now we shall compute a posterior distribution after seeing the same data but instead assuming that $ y_0 $ is drawn from the stationary distribution.\n",
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{
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"cell_type": "code",
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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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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"Please note how the posterior for $ \\rho $ has shifted to the right relative to when we conditioned on $ y_0 $ instead of assuming that $ y_0 $ is drawn from the stationary distribution.\n",
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},
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"## Numpyro Implementation"
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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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"Next, we again compute the posterior under the assumption that $ y_0 $ is drawn from the stationary distribution, so that\n",
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{
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{
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"Look what happened to the posterior!\n",
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}
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"metadata": {
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"date": 1764378007.2089982,
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"date": 1764547274.0689824,
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"filename": "ar1_bayes.md",
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"kernelspec": {
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"display_name": "Python",

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