Applied Bayesian Statistics

Applied Baynesian Stats

Applied Time Series Analysis

Experimental Designs

Methods of Regression Analysis

SAS Programming

Statistical Methods

Theoretical Statistics

Theoretical Statistics Class

User Experience Design

Daniel Eisert Applied Bayesian Statistics April 27, 2020 Lecture 19: Bayesian Model Comparisons & Hypothesis Testing Class Business • Homework V is due on May 1 at 11:59 P.M. 1 Bayes Factor Bayes factor is based on the odds ratio . Recall that the odds ratio is π 1 - π . (1) Bayes factor in favor of a model or hypothesis is the ratio of the posterior odds to the prior odds. Thus, the Bayes factor of model 1 versus model 0 is: BF 1 , 0 = P ( M 1 | y ) P ( M 0 | y ) P ( M 1 ) P ( M 0 ) (2) In 1995, Kass and Raftery recommended these interpretations: ln B 10 B 1 , 0 Evidence against H 0 (or M 0 ) 0-0.5 1-3.2 Not worth more than bare mention 0.5-1 3.2-10 Substantial 1-2 10-100 Strong > 2 > 100 Decisive Bayes factor can also be used for one model and test diﬀerent values of the model parameter or compare two diﬀerent sets in which the parameter(s) could lie. 2 Hypothesis Testing Generically, if θ represents the parameter of interest in a model and Θ 0 , Θ 1 are two nonoverlapping regions that cover the entire possible range of values of θ , then H 0 : θ ∈ Θ 0 H A : θ ∈ Θ 1 , and the following posterior probabilities can be calculated by P ( θ ∈ Θ 0 | y ) P ( θ ∈ Θ 1 | y ) . • Example : Bayes factor in favor of the alternative hypothesis compared to the null hypothesis is BF 1 , 0 = P ( H 1 | y /P ( H 1 | y ) P ( H 1 ) /P ( H 0 ) (3) Recall that the posterior distribution in the wind example was N( μ =9 . 14 ,σ 2 =0 . 276). In R , we can test the hypotheses H 0 : μ< 6 . 94 H A : μ> 6 . 94 Daniel Eisert Page 1 Daniel Eisert Applied Bayesian Statistics April 27, 2020 1 pnorm (6.94, 9.14, sqrt (0.276)) #output: 1.409e-05 P ( μ< 6 . 94) = 1 . 409 × 10 - 5 P ( μ> 6 . 94) = 1 - 1 . 409 × 10 - 5 Now, try ﬁnding Bayes factor in favor of the alternative hypothesis compared to the null hypothesis. This time, be sure to use the prior distribution . 1 pnorm (6.94, 6.2, sqrt (3.31)) #output: 0.6566 P ( μ< 6 . 94) = 0 . 6566 P ( μ> 6 . 94) = 1 - 0 . 6566 Now use Bayes factor BF 1 , 0 = 1 - 1 . 409 × 10 - 5 1 . 409 × 10 - 5 1 - 0 . 6566 0 . 6566 = 135 , 638 . 7 (4) Based on the Kass and Raftery criterion, evidence against the null hypothesis is decisive. Note : OpenBUGS cannot produce the Bayes factor. 3 Deviance Information Criterion (DIC) The goal of the DIC is to compare canddiate models with respect to their ability to predict new data of the sam kind (this can be done in OpenBUGS). When comparing diﬀerent candidate mod- els for the same data, smaller values of DIC suggest better predictive ability. It is not possible to interpret whether a single DIC value for one model in isolation is considered good or bad. The DIC can be used for other copmarisons to including regression models with diﬀerent sets of regressors and generalized linear models with diﬀerent link functions. However, one restriction is that the response variable must have the same form in all the models being compared. Computation requires two quantities which can easily be obtained in OpenBUGS. They can be approximated using the MCMC sample output. DIC = 2 ˆ D avg ( y ) - ˆ D ˆ θ ( y ) (5) where • ˆ D avg ( y ) is D averaged over the posterior distributio of θ , and • ˆ D ˆ θ ( y ) is D evaluated at the posterior mean of θ . Daniel Eisert Page 2

STAT | 4444 | Applied Baynesian Stats | lecture notes | Daniel Eisert Applied Bayesian

STAT-3094_-_Lecture_13_-_Creating_Observations_from_Raw_Data.docx-Daniel T. Eisert STAT-3094 Lecture 13:

