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.

20. Statistics as Inductive Reasoning¶

What this covers¶

Before any statistical method can be judged, a prior question has to be settled: what kind of reasoning is statistics, and why is it inherently risky in a way that a proof is not? This chapter lays out the scope of Stat 210A, distinguishes deductive from inductive reasoning, states Hume's problem of induction, and shows the two ways statistics evades that problem — Bayesian updating and frequentist "inductive behavior" — worked out on one real data set: a large experiment on whether flipped coins are really fair. It assumes only a first course in probability: the binomial distribution, expectation, and the idea of a parameter.

What the theory of statistics is about¶

Statistics is the study of methods that use data to understand the world. Statistical methods are used throughout the natural and social sciences, in machine learning and artificial intelligence, and in engineering — and despite this ubiquity, practitioners are perpetually accused of not actually understanding what they are doing. Statistical theory is, broadly, the attempt to understand what we are doing when we use statistical methods.

Most (not all) statistical methods rest on statistical modeling: treating the data as a realization of some random data-generating process with attributes, usually called parameters, that are a priori unknown. The analyst's job is to use the data to draw accurate inferences about these parameters, or to make accurate predictions about future data. If the modeling has been done well — a very big "if" — the unknown parameters correspond to whatever real-world question motivated the analysis in the first place. Applied courses such as Stat 215A/B ask whether the modeling exercise actually captures something real about the world; this course instead asks how the analyst can use the data most effectively within the mathematical setup once it is fixed. The topics are the structure of statistical models, how to evaluate and design statistical methods, the Bayesian vs. frequentist philosophies, and estimation, confidence intervals, and hypothesis testing — in parametric and nonparametric settings, in finite samples and in asymptotic regimes.

Stat 210A sits among several related Berkeley courses. It focuses on classical statistical contexts — inference in finite samples, or in fixed-dimensional asymptotic regimes. Stat 210B (for which this course is a prerequisite) is more technical and covers empirical process theory and high-dimensional statistics. Statistical Learning Theory (CS 281A / Stat 241A) overlaps somewhat but tilts toward machine learning: more time on predictive modeling, optimization, and signal processing, less on inferential questions such as testing, confidence intervals, and causal inference. Both courses cover estimation and exponential families.

Deductive reasoning¶

Most mathematics courses are entirely concerned with deductive reasoning: drawing conclusions that follow logically from premises. For example:

All real, symmetric matrices have real eigenvalues.
\(A\) is a real, symmetric matrix.
Therefore, \(A\) has real eigenvalues.

Deductive reasoning shows up in everyday life too:

No one in my daughter's preschool class has a nut allergy.
Zoe is in my daughter's preschool class.
Therefore, Zoe is not allergic to peanuts.

This kind of argument is risk-free: as long as the premises are true, the conclusion must hold. The premises could of course be false — I might be confusing my neighbor Zoe with a different Zoe in the class — but that is the only way the conclusion could fail.

Deductive arguments can also involve statements about probability:

This die has six faces labeled 1 through 6.
If I roll it, it is equally likely to land on any face.
Therefore, the chance of rolling a 4 is exactly \(1/6\).

A probability course such as Stat 205A is, in this sense, about deductive arguments like this one: given a fully specified probabilistic premise, it derives further probabilistic facts with certainty.

Inductive reasoning¶

Statistics, by contrast, is the mathematical science of inductive reasoning: reasoning from observations to general claims about the world. Unlike deduction, such arguments are inherently risky — the conclusion can be false even when every premise is true.

(A terminological trap: inductive proofs in mathematics are not inductive reasoning in this sense. An inductive proof is really an example of deductive reasoning, because the inductive step is a logically valid argument that extends the conclusion to the whole class of objects under study. Nothing about it is risky.)

A first example:

I ate a blueberry from the free sample tray at the supermarket.
It was ripe and delicious.
Therefore, if I buy a carton of blueberries, they will probably be ripe and delicious.

The weasel word "probably" is not a rigorous quantitative claim here — it just flags some uncertainty about the conclusion. The argument is more persuasive if the sample is drawn from the carton actually being bought:

I ate five blueberries at random from the carton I intended to buy.
They were all ripe and delicious.
Therefore, if I buy the carton, the rest of the blueberries will probably be ripe and delicious.

We could still be wrong — maybe there were only five good blueberries in the whole carton, and those are exactly the ones we picked — but that is not very likely. Scientists reason inductively all the time:

Water at 1 atm of pressure has been observed to boil at 100°C every time it has been measured in the laboratory.
Therefore, water at 1 atm of pressure probably always boils at 100°C.

