STA 360/602L: Module 2.5

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# STA 360/602L: Module 2.5
## Frequentist vs Bayesian intervals
### Dr. Olanrewaju Michael Akande

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## Frequentist confidence intervals

- Recall that a frequentist confidence interval  `$[l(y), \ u(y)]$` has 95% frequentist coverage for a population parameter `$\theta$` if, before we collect the data,
.block[
`$$\Pr[l(y) < \theta < u(y) | \theta] = 0.95.$$`
]

- This means that 95% of the time, our constructed interval will cover the true parameter, and 5% of the time it won't.

- In any given sample, you don't know whether you're in the lucky 95% or the unlucky 5%.

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## Frequentist confidence intervals

- You just know that either the interval covers the parameter, or it doesn't (useful, but not too helpful clearly).

- There is NOT a 95% chance your interval covers the true parameter once you have collected the data.

- Asking about the definition of a confidence interval is tricky, even for those who know what they're doing.

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## Bayesian intervals

- An interval `$[l(y), \ u(y)]$` has 95% Bayesian coverage for `$\theta$` if
.block[
`$$\Pr[l(y) < \theta < u(y) | Y=y] = 0.95.$$`
]

- This describes our information about where `$\theta$` lies _after_ we observe the data.

- Fantastic!

- This is actually the interpretation people want to give to the frequentist confidence interval.

- Bayesian interval estimates are often generally called .hlight[credible intervals].

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## Bayesian quantile-based interval

- The easiest way to obtain a Bayesian interval estimate is to use posterior quantiles.

- Easy since we either know the posterior densities exactly or can sample from the distributions.

- To make a `$100 \times (1-\alpha)%$` quantile-based credible interval, find numbers (quantiles) `$\theta_{\alpha/2} < \theta_{1-\alpha/2}$` such that

1. `$\Pr(\theta < \theta_{\alpha/2} | Y=y) = \dfrac{\alpha}{2}$`; and
  
  2. `$\Pr(\theta > \theta_{1-\alpha/2} | Y=y) = \dfrac{\alpha}{2}$`.
  
--

- This is an .hlight[equal-tailed interval]. Often when researchers refer to a credible interval, this is what they mean.

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## Equal-tailed quantile-based interval

- This is Figure 3.6 from the Hoff book. Focus on the quantile-based credible interval for now.

- Note that there are values of `$\theta$` outside the quantile-based credible interval, with higher density than some values inside the interval.

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## HPD region

- A `$100 \times (1-\alpha)%$` highest posterior density (HPD) region is a subset `$s(y)$` of the parameter space `$\Theta$` such that

1. `$\Pr(\theta \in s(y) | Y=y) = 1-\alpha$`; and
  
  2. If `$\theta_a \in s(y)$` and `$\theta_b \notin s(y)$`, then `$p(\theta_a | Y=y) > p(\theta_b | Y=y)$`.

- `$\Rightarrow$` **All** points in a HPD region have higher posterior density than points outside the region.

*Note this region is not necessarily a single interval (e.g., in the case of a multimodal posterior).*
  
--

- The basic idea is to gradually move a horizontal line down across the density, including in the HPD region all values of `$\theta$` with a density above the horizontal line.

- Stop moving the line down when the posterior probability of the values of `$\theta$` in the region reaches `$1-\alpha$`.

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## HPD region

Hoff Figure 3.6 shows how to construct an HPD region.

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# What's next?

### Move on to the readings for the next module!