autotelic – Chapters 3

Sampling the imaginary

Example using vampirism test:
- \(P(+|V) = 0.95\), positive test in vampires
- \(P(+|M) = 0.01\), false positive in mortals
- \(P(V) 0.001\) probability of someone being a vampire
- \(P(V|+) = \frac{P(+|V)P(V)}{P(+)}\), probability of vampirism given a positive test
- \(P(+) = P(+|V)P(V) + P(+|M)(1 - P(V))\), total probability of positive test
- \(P(V|+) = 0.087\)
- Most positive tests are false positives, even when true positives are detected correctly in case of rare conditions.
Posterior distributions are probability distributions.
- The posterior defines the expected frequency that different parameters will appear, once we start resampling.
- An integral is just the total probability in some interval, in the Bayesian context.

---
title: "Chapters 3 - 4"
date: now
format:
  html:
    css: /learn/style.css
    code-fold: false
    code-link: true
    code-tools: true
    code-overflow: wrap
    code-line-numbers: false
citation: false
draft: true
haiku:
  - In misted forest,
  - truth unfolds, soft and unseen;
  - posterior calm.
lightbox: true
editor: 
  markdown: 
    wrap: 72
---

```{r setup, include = FALSE}
library(methods)
library(here)
# Use custom font
library(showtext)
font_add_google("Montserrat", "monse")
# Automatically use custom font in every plot
# package version > 0.9 to use with R base graphics
showtext_auto()
font_family <- "monse"


set.seed(2899)

knitr::opts_chunk$set(fig.align = "center", fig.retina = 3,
                      fig.width = 5.5, fig.height = (6 * 0.618),
                      out.width = "90%", collapse = TRUE, 
                      cache = TRUE, comment = "#>",
                      autodep = TRUE)


options(digits = 3,
        scipen = 999,
        width = 300)

options(
  rlang_trace_top_env = rlang::current_env(),
  rlang__backtrace_on_error = "none")
```

### Sampling the imaginary

* Example using vampirism test:
  - $P(+|V) = 0.95$, positive test in vampires
  - $P(+|M) = 0.01$, false positive in mortals
  - $P(V) 0.001$ probability of someone being a vampire
  - $P(V|+) = \frac{P(+|V)P(V)}{P(+)}$, probability of vampirism given a positive test
  - $P(+) = P(+|V)P(V) + P(+|M)(1 - P(V))$, total probability of positive test
  - $P(V|+) = 0.087$
  - Most positive tests are false positives, even when true positives are detected correctly in case of rare conditions.
  
* Posterior distributions are probability distributions. 
  - The posterior defines the expected frequency that different parameters will appear, once we start resampling.
  - An integral is just the total probability in some interval, in the Bayesian context.