From 21f791aa2f782a52f5016d71c0ca6ebe776ee510 Mon Sep 17 00:00:00 2001 From: shiraamitchell Date: Mon, 29 Jul 2019 14:14:55 -0400 Subject: [PATCH] fixed small typos --- divergences_and_bias/divergences_and_bias.Rmd | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/divergences_and_bias/divergences_and_bias.Rmd b/divergences_and_bias/divergences_and_bias.Rmd index db03b19..1c2617c 100644 --- a/divergences_and_bias/divergences_and_bias.Rmd +++ b/divergences_and_bias/divergences_and_bias.Rmd @@ -82,7 +82,7 @@ $$y_{n} \sim \mathcal{N}(\theta_{n}, \sigma_{n}),$$ where $n \in \left\{1, \ldots, 8 \right\}$ and the $\left\{ y_{n}, \sigma_{n} \right\}$ are given as data. -Inferring the hierarchical hyperparameters, $\mu$ and $\sigma$, together with +Inferring the hierarchical hyperparameters, $\mu$ and $\tau$, together with the group-level parameters, $\theta_{1}, \ldots, \theta_{8}$, allows the model to pool data across the groups and reduce their posterior variance. Unfortunately this pooling also squeezes the posterior distribution into a @@ -180,7 +180,7 @@ almost 2% of the iterations in our lone Markov chain ended with a divergence, ```{r, comment=NA} check_div(fit_cp) ``` -Even with a single short chain these divergences are able to identity the bias +Even with a single short chain these divergences are able to identify the bias and advise skepticism of any resulting MCMC estimators. Additionally, because the divergent transitions, here shown here in green, tend @@ -288,7 +288,7 @@ points(div_params_cp80$'theta[1]', log(div_params_cp80$tau), Divergences in Hamiltonian Monte Carlo arise when the Hamiltonian transition encounters regions of extremely large curvature, such as the opening of the -hierarchical funnel. Unable to accurate resolve these regions, the transition +hierarchical funnel. Unable to accurately resolve these regions, the transition malfunctions and flies off towards infinity. With the transitions unable to completely explore these regions of extreme curvature, we lose geometric ergodicity and our MCMC estimators become biased.