Inductive reasoning is the basis of all the empirical sciences. It also applies directly to probability itself:

I flipped this penny 1000 times and got 502 heads.
Therefore, it probably has about a 50% chance of landing heads.

For now we can take for granted what it means for a penny to have a 50% chance of landing heads — something like, its physical properties give it an equal chance of heads or tails (and a negligible chance of landing on its side or flying off into space). There is real controversy among philosophers and statisticians about what probability means in general, though not much of it attaches to ordinary coin flips; this is a thread the course picks up later.

Hume's problem of induction¶

Inductive reasoning is not valid in the sense logicians or mathematicians mean by the word. It doesn't matter how many real symmetric matrices you've seen with real eigenvalues — without a proof, you cannot make the general claim. Mathematics has entertaining examples of patterns that hold for a long stretch and then break, such as the Borwein integral:

\[ \begin{aligned} \int_0^\infty \frac{\sin x}{x}\,dx &= \frac{\pi}{2} \\[4pt] \int_0^\infty \frac{\sin x}{x}\cdot\frac{\sin(x/3)}{x/3}\,dx &= \frac{\pi}{2} \\[4pt] \int_0^\infty \frac{\sin x}{x}\cdot\frac{\sin(x/3)}{x/3}\cdot\frac{\sin(x/5)}{x/5}\,dx &= \frac{\pi}{2} \\[4pt] &\ \ \vdots \\[4pt] \int_0^\infty \frac{\sin x}{x}\cdot\frac{\sin(x/3)}{x/3}\cdots\frac{\sin(x/13)}{x/13}\,dx &= \frac{\pi}{2} \\[4pt] \int_0^\infty \frac{\sin x}{x}\cdot\frac{\sin(x/3)}{x/3}\cdots\frac{\sin(x/15)}{x/15}\,dx &= \frac{\pi}{2} - 2.31 \times 10^{-11} \end{aligned} \]

Each integral in this family equals \(\pi/2\) exactly — until the pattern silently fails at the last one. Bertrand Russell made the same point about induction in ordinary life:

Domestic animals expect food when they see the person who usually feeds them. We know that all these rather crude expectations of uniformity are liable to be misleading. The man who has fed the chicken every day throughout its life at last wrings its neck instead, showing that more refined views as to the uniformity of nature would have been useful to the chicken.

David Hume's A Treatise of Human Nature (1739) first posed the problem of induction: that inductive reasoning presumes — seemingly without justification — that yet-to-be-observed cases will resemble observed ones. This presumption is the uniformity principle, and it is hard to see how to justify it. It cannot be justified by a direct logical argument, since it is not logically valid. It might seem justifiable by past experience — physical properties of the world do generally seem uniform across space and time, at a low enough level — but that argument is circular: the future having resembled the past in the past does not entail that it will resemble the past in the future.

Hume conceded that people have to reason inductively all the time, but he called this a "custom" or "habit" and challenged philosophers to justify it. Almost 300 years later there is still no fully satisfactory answer. Most philosophers of science — Karl Popper among them — accept that inductive reasoning is fallible but hold that there are reasonable ways for scientists to deal with that.

Two evasions of the problem of induction¶

Building a mathematical science of inductive reasoning looks like trouble from the start: the first thing known about induction is that it is not mathematically valid. Statisticians have two main ways of evading this, and they generate the two main frameworks for statistical inference.

Evasion 1 — Bayesian reasoning. Whatever our a priori beliefs about the world are, we at least know how to update them in light of experience, using the mathematics of conditional probability. The prior beliefs themselves may never be justified, but there is (more or less) only one rational way to update them. With enough experience, we may hope these beliefs "wash out," so that observers who started with different priors eventually converge.

Evasion 2 — inductive behavior (frequentist statistics). Instead of justifying the reasoning itself, design methods whose fallibility can be quantified. As long as certain assumptions hold about how the data were collected, it may be possible to prove that a method gives correct conclusions with high probability — without ever certifying that any particular conclusion is correct.

Worked example: are coins really fair?¶

Physical randomizers like coins and dice seem like the firmest possible ground for a theory of probability. Recent work says otherwise: most human flippers have a somewhat greater than 50% chance of seeing a coin land on the same side it started on, apparently due to the physics of a rotating object. This was first hypothesized in a theoretical paper by Diaconis, Holmes, and Montgomery (2007), and confirmed in a large experiment by Bartos et al. (2023), in which 48 human flippers collectively flipped \(n = 350{,}757\) coins, of which \(178{,}079\) (\(50.77\%\)) landed on the same side they started.

