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‎_notebooks/aiyagari.ipynb‎

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‎_notebooks/ak2.ipynb‎

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‎_notebooks/ak_aiyagari.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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"metadata": {},
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"$$\n",
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},
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{
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"cell_type": "markdown",
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"id": "62b09d5e",
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"id": "381c31e6",
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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": "34a963c2",
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"metadata": {},
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"source": [
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"# GPU\n",
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "b9010665",
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"metadata": {
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},
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},
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{
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"cell_type": "markdown",
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"id": "71da1db4",
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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": "b06f4c65",
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"metadata": {},
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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": "8916928b",
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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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"metadata": {
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"hide-output": false
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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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},
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},
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{
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"cell_type": "markdown",
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"id": "66eb762c",
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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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"id": "d0bbc011",
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"metadata": {},
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"source": [
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"## PyMC Implementation\n",
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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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"metadata": {},
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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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"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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"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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"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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"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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"cell_type": "code",
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"execution_count": null,
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"id": "74d3581f",
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"metadata": {
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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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"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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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Numpyro Implementation"
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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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{
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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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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{
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{
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"Look what happened to the posterior!\n",
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}
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],
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"metadata": {
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"date": 1771505064.4662316,
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"date": 1771505815.2329173,
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"filename": "ar1_bayes.md",
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