Lecture notes. Written from the material of Berkeley Stat 210A, licensed CC BY 4.0. These are notes, not a transcript: the material has been reorganised and rewritten. This adaptation carries the same licence, and the original is linked above.

25. Induction and the Coin-Flip Example¶

What this covers¶

This chapter opens the course by asking what separates deductive reasoning, where true premises force a true conclusion, from inductive reasoning, where a finite set of observations is used to support a general claim that could turn out false. It assumes nothing beyond being able to read a probability statement such as a binomial mass function. It ends by using a live dispute about whether a flipped coin is fair to introduce, in miniature, every kind of question the course will spend the semester on: Bayesian versus frequentist frameworks, sufficiency, estimation, testing, and asymptotics.

Deduction: reasoning that cannot go wrong¶

Deduction is drawing inferences that follow logically from premises. If the premises are true and the argument is valid, the conclusion is guaranteed true — the only way a deductive argument can mislead you is if one of its premises was false to begin with. Two examples of the form:

All real, symmetric matrices have real eigenvalues; \(A\) is a symmetric matrix; therefore \(A\) has real eigenvalues.
No one in my daughter's preschool class has a peanut allergy; Zoe is in my daughter's preschool class; therefore Zoe is not allergic to peanuts.

Call this risk-free: valid argument plus true premises gives a true conclusion, full stop. Deduction can also produce probability statements, when a probability appears among the premises: this die has six faces \(1, 2, \dots, 6\), each equally likely, therefore the chance of rolling a \(4\) is \(1/6\). The probability here is deduced, not estimated — it follows from the assumptions about the die exactly as the eigenvalue claim follows from the assumptions about \(A\).

Induction: reasoning that can go wrong¶

Induction moves from observations to a general claim, and it is risky: it can be, and sometimes will be, wrong. Three examples:

I ate a free sample strawberry at the supermarket and it was ripe and delicious; therefore I should (probably) buy a whole carton. (This would be more convincing if the sample had been drawn at random from the carton rather than handed to me by the store.)
Water at \(1\,\text{atm}\) pressure has always been observed to boil at \(100^\circ\text{C}\); therefore all water at \(1\,\text{atm}\) (probably) boils at \(100^\circ\text{C}\).
I flipped this coin \(1000\) times and got \(502\) heads; therefore it (probably) has about a \(50\%\) chance of landing heads.

None of these arguments is deductively valid — in each case the conclusion goes beyond anything the premises logically force. Statistics is defined, for the purposes of this course, as the mathematical science of inductive reasoning: it is the discipline that tries to say precisely how risky an inductive claim is, and how to make it as safe as possible.

The problem of induction¶

Inductive reasoning is not logically valid, and this is not a fixable gap — it is a genuine philosophical problem. Hume's observation is that every inductive argument implicitly presumes that unobserved cases will resemble observed ones, and nothing forces that presumption to be true. So how could that presumption ever be justified?

Not by a logical proof — there is none.
Not by appeal to the fact that induction has worked well in the past — that appeal is itself an inductive argument, so using it to justify induction is circular.

Hume conceded that we cannot help reasoning inductively regardless, and called the habit of doing so "custom" or "habit" rather than something with a rational foundation.

Statistics does not solve Hume's problem — it evades it, and it does so in one of two ways, which are the two frameworks the rest of the course keeps returning to:

Bayesian reasoning. Whatever your prior beliefs happen to be, there is a definite, consistent way to update them in light of new evidence. This does not justify the prior itself; it only guarantees that the updating is done correctly once a prior is granted.
"Inductive behavior" (frequentist statistics). If the observations come from a reasonable experiment, methods can be designed whose conclusions are correct with high probability — and this can be proved. The guarantee here attaches to the method's long-run behavior, not to the truth of any single conclusion it produces.

Case study: is a flipped coin fair?¶

Diaconis, Holmes, and Montgomery (2007) built a physical model of a coin flip based on the coin's precession while spinning in the air, and used it to predict that a coin is a bit more likely to land on the same side it started on — about \(51\%\) for a typical human flipper.

Bartoš et al. (2023) tested this at scale: \(350{,}757\) coin flips, carried out by \(48\) flippers using coins from \(46\) countries, of which \(178{,}079\) landed on the same side they started on — about \(50.8\%\), consistent with the prediction.

Two analyses of the same data¶

Frequentist. Model each flip as independent with the same probability \(\theta\) of a same-side outcome, so the count \(X\) of same-side flips out of \(n\) is Binomial:

\[\mathbb{P}_\theta(X = x) = \binom{n}{x}\theta^x(1-\theta)^{n-x}.\]

This produces a \(95\%\) confidence interval \([50.6\%, 50.9\%]\) for \(\theta\). What that \(95\%\) means is a statement about the procedure, fixed before any data are seen: however the true \(\theta\) sits, the construction guarantees

\[\mathbb{P}_\theta\big(\mathrm{CI}(X) \text{ covers } \theta\big) \ge 95\% \quad (\text{here, almost exactly } 95\%).\]

Bayesian. Same binomial likelihood, but now add a \(\mathrm{Unif}[0,1]\) prior on \(\theta\). This produces a \(95\%\) credible interval — numerically the same \([50.6\%, 50.9\%]\) in this case. The natural question to raise about it: whose prior opinion was actually uniform on \([0,1]\)?

