An Introduction to Bayesian Inference

Michael BetancourtApril 8, 2009

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Introduction to Bayesian Inference

An Example

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Introduction to Bayesian Inference

The General Problem

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Introduction to Bayesian Inference

Estimators

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Introduction to Bayesian Inference

Estimators: An Example

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Introduction to Bayesian Inference

Estimators: The Danger

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Introduction to Bayesian Inference

Estimators: The Limitation

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Introduction to Bayesian Inference

Maximum Likelihood

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Introduction to Bayesian Inference

Maximum Log Likelihood

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Introduction to Bayesian Inference

Minimization

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Introduction to Bayesian Inference

Maximum Likelihood: An Example

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Introduction to Bayesian Inference

Maximum Likelihood Biases

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Introduction to Bayesian Inference

Maximum Likelihood Subtleties

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Introduction to Bayesian Inference

Bimodal Distributions

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Introduction to Bayesian Inference

Back to the Drawing Board

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Introduction to Bayesian Inference

Bayes Theorem

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Introduction to Bayesian Inference

A New Approach?

The are not random variables!

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Introduction to Bayesian Inference

Frequentist Statistics

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Introduction to Bayesian Inference

Uncertainty of Belief

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Introduction to Bayesian Inference

‣ Axiom One

‣ Axiom Two

The Cox Axioms

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Introduction to Bayesian Inference

A New Approach!

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Introduction to Bayesian Inference

Selecting Priors

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Introduction to Bayesian Inference

Selecting Priors

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Introduction to Bayesian Inference

An Example

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Introduction to Bayesian Inference

An Example

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Introduction to Bayesian Inference

An Example Revisted

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Introduction to Bayesian Inference

Marginalization

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Introduction to Bayesian Inference

Marginalization: An Example

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Introduction to Bayesian Inference

Marginalization: An Example

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Introduction to Bayesian Inference

Frequentist vs. Bayesian

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Introduction to Bayesian Inference

Bayesian Inference in Practice

Frequentist ➲ Bayesian

Optimization ➲ Expectation

(Differentiation) ➲ (Integration)

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Introduction to Bayesian Inference

The Opposition

‣“Priors are Subjective, Science Isn’t”

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Introduction to Bayesian Inference

Application: Model Selection

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Introduction to Bayesian Inference

Application: Model Selection

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Introduction to Bayesian Inference

Application: Model Selection

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Introduction to Bayesian Inference

The Bayesian Razor

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Introduction to Bayesian Inference

Application: Systematics

Energy Scale 5%

Background Subtraction 1%

Trigger Bias 2%

Luminosity 4%

... n%

Total 7%

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Introduction to Bayesian Inference

Application: Systematics

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Introduction to Bayesian Inference

Application: Nonparametric Estimation

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Introduction to Bayesian Inference

Further Reading

‣ Information Theory, Inference, and Learning Algorithms,

‣MacKay, http://www.inference.phy.cam.ac.uk/mackay/itila/

‣Data Analysis,

‣ Sivia with Skilling, Oxford 2006

‣ Lectures on Probability, Entropy, and Statistical Physics,

‣ Caticha, http://arxiv.org/abs/0808.0012v1