viewof certain = Inputs.range([0, 100], {label: "Confidence the button works (%)", step: 0.1, value: 90})
dogma_score = function(score)
{
if (score == 100 || score == 0)
return "Dogmatic"
else
return "Not Dogmatic"
}
data = [
{outcome: "Button Functions Normally", percent: certain, dogma: dogma_score(certain)},
{outcome: "Button Malfunctions", percent: 100 - certain, dogma: dogma_score(100 - certain)}
]
Plot.barY(data, {x: "outcome", y: "percent", fill: "dogma"}).plot({
x: {label: null, domain: ["Button Functions Normally", "Button Malfunctions"]},
y: {label: "Confidence (%)", domain: [0, 100]},
color: {domain: ["Not Dogmatic", "Dogmatic"], range: ["#4e79a7", "#e15759"], legend: true}
})2 Dogmatism
2.1 Useful definitions
- Axiom: A claim accepted as true without proof
- Tautology: A statement that is always true
- Contradiction: A statement that is always false
2.2 Priors
In my studies as an undergraduate mathematics major, one of the required courses was Techniques of Proof. This class aims to teach students how to prove things absolutely. After learning formal logic, we were introduced to a system of axioms. For logic to work, we have to have some base assumptions. In our day-to-day lives, priors act as our axioms or base assumptions.
Imagine you’re in an elevator. In order to reach the 5th floor, you likely know, based on past experiences, that you need to press the button with the number 5 inscribed on it.
Consider the possible results from that button press:
| Button Functions Normally | Button Malfunctions | |
|---|---|---|
| result | ? | ? |
Assuming you have mostly only interacted with functional elevators, then you might rate your confidence in each outcome as:
| Button Functions Normally | Button Malfunctions | |
|---|---|---|
| result | \(90\%\) | \(10\%\) |
These numbers are not measurements of how often the button works; they are a rating of how sure you feel.
2.3 Posteriors
The posterior is your updated confidence after seeing some evidence. Keeping the elevator example, imagine you are on the 3rd floor and the light indicating someone is traveling upwards beeps on, and shortly after, a person steps out onto the 3rd floor. That acts as evidence that the elevator is currently functioning normally, and thus it would make sense to change your estimates.
The weight or impact of the evidence here is not important, but you are probably more certain now. This would create your posteriors, and they might look like this:
| Button Functions Normally | Button Malfunctions | |
|---|---|---|
| result | \(95\%\) | \(5\%\) |
Keep in mind these are not frequentist estimates. With these tables you are not saying that in a series of elevator rides you always have a \(95\%\) chance of the button working normally; instead, these are subjective ratings, like how you may tell your doctor that on a scale of one to ten your pain is at a two.
2.4 Dogma
Finally, let’s create the dogmatic prior probability tables:
| Button Functions Normally | Button Malfunctions | |
|---|---|---|
| result | \(0\%\) | \(100\%\) |
or
| Button Functions Normally | Button Malfunctions | |
|---|---|---|
| result | \(100\%\) | \(0\%\) |
In both cases, one of the choices is set to 100% certainty, which, outside of a tautology or contradiction, is considered malformed by most. The reason for this: what happens if the \(0\%\) case turns out to be true? That would contradict the claim carried by \(0\%\) confidence, which is that the outcome can never happen. This would be fairly clear if you shared the elevator with a person who insists that the button won’t work, even after the elevator successfully arrives.
Thus, the dogmatic posteriors are:
| Button Functions Normally | Button Malfunctions | |
|---|---|---|
| result | \(0\%\) | \(100\%\) |
or
| Button Functions Normally | Button Malfunctions | |
|---|---|---|
| result | \(100\%\) | \(0\%\) |
In dogmatism, the priors always equal the posteriors, regardless of evidence.