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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>Bayes Theorem</title>
<meta charset="utf-8" />
<meta name="author" content="Andy Grogan-Kaylor" />
<meta name="date" content="2022-03-10" />
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class: center, middle, inverse, title-slide
# Bayes Theorem
## Applied To Data Analysis
### Andy Grogan-Kaylor
### University of Michigan
### 2022-03-10
---
<style type="text/css">
@import url('https://fonts.googleapis.com/css2?family=Montserrat&display=swap');
.title-slide {
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background-color: #00274C;
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# How To Navigate This Presentation
* Use the <span style="font-size:100px">&#8678;</span> and <span style="font-size:100px">&#8680;</span> keys to move through the presentation.
* Press *o* for *panel overview*.
---
class: animated, fadeIn
# Derivation
--
| | D=1 | D=0 |
|:--------------|:---------------------|:--------------------------------|
| H=1 | `\(P(D, H)\)` | `\(P(\text{not} D, H)\)` |
| H=0 | `\(P(D, \text{not} H)\)` | `\(P(\text{not} D, \text{not} H)\)` |
--
From the definition of conditional probability:
--
`\(P(D|H) = P(D,H) / P(H)\)`
--
`\(P(H|D) = P(D,H) / P(D)\)`
---
class: animated, fadeIn
# Then:
--
`\(P(D|H)P(H) = P(D,H)\)`
--
`\(P(H|D)P(D) = P(D,H)\)`
--
# Then:
`\(P(D|H)P(H) = P(H|D)P(D)\)`
---
class: animated, fadeIn
# Bayes Theorem:
`\(P(H|D) = \frac{P(D|H) P(H)}{P(D)}\)`
--
# In Words:
`\(\text{posterior} \propto \text{likelihood} \times \text{prior}\)`
---
class: animated, fadeIn
# Example
Consider an example using 1,000 hypothetical studies. We imagine that only 10% of interventions are likely to have results. We adopt standard assumptions of adopting an `\(\alpha\)`, or chance of detecting an effect when one is not there of 5%. We similarly assume 80% power `\(\beta\)`, or a 20% chance of failing to detect an effect when it is not present.
--
| Data (D) | D=1 (effect) | D=0 (no effect) |
|:-------------------------|:--------------------|:-------------------|
| Hypothesis (H) | 100 effects | 900 non-effects |
| H=1 (conclude effect) | 80 true positives | 45 false positives |
| H=0 (conclude no effect) | 20 false negatives | 855 true negatives |
--
> With thanks to the [Wikipedia article](https://en.wikipedia.org/wiki/Bayes%27_theorem#Cancer_rate) on this topic for inspiration for this example.
---
class: animated, fadeIn, center, top
# Visualization
<img src="Bayes-theorem-applied-to-data-analysis_files/figure-html/unnamed-chunk-3-1.png" width="80%" />
---
class: animated, fadeIn
# Calculations
--
`$$P(\text{H=1} | \text{D=1}) = \frac{P(\text{D=1} | \text{H=1}) P(\text{H=1})}{P(\text{D})}$$`
--
`$$=\frac{P(\text{D=1} | \text{H=1}) P(\text{H=1})}{P(\text{D=1} | \text{H=1}) P(\text{H=1}) + P(\text{D=0} | \text{H=1}) P(\text{H=1})}$$`
--
`$$P(\text{H=1} | \text{D=1}) = \frac{.8 \times .1}{.08 + .045} = .64$$`
--
> See also [Thinking Through Bayesian Ideas](https://agrogan.shinyapps.io/Thinking-Through-Bayes/)
---
class: animated, fadeIn
# Discussion
--
* Calculations suggest that a true effect is likely in 64% of the cases where one concludes the presence of an effect.
--
* Consequently, calculations suggest that 36% of the time, when one concludes there is an effect, there is actually no effect.
--
* Put another way, despite setting `\(\alpha = .05\)`, 36% of cases result in a false positive.
--
* Notice how the Bayesian approach lets us estimate the probability of different hypotheses given the data. In some cases, this may afford us the opportunity to accept our null hypothesis `\(H_0\)`.
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