The prior model Alicia Johnson Associate Professor, Macalester - - PowerPoint PPT Presentation

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the prior model

The prior model Alicia Johnson Associate Professor, Macalester - - PowerPoint PPT Presentation

Apr 15, 2023 189 likes •566 views

DataCamp Bayesian Modeling with RJAGS BAYESIAN MODELING WITH RJAGS The prior model Alicia Johnson Associate Professor, Macalester College DataCamp Bayesian Modeling with RJAGS Course Goals Explore foundational, generalizable Bayesian models

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DataCamp Bayesian Modeling with RJAGS

The prior model

BAYESIAN MODELING WITH RJAGS

Alicia Johnson

Associate Professor, Macalester College

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DataCamp Bayesian Modeling with RJAGS

Course Goals

Explore foundational, generalizable Bayesian models (eg: Beta-Binomial, Normal- Normal, and Bayesian regression) Define, compile, and simulate Bayesian models using RJAGS Conduct Bayesian posterior inference using RJAGS output

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DataCamp Bayesian Modeling with RJAGS

Bayesian elections: The prior

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DataCamp Bayesian Modeling with RJAGS

Bayesian elections: The prior

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DataCamp Bayesian Modeling with RJAGS

Bayesian elections: The prior

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DataCamp Bayesian Modeling with RJAGS

Bayesian elections: The data

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DataCamp Bayesian Modeling with RJAGS

Bayesian elections: The posterior

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Bayesian elections: New data

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DataCamp Bayesian Modeling with RJAGS

Bayesian elections: New posterior

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Bayesian elections: Newer data

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Bayesian elections: Newer posterior

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Bayesian thinking

A Bayesian posterior model... combines insights from the prior model & observed data evolves as new data come in

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Building a prior model

p = proportion that support you p is between 0 and 1 The prior model for p is a Beta distribution with shape parameters 45 and 55 p ∼ Beta(45,55)

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DataCamp Bayesian Modeling with RJAGS

Tuning the prior

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DataCamp Bayesian Modeling with RJAGS

Let's practice!

BAYESIAN MODELING WITH RJAGS

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DataCamp Bayesian Modeling with RJAGS

Data & the likelihood

BAYESIAN MODELING WITH RJAGS

Alicia Johnson

Associate Professor, Macalester College

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DataCamp Bayesian Modeling with RJAGS

Polling Data

parameter p = proportion that support you data X = 6 of n = 10 polled voters plan to vote for you insights You are more likely to have observed these data if p ≈ 0.6 than if p < 0.5.

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Modeling the dependence of X on p

Poll assumptions: voters are independent p = probability that a voter supports you X = number of n polled voters that support you (count of successes in n independent trials, each having probability of success p) Conditional distribution of X given p: X ∼ Bin(n,p)

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DataCamp Bayesian Modeling with RJAGS

Dependence of X on p

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DataCamp Bayesian Modeling with RJAGS

Dependence of X on p

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DataCamp Bayesian Modeling with RJAGS

Dependence of X on p

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Dependence of X on p

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DataCamp Bayesian Modeling with RJAGS

What's the likelihood?

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Likelihood

The likelihood function summarizes the likelihood of observing polling data X under different values of the underlying support parameter p. It is a function of p. high likelihood ⇒ p is compatible with the data low likelihood ⇒ p is not compatible with the data

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DataCamp Bayesian Modeling with RJAGS

Let's practice!

BAYESIAN MODELING WITH RJAGS

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DataCamp Bayesian Modeling with RJAGS

The posterior model

BAYESIAN MODELING WITH RJAGS

Alicia Johnson

Associate Professor, Macalester College

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DataCamp Bayesian Modeling with RJAGS

Bayesian election model

prior: p ∼ Beta(45,55)

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DataCamp Bayesian Modeling with RJAGS

Bayesian election model

prior: p ∼ Beta(45,55) likelihood: X ∼ Bin(10,p)

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DataCamp Bayesian Modeling with RJAGS

Bayesian election model

prior: p ∼ Beta(45,55) likelihood: X ∼ Bin(10,p)

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DataCamp Bayesian Modeling with RJAGS

Posterior model of p

prior: p ∼ Beta(45,55) likelihood: X ∼ Bin(10,p) Bayes' Rule: posterior ∝ prior * likelihood

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Getting Started with RJAGS

RJAGS combines the power of R with the JAGS (Just Another Gibbs Sampler) engine.

To get started: Download the JAGS program outside R Within R, install the rjags package

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Bayesian Models in RJAGS: DEFINE

X ∼ Bin(n,p) p ∼ Beta(a,b) Warning: the rjags function dbin() is different than base dbinom()

# DEFINE the model vote_model <- "model{ # Likelihood model for X X ~ dbin(p, n) # Prior model for p p ~ dbeta(a, b) }"

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Bayesian Models in RJAGS: COMPILE

# DEFINE the model vote_model <- "model{ # Likelihood model for X X ~ dbin(p, n) # Prior model for p p ~ dbeta(a, b) }" # COMPILE the model vote_jags_A <- jags.model(textConnection(vote_model), data = list(a = 45, b = 55, X = 6, n = 10), inits = list(.RNG.name = "base::Wichmann-Hill", .RNG.seed = 100))

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Bayesian Models in RJAGS: SIMULATE

# DEFINE the model vote_model <- "model{ # Likelihood model for X X ~ dbin(p, n) # Prior model for p p ~ dbeta(a, b) }" # COMPILE the model vote_jags <- jags.model(textConnection(vote_model), data = list(a = 45, b = 55, X = 6, n = 10), inits = list(.RNG.name = "base::Wichmann-Hill", .RNG.seed = 100)) # SIMULATE the posterior vote_sim <- coda.samples(model = vote_jags, variable.names = c("p"), n.iter = 10000)

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Bayesian Models in RJAGS: SIMULATE

# PLOT the simulated posterior plot(vote_sim, trace = FALSE)

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Let's practice!

BAYESIAN MODELING WITH RJAGS