Bayes Theorem

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# Bayes Theorem
## Applied To Data Analysis
### Andy Grogan-Kaylor
### University of Michigan
### 2022-03-10

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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*.

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# 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:

`$P(D|H) = P(D,H) / P(H)$`

`$P(H|D) = P(D,H) / P(D)$`

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# Then:

`$P(D|H)P(H) = P(D,H)$`

`$P(H|D)P(D) = P(D,H)$`

# Then:

`$P(D|H)P(H) = P(H|D)P(D)$`

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# Bayes Theorem:

`$P(H|D) = \frac{P(D|H) P(H)}{P(D)}$`

--
# In Words:

`$\text{posterior} \propto \text{likelihood} \times \text{prior}$`

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# 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.

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# Visualization

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# 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/)

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# Discussion

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* 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 `$H_0$`.