deploy: bc6c2cd

QuantEcon · Jan 29, 2024 · 9e4779f · 9e4779f
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diff --git a/_notebooks/ar1_bayes.ipynb b/_notebooks/ar1_bayes.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "6101b12d",
+   "id": "e3c19ac4",
    "metadata": {},
    "source": [
     "# Posterior Distributions for  AR(1) Parameters\n",
@@ -13,7 +13,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "088fb117",
+   "id": "194738d6",
    "metadata": {
     "hide-output": false
    },
@@ -25,7 +25,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "cbeb1a42",
+   "id": "a810f395",
    "metadata": {
     "hide-output": false
    },
@@ -49,7 +49,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7f68f108",
+   "id": "8903c2ed",
    "metadata": {},
    "source": [
     "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",
@@ -145,7 +145,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "4d968167",
+   "id": "6080c7e5",
    "metadata": {
     "hide-output": false
    },
@@ -175,7 +175,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "2ff374b4",
+   "id": "2ec5bab0",
    "metadata": {
     "hide-output": false
    },
@@ -187,7 +187,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6594b515",
+   "id": "0329be6f",
    "metadata": {},
    "source": [
     "Now we shall use Bayes’ law to construct a posterior distribution, conditioning on the initial value of $ y_0 $.\n",
@@ -199,7 +199,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "00b542c4",
+   "id": "287851ae",
    "metadata": {},
    "source": [
     "## PyMC Implementation\n",
@@ -211,7 +211,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "22bc8a05",
+   "id": "e3ebfdd8",
    "metadata": {
     "hide-output": false
    },
@@ -234,7 +234,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b0f47a54",
+   "id": "90996156",
    "metadata": {},
    "source": [
     "[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:"
@@ -243,7 +243,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "5750f3a9",
+   "id": "356d6214",
    "metadata": {
     "hide-output": false
    },
@@ -256,7 +256,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "8f00983c",
+   "id": "770f0cde",
    "metadata": {
     "hide-output": false
    },
@@ -268,22 +268,22 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c85dd51b",
+   "id": "658500ab",
    "metadata": {},
    "source": [
     "Evidently, the posteriors aren’t centered on the true values of $ .5, 1 $ that we used to generate the data.\n",
     "\n",
-    "This is a symptom of the classic **Hurwicz bias** for first order autoregressive processes (see Leonid Hurwicz [[Hur50](https://python.quantecon.org/zreferences.html#id244)].)\n",
+    "This is a symptom of the classic **Hurwicz bias** for first order autoregressive processes (see Leonid Hurwicz [[Hur50](https://python.quantecon.org/zreferences.html#id246)].)\n",
     "\n",
-    "The Hurwicz bias is worse the smaller is the sample (see [[OW69](https://python.quantecon.org/zreferences.html#id243)]).\n",
+    "The Hurwicz bias is worse the smaller is the sample (see [[OW69](https://python.quantecon.org/zreferences.html#id245)]).\n",
     "\n",
     "Be that as it may, here is more information about the posterior."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "ecc856c2",
+   "id": "a2f4febf",
    "metadata": {
     "hide-output": false
    },
@@ -297,7 +297,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "31e05ee2",
+   "id": "f7465ac3",
    "metadata": {},
    "source": [
     "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",
@@ -314,7 +314,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "eaef1c61",
+   "id": "b629c3d3",
    "metadata": {
     "hide-output": false
    },
@@ -340,7 +340,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "80cd1aa8",
+   "id": "8be43f79",
    "metadata": {
     "hide-output": false
    },
@@ -355,7 +355,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "34b8b76f",
+   "id": "1139dd10",
    "metadata": {
     "hide-output": false
    },
@@ -368,7 +368,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "f1baae3c",
+   "id": "b696cc4e",
    "metadata": {
     "hide-output": false
    },
@@ -382,7 +382,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "00f3dd34",
+   "id": "c4b2d8df",
    "metadata": {},
    "source": [
     "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",
@@ -399,7 +399,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "133d1d92",
+   "id": "728ea887",
    "metadata": {},
    "source": [
     "## Numpyro Implementation"
@@ -408,7 +408,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "49a4da18",
+   "id": "15ad0d76",
    "metadata": {
     "hide-output": false
    },
@@ -444,7 +444,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "fb696090",
+   "id": "5c2dc83b",
    "metadata": {
     "hide-output": false
    },
@@ -465,7 +465,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "d68e8c33",
+   "id": "d04c5e88",
    "metadata": {
     "hide-output": false
    },
@@ -485,7 +485,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "70aff6ac",
+   "id": "9d6d5ac3",
    "metadata": {
     "hide-output": false
    },
@@ -497,7 +497,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "addfc73b",
+   "id": "b365d10f",
    "metadata": {
     "hide-output": false
    },
@@ -508,7 +508,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b4d12b8e",
+   "id": "1f29a3d6",
    "metadata": {},
    "source": [
     "Next, we again compute the posterior under the assumption that $ y_0 $ is drawn from the stationary distribution, so that\n",
@@ -523,7 +523,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "c0dfc979",
+   "id": "236f75ac",
    "metadata": {
     "hide-output": false
    },
@@ -548,7 +548,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "69a309ec",
+   "id": "434a6c10",
    "metadata": {
     "hide-output": false
    },
@@ -568,7 +568,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "236344e9",
+   "id": "04f5d399",
    "metadata": {
     "hide-output": false
    },
@@ -580,7 +580,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "576d58cf",
+   "id": "5b948fb9",
    "metadata": {
     "hide-output": false
    },
@@ -591,7 +591,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6248f9d1",
+   "id": "072e44cf",
    "metadata": {},
    "source": [
     "Look what happened to the posterior!\n",
@@ -606,7 +606,7 @@
   }
  ],
  "metadata": {
-  "date": 1706246574.4990308,
+  "date": 1706493928.679338,
   "filename": "ar1_bayes.md",
   "kernelspec": {
    "display_name": "Python",