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@mmcky
mmcky committed Jan 31, 2024
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66 changes: 33 additions & 33 deletions _notebooks/ar1_bayes.ipynb
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"# Posterior Distributions for AR(1) Parameters\n",
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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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},
{
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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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"## PyMC Implementation\n",
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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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"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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},
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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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{
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{
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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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"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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"Look what happened to the posterior!\n",
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}
],
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"date": 1706658080.0212908,
"date": 1706679395.7876425,
"filename": "ar1_bayes.md",
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"display_name": "Python",
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