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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>Generative models and multilevel models</title>
<meta charset="utf-8" />
<meta name="author" content="Dominique Gravel & Andrew MacDonald" />
<script src="assets/header-attrs-2.8/header-attrs.js"></script>
<link href="assets/remark-css-0.0.1/default.css" rel="stylesheet" />
<link href="assets/remark-css-0.0.1/hygge.css" rel="stylesheet" />
<link rel="stylesheet" href="assets/ecl707.css" type="text/css" />
</head>
<body>
<textarea id="source">
class: title-slide, middle
<style type="text/css">
.title-slide {
background-image: url('assets/img/bg.jpg');
background-color: #23373B;
background-size: contain;
border: 0px;
background-position: 600px 0;
line-height: 1;
}
</style>
# Model evaluation with ecological data
<hr width="65%" align="left" size="0.3" color="orange"></hr>
## Generative models and multilevel models
<hr width="65%" align="left" size="0.3" color="orange" style="margin-bottom:40px;" alt="@Martin Sanchez"></hr>
.instructors[
Andrew MacDonald, Dominique Gravel
]
<img src="assets/img/logo.png" width="25%" style="margin-top:20px;"></img>
---
# Generative models
Bayesian models can generate data. This is part of what makes them useful in ecology:
* Easy to make predictions
* Straightforward to test the model
---
# Simple Generative model
We've already seen a simple model which is generative: our Binomial model with a Beta prior:
We start with this model
$$
`\begin{align}
y \sim \text{Binomial}(6, \theta) \\
\theta \sim Beta(2,2)
\end{align}`
$$
and we have data `4, 5, 4, 4, 3`
* So total successes is `\(4 + 5 + 4 + 4 + 3 = 20\)`
* And total failures is `\(6 * 5 - 20 = 10\)`
The posterior becomes:
$$
\theta_{post} \sim \text{Beta}(22, 12)
$$
---
# Generating data with this simple model
Quick pseudocode (_Note_ this is also called the Beta-Binomial distribution) :
* generate values of `\(\theta\)` from `\(\text{Beta}(22, 12)\)`
* for each value, generate an observation of surviving eggs using:
$$
y \sim \text{Binomial}(6, \theta)
$$
.pull-left[
```r
theta_post <- rbeta(200,
22, 12)
obs <- rbinom(n = 200,
size = 6,
theta_post)
hist(obs, breaks = 20)
```
]
.pull-right[
<img src="generative_files/figure-html/unnamed-chunk-1-1.png" width="504" />
]
---
# nonlinear tree growth -- write the model
$$
`\begin{align}
\text{height} &\sim \text{LogNormal}(\text{log}(\mu), \sigma_{height}) \\
\mu &= \frac{aL}{a/s + L} \\
\end{align}`
$$
To make it Bayesian, we specify distributions for every parameter:
--
$$
`\begin{align}
\sigma_{height} & \sim \text{Exponential}(0.5)\\
a & \sim \text{Normal}(200, 15)\\
s & \sim \text{Normal}(15, 2)\\
\end{align}`
$$
--
* _not_ the only choices for these parameters! Not even particularly good ones
--
* use prior predictive checks to see if these priors make sense
---
# Nonlinear light data -- simulation
Draw parameters from your priors.
$$
`\begin{align}
\text{height} &\sim \text{LogNormal}(\text{log}(\mu), \sigma_{height}) \\
\mu &= \frac{aL}{a/s + L} \\
\sigma_{height} & \sim \text{Exponential}(2)\\
a & \sim \text{Normal}(200, 15)\\
s & \sim \text{Normal}(15, 2)\\
\end{align}`
$$
---
# Nonlinear light data -- simulation
Draw parameters from your priors.
$$
`\begin{align}
\text{height} &\sim \text{LogNormal}(\text{log}(\mu), \sigma_{height}) \\
\mu &= \frac{aL}{a/s + L} \\
\sigma_{height} & = 0.36\\
a & = 196.4\\
s & = 16.1\\
\end{align}`
$$
---
# Nonlinear light data -- simulation
Draw parameters from your priors.
$$
`\begin{align}
\text{height} &\sim \text{LogNormal}(\text{log}(\mu), \sigma_{height}) \\
\mu &= \frac{aL}{a/s + L} \\
\sigma_{height} & = 0.36\\
a & = 196.4\\
s & = 16.1\\
\end{align}`
$$
--
* keep these fixed for the rest of the simulation! You can go back and try different ones later!
--
* **pop quiz** is `\(\mu\)` a parameter??
---
# Nonlinear light data -- simulation
$$
`\begin{align}
\text{height} &\sim \text{LogNormal}\left(\text{log}\left(\frac{aL}{a/s + L} \right), \sigma_{height}\right) \\
\sigma_{height} & = 0.36\\
a & = 196.4\\
s & = 16.1\\
\end{align}`
$$
<br><br>
**NO**. `\(\mu\)` is not a parameter, it is simply a handy way to rewrite this equation so it is easier to read.
`\(\mu\)` might be a very interesting thing to calculate after the model is run!
---
class: center
# Simple simulation
<img src="generative_files/figure-html/unnamed-chunk-2-1.png" width="504" />
---
# Other ways of talking about the same model
let's go back to this representation of the model:
$$
`\begin{align}
\text{height} &\sim \text{LogNormal}(\text{log}(\mu), \sigma_{height}) \\
\mu &= \frac{aL}{a/s + L} \\
\sigma_{height} & \sim \text{Exponential}(2)\\
a & \sim \text{Normal}(200, 15)\\
s & \sim \text{Normal}(15, 2)\\
\end{align}`
$$
* the deterministic part of the model is indicated with `\(=\)`
* the stochastic parts are all indicated with `\(\sim\)`
When one distribution has a parameter which is also a distribution, we say that the first is *conditional* on the second.
