08_Ch9_Solutions

9E5. Which value should Rhat approach, when a chain is sampling the posterior distribution correctly?

Rhat should approach 1. How close should it get? People disagree, but it is common to judge that any value less than 1.1 indicates convergence. But like all heuristic indicators, Rhat can be fooled.

9H2. Recall the divorce rate example from Chapter 5. Repeat that analysis, using ulam this time, fitting models m5.1, m5.2, and m5.3. Use compare to compare the models on the basis of WAIC or PSIS. To use WAIC or PSIS with ulam, you need add the argument log_log=TRUE. Explain the model comparison results.

library(rethinking)

## Loading required package: cmdstanr

## This is cmdstanr version 0.8.0

## - CmdStanR documentation and vignettes: mc-stan.org/cmdstanr

## - CmdStan path: /Users/brianbeckage/.cmdstan/cmdstan-2.36.0

## - CmdStan version: 2.36.0

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data(WaffleDivorce)
d <- WaffleDivorce

# standardize variables
d$D <- standardize( d$Divorce )
d$M <- standardize( d$Marriage )
d$A <- standardize( d$MedianAgeMarriage )
d_trim <- list(D=d$D,M=d$M,A=d$A)

Here are models m5.1, m5.2, and m5.3, fit using ulam:

m5.1_stan <- ulam(
    alist(
        D ~ dnorm(mu, sigma),
        mu <- a + bA * A,
        a ~ dnorm(0,0.2),
        bA ~ dnorm(0,0.5),
        sigma ~ dexp(1)
    ) , data = d_trim,chains=4,log_lik=TRUE )

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m5.2_stan <- ulam(
    alist(
        D ~ dnorm( mu , sigma ) ,
        mu <- a + bM * M ,
        a ~ dnorm( 0 , 0.2 ) ,
        bM ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = d_trim, chains=4,log_lik=TRUE )

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m5.3_stan <- ulam(
    alist(
        D ~ dnorm( mu , sigma ) ,
        mu <- a + bM*M + bA*A ,
        a ~ dnorm( 0 , 0.2 ) ,
        bM ~ dnorm( 0 , 0.5 ) ,
        bA ~ dnorm( 0 , 0.5 ) ,
        sigma ~ dexp( 1 )
    ) , data = d_trim, chains=4,log_lik=TRUE)

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precis(m5.1_stan)

precis(m5.2_stan)

precis(m5.3_stan)

compare(m5.1_stan, m5.2_stan, m5.3_stan, func=PSIS)

## Some Pareto k values are high (>0.5). Set pointwise=TRUE to inspect individual points.
## Some Pareto k values are high (>0.5). Set pointwise=TRUE to inspect individual points.

compare(m5.1_stan, m5.2_stan, m5.3_stan, func=WAIC)

The model with only age-at-marriage (m5.1) comes out on top, although the model with both predictors (m5.3) does nearly as well.