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Unformatted text preview: Outline Bayesian Inference November 22, 2011 Bayesian Inference Outline Outline 1 Brief Introduction 2 Two Examples 3 Judging Bayes vs Frequentist Bayesian Inference Brief Introduction Two Examples Judging Bayes vs Frequentist Comparison Priors Outline 1 Brief Introduction 2 Two Examples 3 Judging Bayes vs Frequentist Bayesian Inference Brief Introduction Two Examples Judging Bayes vs Frequentist Comparison Priors The hard part of frequentist inference is coping with the logic of the approach The truth is fixed The data have a distribution Our estimator, which we use to hopefully estimate the truth, has a distribution But at the end of one study we cant really say a whole lot about where the truth is In this regard, Bayesian inference is easy Bayesian Inference Brief Introduction Two Examples Judging Bayes vs Frequentist Comparison Priors Bayes is great because the inferences seem natural Theres a distribution on the true parameter estimate Maybe this makes sense in quantum dynamics In public health, this distribution represents our uncertainty in where the truth lies Bayesian inference lets us make statements about where we think the truth is after we conduct our study Bayesian Inference Brief Introduction Two Examples Judging Bayes vs Frequentist Comparison Priors Similarities Both frequentists and Bayesian use likelihoods Frequentists base their inference completely on the likelihood from one study Choose estimator that maximizes likelihood This has downsides: what if your study randomly got an aberrant result? The discussion section of their papers will typically compare and combine their estimates with previous estimates in an ad hoc, informal manner Bayesian inference also uses likelihoods . . . Bayesian Inference Brief Introduction Two Examples Judging Bayes vs Frequentist Comparison Priors Differences However, Bayesians combine the likelihood (the information gathered in the current study) with a prior distribution The prior distribution is a mathematical representation of our belief about the unknown parameters The prior distribution should represent the current scientific state of knowledge Can incoroporate past data and/or expert opinion Bayesian Inference Brief Introduction Two Examples Judging Bayes vs Frequentist Comparison Priors In Bayesian analysis the prior and likelihood are combined to produce a posterior distribution The posterior distribution represents our belief in the parameter size after weve completed our study Bayesian analysis is a coherent way to update your prior knowledge with based upon your current study Bayesian Inference Brief Introduction Two Examples Judging Bayes vs Frequentist Comparison Priors Priors Bayesian analysis lives and dies by the prior If your prior is good (Ill be more specific about this later), well see that Bayesian statistics can handily beat (and more specific about this) frequentist statistics If your prior is bad . . ....
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 Spring '11
 RichMacLehose

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