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# slides1-9-28 - Stat C180/C236/Sanchez Introduction to...

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Stat C180/C236/Sanchez Introduction to Bayesian Statistics. Introduction and Examples Reading: Hoff, chapter 1 and Back Matter ( list of probability distributions) Fall 2010 Reading: Hoff, chapter 1 and Back Matter ( list of probability distributions) Stat C180/C236/Sanchez Introduction to Bayesian Statistics. Introductio

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Bayesian Learning (1) 1. Statistical induction is the process of learning about the general characteristics of a population from a subset of members of that population. 2. Numerical values of population characteristics are typically expressed in terms of a parameter θ 3. Numerical descriptions of the subset make up a dataset y Reading: Hoff, chapter 1 and Back Matter ( list of probability distributions) Stat C180/C236/Sanchez Introduction to Bayesian Statistics. Introductio
Bayesian Learning (2) I Before a dataset is obtained, the numerical values of both the population characteristics and the dataset are uncertain. I After a dataset y is obtained, the information it contains can be used to decrease our uncertainty about the population characteristics. I Quantifying this change in uncertainty is the purpose of Bayesian inference. Reading: Hoff, chapter 1 and Back Matter ( list of probability distributions) Stat C180/C236/Sanchez Introduction to Bayesian Statistics. Introductio

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Bayesian Learning (3) I The sample space Y is the set of all possible datasets, from which a single dataset y will result. I The parameter space Θ is the set of possible parameter values, from which we hope to identify the value that best represents the true population characteristics. I The idealized form of Bayesian learning begins with a numerical formulation of joint beliefs about y and θ , expressed in terms of probability distributions over Y and Θ. Reading: Hoff, chapter 1 and Back Matter ( list of probability distributions) Stat C180/C236/Sanchez Introduction to Bayesian Statistics. Introductio
Bayesian Learning (4) 1. For each numerical value θ Θ, our prior distribution p ( θ ) describes our belief that θ represents the true population characteristics. 2. For each θ Θ and y Y , our sampling model p ( y | θ ) describes our belief that y would be the outcome of our study if we knew θ to be true. Once we obtain the data y , the last step is to update our beliefs about θ : (3) For each numerical value of θ Θ, our posterior distribution p ( θ | y ) describes our belief that θ is the true value, having observed dataset y . Reading: Hoff, chapter 1 and Back Matter ( list of probability distributions) Stat C180/C236/Sanchez Introduction to Bayesian Statistics. Introductio

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Bayesian Learning (5) The posterior distribution is obtained from the prior distribution and sampling model via Bayes rule : p ( θ | y ) = p ( y | θ ) p ( θ ) R Θ p ( y | ˜ θ ) p ( ˜ θ ) d ˜ θ (1) It is important to note that Bayes rule does not tell us what our beliefs should be, it tells us how they should change after seeing new information.
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slides1-9-28 - Stat C180/C236/Sanchez Introduction to...

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