Bartos et al. actually collected far more than this one number: each of the 48 flippers recorded the full sequence of their flips, the type of coin (46 countries' worth of coins were used), and even video of themselves flipping. The analysis below, though, uses only the summary statistic \(X = 178{,}079\), the number of flips landing same-side-up.

Frequentist analysis in the binomial model¶

The data are easy to analyze under two seemingly innocuous assumptions: the flips are statistically independent, and every flip has the same probability \(\theta\) of landing on the side it started. Under these assumptions it follows deductively that the probability of exactly \(x\) same-side landings out of \(n\) flips is

\[ \binom{n}{x}\theta^x(1-\theta)^{n-x}, \qquad x = 0, 1, \dots, n. \]

This is the one-parameter binomial model, written \(X \sim \mathrm{Binom}(n, \theta)\). If Diaconis–Holmes–Montgomery's prediction is right, \(\theta > 0.5\); if the starting side makes no difference, \(\theta = 0.5\).

The frequentist estimator of \(\theta\) is \(\hat\theta = X/n\), here \(0.5077\). One thing this course will prove is that \(X/n\) is the best estimator of \(\theta\) among all unbiased estimators — those satisfying \(\mathbb{E}_\theta\hat\theta = \theta\) for every possible value of \(\theta \in [0,1]\). This is an instance of inductive behavior: using \(X/n\) guarantees getting the answer right on average, as precisely as any unbiased estimator can. It is also possible to say how variable the estimator is: its standard error is \(\sqrt{\theta(1-\theta)/n}\). Substituting \(\hat\theta\) for \(\theta\) gives a typical error of about \(0.00084\), so there is good reason to believe \(\hat\theta = 0.5077\) is about that close to the truth — but not a guarantee for this particular experiment; we could have gotten unlucky.

Can we conclude, inductively, that a same-side bias really exists (\(\theta > 0.5\))? Even if \(\theta = 0.5\) exactly, \(X/n\) would be expected to deviate somewhat from \(0.5\) just by chance. Frequentist analysis evades this question and substitutes another for it — testing the null hypothesis \(\theta = 0.5\) against the alternative \(\theta > 0.5\), and reporting a confidence interval for \(\theta\):

binom.test(x = 178079, n = 350757, p = 0.5, alternative = "greater")

    Exact binomial test

data:  178079 and 350757
number of successes = 178079, number of trials = 350757, p-value <
2.2e-16
alternative hypothesis: true probability of success is greater than 0.5
95 percent confidence interval:
 0.5063091 1.0000000
sample estimates:
probability of success
             0.5076991

The \(p\)-value is the likelihood of observing at least as many same-side flips as were seen, if \(\theta\) really were \(0.5\). It is so small here (\(p < 2.2 \times 10^{-16}\)) that R does not bother computing its exact value. The probability behind this calculation follows deductively from the binomial assumptions, but most reasonable people would accept it as sufficient evidence to reject the null hypothesis — to reach the inductive conclusion that \(\theta \neq 0.5\). This conclusion is risky: even a much more conservative threshold, say rejecting only when \(p < 10^{-10}\), still leaves some chance of error in any given experiment. What has been gained is that this chance can be quantified and controlled, rather than eliminated.

Beyond detecting that a bias exists, binom.test also returns a \(95\%\) confidence interval, \([50.6\%, 50.9\%]\), for \(\theta\). (Confidence intervals are developed properly later in the course; for now, the interval is constructed so that it covers the true \(\theta\) with probability at least \(95\%\), whatever value \(\theta\) actually takes.) This is again inductive behavior: producing the interval risks the conclusion "\(\theta\) lies between \(0.506\) and \(0.509\)" being wrong, but the procedure is built so that risk can be quantified and limited in advance.

Bayesian analysis¶

A more Bayesian route introduces an explicit distribution for \(\theta\) itself — say, \(\theta \sim \mathrm{Unif}[0,1]\). This is a stronger assumption than anything made above: there is only one draw from the distribution of \(\theta\), observed only indirectly through the coin flips, so it is hard to test. It turns out not to matter very much which prior is chosen here — many different priors would lead to almost the same posterior.

Given this prior, the posterior distribution of \(\theta\) after observing \(X = x\) can be computed directly:

\[ \theta \mid X = x \;\sim\; \frac{(n+1)!}{x!\,(n-x)!}\,\theta^x(1-\theta)^{n-x}, \qquad \theta \in [0,1]. \]

This is the Beta distribution with parameters \(\alpha = x+1\), \(\beta = n-x+1\) — a density for the parameter \(\theta\), not a distribution for the data \(X\). Once the posterior is known, direct probability statements about \(\theta\) become available: for instance, after seeing the data, there is only a \(3.8 \times 10^{-20}\) chance that \(\theta \le 0.5\). A \(95\%\) Bayesian credible interval — an interval containing \(95\%\) of the posterior mass — comes out to \([50.6\%, 50.9\%]\), coinciding with the frequentist confidence interval to several decimal places here. But the two intervals mean different things: the credible interval claims, in this experiment, a \(95\%\) chance that \(\theta\) actually falls in that range, whereas the confidence interval only claims that the procedure covers the true value at least \(95\%\) of the time across repetitions.

