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Normal-Distribution-slides-update3

# Normal-Distribution-slides-update3 - The Normal Model Stat...

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The Normal Model Stat C180/C236 Intro Bayesian Statistics Juana Sanchez [email protected] UCLA Department of Statistics Stat C180/C236 Intro Bayesian Statistics/J. Sanchez The Normal Model

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The Normal Model (1) The Normal density function. Review. Properties of the Normal Density. Joint Distribution of n iid Normal random variables (2) Likelihood Interpretation of joint distribution (3) Case 1. Normal model, mean unknown, variance known. Conjugate prior distribution for θ . n=1. Interpretation of the posterior mean of θ . Likelihood, prior and posterior for θ , several examples. R code for ﬁgure. (4) Case 2. Normal model n > 1, θ unknown, σ 2 known. Example-midge wing length. Interpretation of the posterior distribution parameters. (5) Case 3. Normal model, variance unknown, mean known (6) Multiparameter models. Averaging over nuisance parameters. Two parameter case example. Interpretation of the marginal posterior distribution for θ . Posterior inference by simulation of the joint posterior distribution. (7) Case 4 Two parameter Normal model with non informative prior. Obtaining marginal posterior distribution for θ . Example: experiment on magnetic ﬁelds. (8) Case 5. Joint Inference for the mean and variance with informative priors Stat C180/C236 Intro Bayesian Statistics/J. Sanchez The Normal Model
The Normal density function. Review A random variable Y is said to be normally distributed with mean θ and variance σ 2 > 0 if the density of Y is given by p ( y | θ, σ 2 ) = 1 2 πσ 2 e - 1 2 ( y - θ σ ) 2 - ∞ < y < 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 y p(y | θ , σ 2 ) θ = 2, σ 2 = 0.25 θ = 5, σ 2 = 4 θ = 7, σ 2 = 1 Figure: Some Normal Densities for a few values of θ and σ Stat C180/C236 Intro Bayesian Statistics/J. Sanchez The Normal Model

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Properties of the Normal Density 1 The distribution is symmetric about θ , and the mode, median and mean are all equal to θ 2 95% of the population lies within two standard deviations of the mean (68-95-99.7 rule). 3 If X normal ( μ, τ 2 ), Y normal ( θ, σ 2 ) and X and Y are independent, then aX + bY normal ( a μ + b θ, a 2 τ 2 + b 2 σ 2 ); 4 The dnorm, rnorm, pnorm and qnorm commands in R take the standard deviation σ as their argument, not the variance σ 2 . Be careful about this. Stat C180/C236 Intro Bayesian Statistics/J. Sanchez The Normal Model
Joint Distribution of n iid Normal Random Variables Suppose we have a random sample of i.i.d. y 1 , .... , y n , where y i N ( θ, σ 2 ) Then the joint distribution is p ( y 1 , .... , y n | θ, σ 2 ) = n Y i =1 p ( y i | θ, σ 2 ) = n Y i =1 1 2 πσ 2 e - 1 2 ( y - θ σ ) 2 = (2 2 ) - n 2 exp ( - 1 2 X ± y - θ σ ² 2 ) where ( - 1 2 X ± y - θ σ ² 2 = 1 σ 2 X y 2 i - 2 θ σ 2 X y i + n θ 2 σ 2 ) Stat C180/C236 Intro Bayesian Statistics/J. Sanchez The Normal Model

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Likelihood interpretation of joint Distribution and Nice properties 1 As a function of the parameters θ and σ 2 , the joint distribution is the likelihood of the data for diﬀerent values of the parameters. Recall in Mathematical Statistics we maximized the likelihood with respect to θ and σ 2 to obtain the maximum Likelihood estimators of θ and σ 2 .
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Normal-Distribution-slides-update3 - The Normal Model Stat...

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