Daniel Eisert Applied Bayesian Statistics March 25, 2020 Lecture 13: Estimation & Inference of a Population Mean Class Business • Homework III due on April 5 at 11:59 P.M. 1 Bayesian Framework Let’s recall our example from Lecture 12. • EXAMPLE : What is the population mean of the average wind speed in New York City during the summer of 1973? 1.1 Data Distribution (Likelihood) Let y 1 ,y 2 , ..., y n be randomly sampled from a Normal distribution. First, identify the density function. p ( y | μ, σ 2 )= 1 √ 2 πσ 2 e - ( y - μ ) 2 2 σ 2 Now, ﬁnd the likelihood function. Remember that the likelihood function is a product of all of the values. p (ˆ y | μ, σ )= n Y i =1 1 (2 πσ 2 ) n 2 e - ∑ n i =1 ( y - μ ) 2 2 σ 2 Write the sum within the exponential into two components. Recall that this property states the following: ∑ ( y i - μ 2 )= ∑ ( y i - ¯ y ) 2 + n (¯ y - μ ) 2 (see property 1 in Lecture 12 to see this derived). p (ˆ y | μ, σ )= 1 (2 πσ 2 ) n 2 e - ∑ n i =1 ( y i - ¯ y ) 2 2 σ 2 e - n (¯ y - μ ) 2 2 σ 2 If σ 2 is assumed to be a known constant (which it is), then the likelihood function can be written as a likelihood of the only unknown parameter, μ . L ( μ | y ) ∝ e - n (¯ y - μ ) 2 2 σ 2 ∝ e - ( μ - ¯ y ) 2 2( σ 2 /n ) The above expression is a function of μ, ¯ y , and σ 2 are known constants. The other terms do not contain μ . They are just constants and can be dropped. This looks very similar to the kernel of a normal distribution (shown below). e - ( y - μ ) 2 2 σ 2 So, the mean μ =¯ y , and the variance is σ 2 = σ 2 /n . When a statistic contains all the information in the data that is useful in estimating the un- known parameter of interest, the statistic is called a suﬃcient statistic . These make Bayesian computation signiﬁcantly easier. Daniel Eisert Page 1 Daniel Eisert Applied Bayesian Statistics March 25, 2020 1.2 Finding a Conjugate Prior There are an inﬁnite number of ways to specify a prior for μ , but a conjugate prior simpliﬁes posterior calculations. Therefore, we will identify the parametric family that is conjugate to the likelihood for a normal mean and see whether there is a member of the conjugate family that adequately expresses our prior information. • The normal density is the conjugate prior for μ in the normal likelihood when σ 2 is assumed known. • If we express the likelihood in terms of the suﬃcient statistic ¯ y , then we can write the Bayesian Normal model as ¯ y follows N ( μ, σ 2 /n ) or μ follows (likelihood) N (¯ y,σ 2 /n ) μ follows (prior) N ( μ 0 ,σ 2 0 ) 1.3 Update the Prior (Posterior) • Likelihood N (¯ y,σ 2 /n ) • Prior N ( μ 0 ,σ 2 0 ) Deﬁnition : Precision is the inverse of the variance. The more spread out a distribution is, the less precise it is. When reading a Bayesian study, make sure whether the normal distributions are pa- rameterized in terms of the variance or precision. Most software uses precision rather than variance for ease of calculation. For our example of a normal likelihood and prior, we will use precisions to let τ 2 = 1 σ 2 τ 2 0 = 1 σ 2 0 So our new data likelihood for μ | ¯ y,τ 2 will follow N  μ, 1 nτ 2  and the prior distribution of the population mean μ will follow N  μ 0 , 1 τ 2 0  Daniel Eisert Page 2

STAT | 4214 | Methods of Regression Analysis | study guides | Daniel Eisert Applied Bayesian

1 Daniel Eisert STAT-4444 Lecture I: Introduction to Bayesian Statistics STAT-4444: Applied Bayesian Statistics January 27, 2020 (Week 2) Basics of Bayesian Statistics Frequentist approach only uses a likelihood function (some distribution). Bayesian approach uses: - Likelihood function - Prior function The likelihood and prior functions together give you a better estimation. The Bayesian approach does not use a p-value (but still uses probability). The Bayesian approach will also give you a probability for the confidence interval (e.g., 95% probability that the interval is between a and b ) unlike the frequentist approach. The main difference between the two frameworks come from the way that the concept of probability itself is interpreted. Prior Function× Likelihood Function = Posterior Distribution Before the 1990s, applying Bayesian methods was very challenging due to the complicated mathematical computations (usually required by a computer). Scientific Method 1. Define a question/problem to solve. 2. Assess information already available ( prior information). o If it is sufficient, make conclusions, decisions, and take appropriate action. o If it is not sufficient, collect more information using an observational or experimental study. Use the data obtained to update what was previously known ( posterior information). Bayesian is not about collecting more data, but rather using previous and current data to update a decision. Example: Estimating Heights A Bayesian would represent the uncertainty in the mean probabilistically. They would define a probability distribution over the possible values of the mean and use sample data to update the distribution. The newly collected data would make the probability distribution over the parameter narrower . The Bayesian now has a full posterior distribution over the possible parameter values allowing them to take into account the uncertainty in the estimate by integrating the full posterior distribution instead of basing the prediction just on the most likely value (the latter something a frequentist would do). Alpha-Level The chance of a false positive (Type I Error). The smaller the alpha, the higher the chance of a Type II Error (false negative). Increasing the sample size allows both errors to decrease. Quantifying Uncertainty Subjective Probability – a probability of an even or of the truth of a statement is a number between 0 and 1 that quantifies a particular person’s subjective opinion as to how likely that event is to occur. Remember that the probability of any event X is 0 ≤X≤ 1 . Bayes’ Rule P ( Model ∨ Data ) ∝ P ( Model ) ×P ( Data ∨ Model ) Posterior ∝ Prior × Data

STAT | 4214 | Methods of Regression Analysis | lecture notes | 1 Daniel Eisert STAT-4444 Lect

Daniel Eisert Theoretical Statistics I April 2, 2020 Test II Review 3 Discrete Random Variables E [ Y ]= X all y yp ( y ) E [ g ( y )] = X all y g ( y ) p ( y ) E [ Y 2 ]= X all y y 2 p ( y ) V ar [ Y ]= E [ Y 2 ] - ( E [ Y ]) 2 3.4 Binomial Distribution p ( y )=  n y  p y q n - y μ = E [ Y ]= np σ 2 = V ar [ Y ]= npq 3.5 Geometric Distribution p ( y )= q y - 1 p μ = E [ Y ]= 1 p σ 2 = V ar [ Y ]= 1 - p p 3.7 Hypergeometric Distribution

All courses in STAT

Popular Documents