lesson5.Rmd

---
title: "Lesson 5"
output: html_document
---


## Brms internals

Core brms abstraction: one row of data -> one target increment

```{r, include=FALSE}
library(brms)
df <- data.frame(y = rnorm(100), x = rnorm(100), g = sample(LETTERS[1:3], size = 100, replace = TRUE))
```

```
make_stancode(y ~ x, df)
```

```{r, results = "asis", echo=FALSE}
cat("```stan\n")
cat(make_stancode(y ~ x, df))
cat("```")
```

```
make_stancode(y ~ x + (1 | g), df)
```

```{r, results = "asis", echo=FALSE}
cat("```stan\n")
cat(make_stancode(y ~ x + (1 | g), df))
cat("```")
```

```
make_stancode(y ~ x + (1 | g), df)
```

```{r, results = "asis", echo=FALSE}
cat("```stan\n")
cat(make_stancode(y ~ (1 + x | g), df))
cat("```")
```


Stanvars!

## Basics of numerical mathematics

- [IEEE 754 Floating point](https://en.wikipedia.org/wiki/IEEE_754)
  - The worst possible way to represent reals (except all the others we tried)
  - Sign, exponent, Mantissa
  - Special values: +/-Inf, NaN
  - Signed zero
    $x = y \centernot\implies \frac{1}{x} = \frac{1}{y}$
  - [Subnormal/Denormals](https://en.wikipedia.org/wiki/Subnormal_number)
    - The subnormal floats are a linearly spaced set of values, which span the gap between the negative and positive normal floats. 
- Small value + large value -> lost precision
- Much more precision around 0 than around 1
- Special functions
  - `log1p`, `log1p_exp`, `log_sum_exp`, `log_diff_exp`