The model was wrong¶

Here is the twist: the pooled binomial model does not actually hold for this data set. Different flippers had different personal values of \(\theta\), and most flippers' values moved closer to \(50\%\) as they practiced over the course of the experiment. So both intervals above — the confidence interval and the credible interval — were computed under an assumption the data itself violates.

The questions this course is organized around¶

The coin-flipping example is used to introduce, in one setting, the whole vocabulary of the course.

Frameworks. What are the pros and cons of the Bayesian and frequentist approaches? Where does a prior actually come from, and does that choice matter to the final answer? What does "probability" even mean under each framework?

Sufficiency. Both analyses above threw away the order and identity of the \(350{,}757\) individual flips and summarized the whole data set as the single number \(X = 178{,}079\). Did that summary lose anything? (It did not, under the binomial model.) What is it about the structure of a model that licenses this kind of compression?

Estimation. "What's a good way to estimate..." turns out to have several nested versions:

the overall \(\theta\) in the pooled binomial model — there is no single estimator that is best for every possible \(\theta\); \(X/n\) is the "obvious" choice when \(n\) is as large as \(350{,}000\) (unless \(\theta\) itself happens to be tiny, e.g. \(10^{-6}\)), and a much less obvious question when \(n\) is small, e.g. \(n = 35\);
\(\theta_i\) for an individual flipper \(i\), under the finer model \(X_i \overset{\text{ind.}}{\sim} \mathrm{Binom}(n_i,\theta_i)\) for \(i = 1,\dots,m\) — how should data from the other flippers be used to help estimate flipper \(i\)'s own \(\theta_i\)?
how quickly \(\theta_{i,t}\), a single flipper's bias, is changing between flip \(1\) and flip \(n_i\) — a variety of models, parametric and nonparametric, could describe this drift.

Testing. How do we efficiently test hypotheses like:

\(H_0: \theta \le 50\%\) vs \(H_1: \theta > 50\%\) (one-sided) — a uniquely best test exists;
\(H_0: \theta = 50\%\) vs \(H_1: \theta \ne 50\%\) (two-sided) — a natural choice exists;
\(H_0: \theta_i = \theta\) for all flippers vs \(H_1\): the \(\theta_i\) vary — here the common value \(\theta\), under the null, is a nuisance parameter: it affects the null distribution of any test statistic, and there are many different ways the \(\theta_i\) could fail to be equal, and which of those you actually have in mind should affect which test you choose;
\(H_0\): each flipper's \(\theta_i\) is constant through their own sequence of flips vs \(H_1: \theta_{i,t}\) tends to be decreasing in \(t\) — this one can be tested nonparametrically.

Asymptotics. No one calculated \(350{,}757!\) to get an exact binomial probability. In practice, a count this large is handled through a normal approximation,

\[X \sim N\big(n\theta,\, n\theta(1-\theta)\big) \quad \text{or, per flipper,} \quad X_i \overset{\text{ind.}}{\sim} N\big(n_i\theta_i,\, n_i\theta_i(1-\theta_i)\big),\]

and the course cares about when such approximations are good, and how to build comparable approximations for the more complicated models that show up later.

Sources¶

This chapter is drawn entirely from the model-reconstructed handwritten slides for STAT210A's first lecture (no transcript or slide deck in the ordinary sense was supplied for this task):

docs/statistics/berkeley/stat210a/fall-2024/handwritten/lecture01-intro/01-course-introduction.md and 02-coin-flipping.md (source PDF handwritten/lecture01-intro.pdf, berkeley-stat210a fall-2024, CC BY 4.0).
The same two pages recur, with identical content, as fall-2025/units/handwritten/lecture01-intro/01-course-introduction.md / 02-coin-flipping.md, and, in a second wording, as fall-2025/handwritten/lecture01-intro/01-deductive-vs-inductive-reasoning.md / 02-coin-flipping.md and fall-2026/handwritten/lecture01-intro/01-deductive-vs-inductive-reasoning.md / 02-coin-flipping.md. All four pairs cover the same lecture; only the file names and provenance metadata differ.

Both source pages are flagged fidelity: reconstructed — a model read PDF pages with no text layer, so equations are unverified. The binomial mass function above is taken from the fall-2024 rendering, \(\theta^x(1-\theta)^{n-x}\); the fall-2025/fall-2026 rendering has \(x^\theta(1-\theta)^{n-x}\) at the same spot, which is evidently a transcription slip rather than a different formula.

The lecture's own outline lists "Syllabus" as its first item; no syllabus content appears in either rendering of the notes, so none is reproduced here.

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