---
class: center
# Graphing conditional distributions
<img src="generative_files/figure-html/dag-fig-1.png" width="504" />
---
class: center
# Converting the graph to probability notation
<img src="generative_files/figure-html/unnamed-chunk-3-1.png" width="504" />
<p>
$$
[a, s, \sigma_{\text{height}}|y ] \propto
[y |g(L, a, s), \sigma_{\text{height}}]
[a]
[s] [\sigma_{\text{height}}]
$$
</p>
---
# Converting the graph to probability notation
<p>
$$
[a, s, \sigma_{\text{height}}|y ] \propto
[y |g(L, a, s), \sigma_{\text{height}}]
[a]
[s] [\sigma_{\text{height}}]
$$
</p>
--
* the vertical bar `\(|\)` reads "conditioned on" or "dependent on"
--
* `\(L\)` is observed and therefore it appears *only* of the right-hand side of `\(|\)`
--
* `\(a\)`, `\(s\)` and `\(L\)` are parameters and therefore need distributions
--
* `\([y |g(L, a, s), \sigma_y]\)` is a likelihood! This means all the observations multiplied together:
<p>
$$
[y |g(L, a, s), \sigma_y] = \prod_{i=1}^{300} \text{LogNormal}\left(y_i, \text{log}\left(\frac{aL}{a/x + L} \right), \sigma_{height}\right)
$$
</p>
---
# Hierarchical models
$$
`\begin{align}
\text{height} &\sim \text{LogNormal}(\text{log}(\mu), \sigma_{height}) \\
\mu &= \frac{aL}{a/s + L} \\
\sigma_{height} & \sim \text{Exponential}(2)\\
a & \sim \text{Normal}(200, 15)\\
s & \sim \text{Normal}(15, 2)\\
\end{align}`
$$
We replace a parameter with a linear model, complete with a distribution:
---
# Hierarchical models
$$
`\begin{align}
\text{height} &\sim \text{LogNormal}(\text{log}(\mu_n), \sigma_{height}) \\
\mu_n &= \frac{a_nL}{a_n/s + L} \\
a_n &= \bar{a} + a_{j[n]}\\
a_j &\sim \text{Normal}(0, \sigma_a)\\
\bar{a} & \sim \text{Normal}(200, 15)\\
\sigma_{a} & \sim \text{Exponential}(2)\\
\sigma_{height} & \sim \text{Exponential}(4)\\
s & \sim \text{Normal}(15, 2)\\
\end{align}`
$$
* For observation number `\(n\)`, between `\(1\)` and `\(N\)` total observations.
* For species number `\(j\)` , which is between `\(1\)` and `\(S\)` total species (in our example `\(S\)` is 5)
* There are **5 values** of `\(a_j\)` : `\(a_1\)`, `\(a_2\)`, `\(a_3\)`, `\(a_4\)`, `\(a_5\)`
---
# Quick word about nested indexing
`\(a_{j[n]}\)` This means the value of `\(a\)` for observation `\(n\)` -- whichever of the `\(j\)` species it is in.
```r
sp_ids <- c(fir = 1, maple = 2)
fake_data <- expand.grid(light = c(5,10, 60),
spp = c("fir", "maple"))
*fake_data$id <- sp_ids[fake_data$spp]
knitr::kable(fake_data)
```
| light|spp | id|
|-----:|:-----|--:|
| 5|fir | 1|
| 10|fir | 1|
| 60|fir | 1|
| 5|maple | 2|
| 10|maple | 2|
| 60|maple | 2|
---
# Hierarchical models -- two ways to write
$$
`\begin{align}
\text{height} &\sim \text{LogNormal}(\text{log}(\mu_n), \sigma_{height}) \\
\mu_n &= \frac{a_nL}{a_n/s + L} \\
a_n &= \bar{a} + a_{j[n]}\\
a_j &\sim \text{Normal}(0, \sigma_a)\\
\bar{a} & \sim \text{Normal}(200, 15)\\
\sigma_{a} & \sim \text{Exponential}(2)\\
\sigma_{height} & \sim \text{Exponential}(4)\\
s & \sim \text{Normal}(15, 2)\\
\end{align}`
$$
---
# Hierarchical models -- two ways to write
$$
`\begin{align}
\text{height} &\sim \text{LogNormal}(\text{log}(\mu), \sigma_{height}) \\
\mu_n &= \frac{a_{j[n]}L}{a_{j[n]}/s + L} \\
a_j &\sim \text{Normal}(\bar{a}, \sigma_a)\\
\bar{a} & \sim \text{Normal}(200, 15)\\
\sigma_{a} & \sim \text{Exponential}(2)\\
\sigma_{height} & \sim \text{Exponential}(4)\\
s & \sim \text{Normal}(15, 2)\\
\end{align}`
$$
--
BOTH will have the same 5 values for `\(a_j\)`
---
class: center
# Different species `\(a\)` simulation
<img src="generative_files/figure-html/unnamed-chunk-5-1.png" width="504" />
---
# DAG for Multilevel models
<img src="generative_files/figure-html/unnamed-chunk-6-1.png" width="504" />
Note: we have **arrows pointing in** AND **arrows pointing out**
---
# Probability notation
<p>
$$
`\begin{align}
[a_j,\bar{a}, \sigma_a, s, \sigma_{\text{height}}|y ] &\propto \\
&[y |g(L, a_j, s), \sigma_{\text{height}}] \times \\
&[a_j |\bar{a}, \sigma_a] \times \\
&[\bar{a}] \times \\
&[\sigma_a] \times \\
&[s] \times \\
&[\sigma_{\text{height}}]
\end{align}`
$$
</p>
</textarea>
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