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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>Visualizing priors</title>
<meta charset="utf-8" />
<meta name="author" content="Andrew MacDonald" />
<meta name="date" content="2020-06-20" />
<script src="libs/header-attrs/header-attrs.js"></script>
<link href="libs/remark-css/default.css" rel="stylesheet" />
<link href="libs/remark-css/default-fonts.css" rel="stylesheet" />
</head>
<body>
<textarea id="source">
class: center, middle, inverse, title-slide
# Visualizing priors
## <br/>With great power comes great responsibility: Stan for modern ecological modelling
### Andrew MacDonald
### 2020-06-20
---
# Bayesian models
$$
`\begin{align}
F_i & \sim \text{Poisson}(\lambda_i) \\
\text{log}(\lambda_i) &= \alpha + \beta x \\
\alpha & \sim \text{Normal}(??, ??) \\
\beta & \sim \text{Normal}(??, ??)
\end{align}`
$$
---
# Bayesian models
$$
`\begin{align}
F_i & \sim \text{Poisson}(\lambda_i) \\
\text{log}(\lambda_i) &= \alpha + \beta x \\
\alpha & \sim \text{Normal}(0, 1000) \\
\beta & \sim \text{Normal}(0, 1000)
\end{align}`
$$
.footnote[as seen in Kéry & Royle 2016 p 188]
---
class: inverse, center, middle
# So what _is_ a good prior?
---
.footnote[
Gabry, Jonah, Daniel Simpson, Aki Vehtari, Michael Betancourt, and Andrew Gelman. « Visualization in Bayesian Workflow ». Journal of the Royal Statistical Society: Series A (Statistics in Society) 182 (2): 389‑402. https://doi.org/10.1111/rssa.12378.
]
---
# what does this prior mean?
$$
`\begin{align}
F_i & \sim \text{Poisson}(\lambda_i) \\
\text{log}(\lambda_i) &= \alpha + \beta x \\
\alpha & \sim \text{Normal}(0, 1000) \\
\beta & \sim \text{Normal}(0, 1000)
\end{align}`
$$
---
# what does this prior mean?
lets do an example about my favourites: fly larvae!
.pull-left[
<img src="img/lil_larv.png" style="width: 70%" />
]
.pull-right[
$$
`\begin{align}
F_i & \sim \text{Poisson}(\lambda_i) \\
\text{log}(\lambda_i) &= \alpha + \beta x \\
\alpha & \sim \text{Normal}(0, 1000) \\
\beta & \sim \text{Normal}(0, 1000)
\end{align}`
$$
where `\(x\)` is some kind of standardized environmental variable
]
---
# A quick look at the math for those curious
$$
`\begin{align}
\text{log}(\lambda) &= \alpha + \beta x \\
\lambda &= e^{\alpha + \beta x} \\
\lambda &= e^{\alpha}e^{\beta x} \\
\end{align}`
$$
And since `\(N(0,1000)\)` implies that -1000 and +1000 are entirely reasonable..
.pull-left[
* `\(2.72^{-1000}\times 2.72^{-1000}\)`
* (practically zero)
]
.pull-right[
* `\(2.72^{1000}\times 2.72^{1000}\)`
* (..kind of a lot)
]
---
# either no flies at all.. or a huge planet of maggots
<img src="img/many_insects.png" style="width: 70%" />
for comparison, the planet Saturn weighs 5.7 * 10^29 grams
---
# syntax: brms
```r
# define formula
insects_bf <- bf(
abundance ~ 1 + env,
family = poisson()
)
```
---
# syntax: setting priors
```r
get_prior(insects_bf, data = insect_data)
insect_priors <- c(
prior(normal(0,100), class = "b", coef = "env"),
prior(normal(0,100), class = "Intercept")
)
```
---
# syntax: sampling the model
```r
insect_samples <- brm(insects_bf,
data = insect_data, # does nothing
prior = insect_priors,
sample_prior = "only")
```
---
# Exercise I -- Fly larvae
* see the file `insects.R` on [github](https://github.com/aammd/ISEC_stan_course/blob/996eb5b2f49c52b0290c63a6635f925104bb623b/insects.R)
* on your own or in a small group, experiment with setting priors for insect abundances.
* AND/OR consider an animal or plant which is more relevant to you
* or stay with insects even if that is not your specialty, and set "vague" priors!
---
# exercises -- les poissons
using data from [Kaggle, about fish](https://www.kaggle.com/aungpyaeap/fish-market)
```r
library(readr)
fish <- read_csv("https://raw.githubusercontent.com/aammd/ISEC_stan_course/master/Fish.csv")
knitr::kable(head(fish), format = 'html')
```
<table>
<thead>
<tr>
<th style="text-align:left;"> Species </th>
<th style="text-align:right;"> Weight </th>
<th style="text-align:right;"> Length1 </th>
<th style="text-align:right;"> Length2 </th>
<th style="text-align:right;"> Length3 </th>
<th style="text-align:right;"> Height </th>
<th style="text-align:right;"> Width </th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;"> Bream </td>
<td style="text-align:right;"> 242 </td>
<td style="text-align:right;"> 23.2 </td>
<td style="text-align:right;"> 25.4 </td>
<td style="text-align:right;"> 30.0 </td>
<td style="text-align:right;"> 11.5200 </td>
<td style="text-align:right;"> 4.0200 </td>
</tr>
<tr>
<td style="text-align:left;"> Bream </td>
<td style="text-align:right;"> 290 </td>
<td style="text-align:right;"> 24.0 </td>
<td style="text-align:right;"> 26.3 </td>
<td style="text-align:right;"> 31.2 </td>
<td style="text-align:right;"> 12.4800 </td>
<td style="text-align:right;"> 4.3056 </td>
</tr>
<tr>
<td style="text-align:left;"> Bream </td>
<td style="text-align:right;"> 340 </td>
<td style="text-align:right;"> 23.9 </td>
<td style="text-align:right;"> 26.5 </td>
<td style="text-align:right;"> 31.1 </td>
<td style="text-align:right;"> 12.3778 </td>
<td style="text-align:right;"> 4.6961 </td>
</tr>
<tr>
<td style="text-align:left;"> Bream </td>
<td style="text-align:right;"> 363 </td>
<td style="text-align:right;"> 26.3 </td>
<td style="text-align:right;"> 29.0 </td>
<td style="text-align:right;"> 33.5 </td>
<td style="text-align:right;"> 12.7300 </td>
<td style="text-align:right;"> 4.4555 </td>
</tr>
<tr>
<td style="text-align:left;"> Bream </td>
<td style="text-align:right;"> 430 </td>
<td style="text-align:right;"> 26.5 </td>
<td style="text-align:right;"> 29.0 </td>
<td style="text-align:right;"> 34.0 </td>
<td style="text-align:right;"> 12.4440 </td>
<td style="text-align:right;"> 5.1340 </td>
</tr>
<tr>
<td style="text-align:left;"> Bream </td>
<td style="text-align:right;"> 450 </td>
<td style="text-align:right;"> 26.8 </td>
<td style="text-align:right;"> 29.7 </td>
<td style="text-align:right;"> 34.7 </td>
<td style="text-align:right;"> 13.6024 </td>
<td style="text-align:right;"> 4.9274 </td>
</tr>
</tbody>
</table>
---
---
</textarea>
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