Questioning the binomial model¶

Bartos et al. themselves noted that the binomial model is not quite right — some flippers showed more same-side bias than others. The model can be extended to give each flipper \(i\) their own probability \(\theta_i\) over their \(n_i\) flips (with \(\sum_i n_i = n\)); retaining independence gives \(X_i \sim \mathrm{Binom}(n_i, \theta_i)\) independently for \(i = 1,\dots,48\).

There was also evidence that individual flippers improved over time. Accommodating that means expanding further, to \(X_{i,t} \sim \mathrm{Bernoulli}(\theta_{i,t})\) independently for \(i = 1,\dots,48\) and \(t = 1,\dots,n_i\), perhaps with the monotonicity constraint \(\theta_{i,1} \ge \theta_{i,2} \ge \cdots \ge \theta_{i,n_i}\) for each \(i\). With \(n = 350{,}757\) parameters — one per data point — this is effectively a nonparametric model.

Questions the course returns to¶

This one example already previews several recurring questions.

Bayesian vs. frequentist frameworks. What are the pros and cons of each? Where does a prior come from, and how much does the choice of prior matter to the analysis?
Sufficiency. The first (binomial) analysis summarized the whole data set by the single number \(X\), even though far more was recorded (the full sequence of flips for every flipper, in particular). It turns out that, under the binomial model, nothing is lost by reducing the data to \(X\) alone. What is it about the structure of the binomial model that makes this reduction lossless?
Estimation. What is a good way to estimate the parameters of each of these models? In the model with a separate \(\theta_i\) per flipper, how should — or shouldn't — the estimate for one flipper be informed by data from the other 47? In the time-varying model, what functional form for \(\theta_{i,t}\) would be reasonable, and how would it be estimated?
Testing. In the basic binomial model, testing \(H_0: \theta \le 0.5\) against \(H_1: \theta > 0.5\) turns out to have a unique best solution: reject when \(X\) is large. Testing whether the flippers really need different \(\theta_i\)'s — \(H_0: \theta_1 = \cdots = \theta_{48}\) against "not all equal" — is a harder problem, for two reasons: the null hypothesis has a nuisance parameter that can affect the null distribution of any test statistic, and the \(48\)-dimensional alternative can move away from the \(1\)-dimensional null in many different directions, some of which a given test will be much better at detecting than others.
Asymptotics. Nothing in the actual computation ever evaluates \(350{,}757!\), even though such quantities appear in the formulas. In practice the binomial model is replaced by an appropriate Normal approximation, \(X \sim \mathcal{N}(n\theta, n\theta(1-\theta))\) — an immediate consequence of the central limit theorem here, but a technique that extends to many settings where the approximation is far less obvious in advance.

Sources¶

This chapter merges the introduction to the Stat 210A course reader, which is offered under CC BY 4.0 and appears with essentially identical prose across three converted terms of the course. The base text follows the fall-2024 and fall-2026 conversions (identical to each other, converted losslessly from the underlying .qmd source):

berkeley-stat210a/fall-2024/reader/introduction/01-about-stat-210a.md
berkeley-stat210a/fall-2024/reader/introduction/02-deductive-vs-inductive-reasoning.md
berkeley-stat210a/fall-2024/reader/introduction/03-the-problem-of-induction.md
berkeley-stat210a/fall-2024/reader/introduction/04-statistical-evasions-in-practice-coin-flipping.md (and their fall-2026 duplicates).

The R console output shown under "Frequentist analysis in the binomial model" is reproduced from the fall-2025 conversion of the same reader page (berkeley-stat210a/fall-2025/reader/introduction/04-..., converted from rendered .html rather than raw .qmd), which captured the executed code-chunk output that the plain .qmd source does not itself contain. The fall-2025/units/reader/... copies supplied alongside these are duplicates of the fall-2025 reader files and were not used separately. No slide deck or lecture transcript was supplied for this chapter — the course reader is the primary source, and the display of the Borwein integral has been re-set from a source formatting artifact ([10pt] spacing markers rendered as literal text) into a standard aligned display; the mathematical content is unchanged.

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