# The Prior Can Often Only Be Understood in the Context of the Likelihood

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## Abstract

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## 1. The Role of the Prior Distribution in a Bayesian Analysis

#### 1.1. The Practical Consequences of A Prior Can Depend on the Data

#### 1.2. Existing Methods for Setting Priors Already Depend on the Likelihood

#### 1.3. The Role of the Prior in Generative and Predictive Modeling

#### 1.4. Coherence and Cheating

## 2. A Simple Motivating Example

#### 2.1. Bayesian Analysis under Different Priors

#### 2.2. Understanding the Problem

## 3. When Exactly Is the Prior Irrelevant in Practice?

#### 3.1. Uniform Priors Are Not A Panacea and Can Do Unbounded Damage

#### 3.2. Asymptotics: so Close, yet so Far Away

#### 3.3. For Complex Models, Certain Aspects of the Prior Will Always Be Relevant

## 4. A Prior Is More than Just A Probability Measure, So We Need to Start Thinking Generatively

#### 4.1. When Is A Probability Distribution A Prior?

#### 4.2. Prior Choice Is Especially Important in High Dimensions

#### 4.3. Sensitivity of the Marginal Likelihood to the Prior

## 5. Generative Priors Need to Be Prediction Focused

#### 5.1. In the Sea of Complex Models, the Leviathan Is Overfitting

#### 5.2. Overfitting Leads to Poor Posterior Predictive Performance

#### 5.3. Don’t Forget Your Roots: Predictive Priors Aren’t Always Generative

## 6. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Gelman, A.; Simpson, D.; Betancourt, M.
The Prior Can Often Only Be Understood in the Context of the Likelihood. *Entropy* **2017**, *19*, 555.
https://doi.org/10.3390/e19100555

**AMA Style**

Gelman A, Simpson D, Betancourt M.
The Prior Can Often Only Be Understood in the Context of the Likelihood. *Entropy*. 2017; 19(10):555.
https://doi.org/10.3390/e19100555

**Chicago/Turabian Style**

Gelman, Andrew, Daniel Simpson, and Michael Betancourt.
2017. "The Prior Can Often Only Be Understood in the Context of the Likelihood" *Entropy* 19, no. 10: 555.
https://doi.org/10.3390/e19100555