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fixed small typos #17

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fixed small typos #17

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6 changes: 3 additions & 3 deletions divergences_and_bias/divergences_and_bias.Rmd
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Expand Up @@ -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
Expand Down Expand Up @@ -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
Expand Down Expand Up @@ -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.
Expand Down