---
tag: 75P3EJPH81
kind: counterexample
name: Lindley's paradox
msc: [62F15, 62F03]
uses: [5ZEGZVZBTJ, 0EHYZFZ4PK, 07RXBVTVZB]
sources:
  - url: https://en.wikipedia.org/wiki/Lindley%27s_paradox
same_as:
  wikidata: Q6552538
  wikipedia: Lindley's paradox
---

## Counterexample

Let $X_1, \dots, X_n$ be i.i.d. [[normal|07RXBVTVZB]] $\mathcal{N}(\theta, 1)$, so that $\bar{X}_n \sim \mathcal{N}(\theta, 1/n)$.
Test $H_0 \colon \theta = 0$ against $H_1 \colon \theta \ne 0$.
Suppose the data give $\sqrt{n}\,\bar{x}_n = z$ with $z = 1.96$, for every $n$.
The two-sided [[p-value|0EHYZFZ4PK]] is then $P(\abs{Z} \ge 1.96) \approx 0.05$ for $Z \sim \mathcal{N}(0, 1)$, whatever $n$ is.
So a test of level $0.05$ is at the edge of rejecting $H_0$ for every $n$.
For the Bayesian comparison, give $\theta$ the prior $\mathcal{N}(0, \tau^2)$ under $H_1$, with $\tau > 0$ fixed.
Under $H_0$ the density of $\bar{X}_n$ is that of $\mathcal{N}(0, 1/n)$, and under $H_1$ it is that of $\mathcal{N}(0, \tau^2 + 1/n)$.
A direct computation gives the [[Bayes factor|NSBYE143KS]] of $H_0$ against $H_1$:
$$B_{01} = \sqrt{1 + n\tau^2} \, \exp\left(-\frac{z^2}{2} \cdot \frac{n\tau^2}{1 + n\tau^2}\right).$$
As $n \to \infty$ with $z$ fixed, $B_{01} \to \infty$.
So for any prior probability $P(H_0) \in (0, 1)$, the posterior probability of $H_0$ tends to $1$, while the p-value stays at $0.05$.

## Remarks

The p-value and the posterior probability answer different questions, and for large samples they can point in opposite directions.
For large $n$, a fixed $z$ means that $\bar{x}_n$ is very close to $0$.
Such a value is much better explained by $\theta = 0$ than by a $\theta$ spread over a range of width about $\tau$.
The conflict also depends on the prior: as $\tau \to \infty$ with $n$ fixed, $B_{01} \to \infty$ as well, so a vague prior under $H_1$ favours $H_0$.

## Explanations

### L1 human

With a huge sample, a small but "significant" deviation from zero can appear.
A classical test then says "reject the null hypothesis at the $5\%$ level".
A Bayesian comparison of "the effect is exactly zero" with "the effect is some reasonable size" can say the opposite: the data favour exactly zero.
The paradox shows that significance and evidence are not the same thing.