Continuous priors & posterior analysis

# Continuous priors & posterior analysis
### Math 315: Bayesian Statistics

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## 1. Updating a continuous prior

## 2. Summarizing the posterior

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class: middle, inverse

# Updating a continuous prior

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# Design, redux

.Large[

**Data:** N N N N B B N N N B N N N N N N N (14 Ns; 3 Bs)

**Data model:**

Some true proportion of guesses, `\(\theta\)`

Toss a coin with probability of heads, `\(\theta\)`

**Belief about `\(\theta\)`:**

Uniform over (0, 1)

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class: inverse

# Your turn

.Large[
- Break into groups

- Derive the posteriors

- Are you working with a conjugate family?
]

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class: middle, inverse

# Summarizing the posterior

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# Posterior analysis

.large[
> To a Bayesian, the best information one can ever have about `\(\theta\)` is to know the posterior density.
>
> — Christensen, et al; *Bayesian Ideas and Data Analysis*, p. 31
]

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# Point estimates

.Large[
- Posterior mean 
- Posterior median 
- Posterior mode — i.e. *maximum a posteriori* (MAP) estimate
]

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# Credible intervals

```r
# q*() functions calculate quantiles from the specified distribution
c(lower = qbeta(0.055, 14, 4), upper = qbeta(1 - 0.055, 14, 4))
```

```
##     lower     upper 
## 0.6100397 0.9126956
```

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# Credible intervals are not unique

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# Highest Posterior Density Interval (HPDI)

![](03-posterior-analysis_files/figure-html/hpdi-1.svg)
]

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class: center
![](03-posterior-analysis_files/figure-html/ci1-1.svg)

![](03-posterior-analysis_files/figure-html/hpdi-1.svg)

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# Testing a hypothesis

`\(H_0: \theta \le 0.5\)`

`\(P(\theta \le 0.5 | Y = 14) = 0.004\)`

![](03-posterior-analysis_files/figure-html/unnamed-chunk-5-1.svg)

]

`\(H_1: \theta > 0.5\)`

`\(P(\theta > 0.5 | Y = 14) = 0.996\)`

![](03-posterior-analysis_files/figure-html/unnamed-chunk-6-1.svg)
]

]