---
type: claim
domain: mechanisms
description: "A Gaussian prior produces mean regression, a power-law prior produces multiplicative extrapolation -- using the wrong prior on the right data degrades prediction systematically"
confidence: likely
source: "Jaynes (2003), Gelman et al. (Bayesian Data Analysis, 2013), Taleb (fat tails, 2007), Mandelbrot (1963)"
created: 2026-04-21
secondary_domains: [internet-finance, grand-strategy]
related_claims:
  - "the-efficient-market-hypothesis-fails-because-its-three-core-assumptions-rational-investors-independence-and-normal-distributions-all-fail-empirically"
  - "information-cascades-produce-rational-bubbles-where-every-individual-acts-reasonably-but-the-group-outcome-is-catastrophic"
---

# The shape of the prior distribution determines the prediction rule and getting the prior wrong produces worse predictions than having less data with the right prior

Bayesian inference combines prior beliefs with observed data to produce posterior predictions. The standard teaching emphasizes that with enough data, the prior washes out. This is true for well-behaved (thin-tailed) distributions. It is catastrophically false for fat-tailed distributions, which characterize most quantities that matter: wealth, city sizes, earthquake magnitudes, market crashes, pandemic severity, war casualties.

The mechanism: when a variable follows a Gaussian (thin-tailed) distribution, extreme observations are vanishingly unlikely, so the optimal prediction for any individual regresses toward the mean. A basketball player who scores 50 points will probably score closer to 25 next game. But when a variable follows a power-law (fat-tailed) distribution, extreme observations carry information that the generating process can produce extreme values. A city that has 10 million people is more likely to grow to 20 million than a city of 100,000 is -- because the process that produced a 10-million city is different from the process that produced a 100,000 city.

The prediction rule changes completely. Under a Gaussian prior, you predict regression to the mean. Under a power-law prior, you predict multiplicative extrapolation from the observed value. Using a Gaussian prior on power-law data produces predictions that are systematically wrong in the direction of underestimating extremes -- which is exactly where the stakes are highest.

This is why financial risk models failed in 2008: they used Gaussian priors on fat-tailed data. Why pandemic models underestimated COVID: they used thin-tailed priors for a process with super-spreader dynamics. Why VC returns are misunderstood: a single outlier return in a power-law portfolio contains more expected value than all other investments combined, but Gaussian-trained intuitions treat it as an anomaly to regress away from.

The implication for any knowledge system: before evaluating evidence, you must ask what generating process produced it. The prior is not a subjective belief to be minimized -- it is a structural claim about reality that determines the correct inference rule. Getting the prior wrong is worse than having less data, because more data with the wrong prior converges to a confident wrong answer.

## Evidence
- Mandelbrot (1963) -- cotton prices follow Levy stable distributions, not Gaussians
- Taleb (2007) -- "The Black Swan" documents systematic Gaussian-prior errors in finance, risk, and prediction
- Clauset, Shalizi, Newman (2009) -- rigorous statistical methods for distinguishing power-law from other heavy-tailed distributions in empirical data
- COVID-19 super-spreader events -- 80/20 rule (20% of infected produced 80% of transmission) follows power-law dispersion, not Gaussian
- VC returns (Horsley Bridge data) -- fund returns follow power law; top company in portfolio generates more return than rest combined

## Challenges
- Distinguishing power-law from log-normal empirically is extremely difficult (Clauset et al. 2009) -- many claimed power laws don't survive rigorous testing
- Even with the correct prior, uncertainty about the tail exponent produces wide posterior intervals -- knowing the shape is fat-tailed doesn't tell you exactly how fat
