Three Inference Paradigms:�a Brief Introduction

Floyd Bullard, PhD

The NC School of Science and Mathematics

Teaching Contemporary Mathematics Conference�24 February 2024

Jumping Paper Frogs

Approach 1: Frequentist (or “Classical”)

What makes Frequentist inference “frequentist”?

• Early 20^th century statisticians, including Fisher, Neyman, and Pearson, believed that probability must be an objective thing, residing in the world outside and not in our heads.
• Probability is defined to be the long-run relative frequency with which a random event occurs over many trials.
• Frequentist methods (hypothesis testing, confidence intervals) dominate most published scientific research today.
• A 95% confidence interval is one result of a method that has a 95% “capture rate”.

Frequentism has its problems…

Approach 2: Bayes

Bayes (continued)

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Joint probabilities

Prior probabilities

Likelihood probabilities

who cares?

who cares?

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who cares?

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who cares?

who cares?

who cares?

Bayes (continued)

This is called a “posterior probability”. Here are all four of them:

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Bayes (continued)

Bayes Sandbox as a Google Sheet

Bayes (continued)

This graph in Desmos

Bayes (continued)

95%

(Is this a credible frog?)

Bayes background

Maybe or maybe not actually a picture of�Thomas Bayes.

Clandestine Bayes

• While the Bayes paradigm was scorned by 20^th century statisticians like Fisher and Neyman, it was used independently, and productively, by numerous problem-solvers of the time—though they often developed their methods independently.
• These include Alan Turing. His machine that cracked the Enigma code relied heavily on Bayes. Initial guesses (“the first word in this submarine officer’s report may be “cold”) were tried and refined using Bayes’s Theorem.
• For the interested, I recommend this history book:

The Theory That Would Not Die,�by Sharon Bertsch Mcgrayne

Bayes has its problems

• The Bayesian paradigm is criticized most often for employing—brazenly!—a subjective definition of probability.
• But even its advocates (including yours truly) admit that there are many situations when a true “prior belief” is impossible to elicit, and that imposing “uninformed” prior distributions is inconsistent.
• (For example, a probability model can often be parameterized in different ways, such as opting for a frequency parameter instead of a period parameter. But a uniform prior distribution on one is not equivalent to a uniform prior distribution on the other.)

Method 3: Likelihood

• Properly speaking, this method isn’t really an inference paradigm, but an evidence paradigm. To “infer” is to draw a conclusion, and the likelihood paradigm eschews conclusions and decision-making in favor of evidence evaluation.

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Likelihood (continued)

Likelihood (continued)

Likelihood (continued)

Likelihood (continued)

Likelihood (continued)

Likelihood: where is it going?

• Who knows?
• Perhaps strangely, given the clamor for objectivity, likelihood evaluation of evidence has few adherents.
• It is my view that the way we learn from data has been influenced by quirks of history more than by rational debate and study.
• It is also my view that likelihood evaluation of evidence is the best way to communicate to both researchers and laypeople alike what the data say, and to let them weigh the evidence against their own preconceptions.

Summary

Coda

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Thank you for coming!

bullard@ncssm.edu

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