R Tutorial: The prior model

hi welcome to Bayesian modeling with our Jake's I'm Alisha Johnson I'm an associate professor of statistics at Macalester College and will be the instructor of this course I assume that you've worked through the previous course in Bayesian data analysis that's are familiar with the fundamental ideas behind Bayesian analysis and inference in this course you'll generalize these logical flexible and intuitive fundamentals to more advanced Bayesian model settings specifically you'll explore foundational Bayesian models such as the beta binomial normal normal and Bayesian regression models that are easily generalized to broader settings you will learn how to define compile and simulate these models using the RJ's package and are finally you will learn how to use RJ simulation output to conduct Bayesian posterior inference let's start with a review suppose you're running in an election for public office all their polls suggest that you have the support of 45 percent of the voters however due to polling errors and fluctuations in support this figure is uncertain engineered from past polling and election data the prior probability model shown here captures this uncertainty you'll most likely receive around 45 percent of the vote it's also unlikely though possible that you'll receive as little as thirty percent or as great as sixty percent of the vote to gain better insight your campaign conducts a small pool of ten voters among them six or sixty percent plan to vote for you the posterior model combines insights from the prior and these small polling data mainly in light of the pool the updated or posterior model of your election support is slightly more optimistic than the prior model you continue to collect data in a new pool 48 of 90 polled voters or 53% plan to vote for you in light of these new data the posterior optimism about your election chances inches up once again in a final pool 166 of 300 or 55% of voters support you compelled by the information in such a large sample your posterior optimism about receiving more than 50% of the votes hence winning the election is very high this election example highlights the power of Bayesian models not only does a Bayesian posterior model combine insights from the prior model and observe data it continues to evolve as new data come in and chapter 1 you'll explore the three fundamental pieces of Bayesian models the prior likelihood and posterior let's start with the prior engineering and communicating a prior model requires some notation let P denote the proportion of voters that support you thus P is a value between 0 & 1 in a Bayesian analysis we treat parameter P as a random variable thus the prior model of P is simply a probability distribution the beta distribution which also lives on 0 to 1 is a natural choice here the original prior model for P shown here corresponds to the beta distribution with shape parameters 45 and 55 we communicate this model using mathematical notation that specifies the name of the distribution beta and the parameter values upon which it depends 45 and 55 tuning the beta shape parameters produces alternative prior models of P just a few of which are shown here these range from models that reflect more pessimism about your election chances here the beta 1 5 in green 2 models that reflect a complete lack of certainty about your chances here the beta 1 1 in red and the following exercises you'll use simulation techniques to approximate explore and interpret the beta prior model