Simulare il Normal-Normal in RJAGS

Modeling bayesiano con RJAGS

Alicia Johnson

Associate Professor, Macalester College

Studio sul sonno

$Y_i$ = variazione del tempo di reazione (ms) dopo 3 giorni di deprivazione di sonno

$Y_i \sim N(m, s^2)$

Cosa dicono i priori

Cosa dicono i dati (e la verosimiglianza)

mean(sleep_study$diff_3)
sd(sleep_study$diff_3)

26.34021
37.20764

Assumendo che questi dati provengano da $Y_i \sim N(m, s^2)$, sono più probabili se...

$m \approx 26$ ms
$s \approx 37$ ms

Cosa dice la posteriore

DEFINIRE il Normal-Normal

sleep_model <- "model{
    # Likelihood model for Y[i]




    # Prior models for m and s


}"

DEFINIRE il Normal-Normal

sleep_model <- "model{
    # Likelihood model for Y[i]




    # Prior models for m and s


}"

$Y_i \sim N(m, s^2)$ per $i$ in ${1,2,\ldots,18}$

DEFINIRE il Normal-Normal

sleep_model <- "model{
    # Likelihood model for Y[i]
    for(i in 1:length(Y)) {

    }

    # Prior models for m and s


}"

$Y_i \sim N(m, s^2)$ per $i$ in ${1,2,\ldots,18}$

DEFINIRE il Normal-Normal

sleep_model <- "model{
    # Likelihood model for Y[i]
    for(i in 1:length(Y)) {
        Y[i] ~ dnorm(m, s^(-2))
    }

    # Prior models for m and s


}"

$Y_i \sim N(m, s^2)$ per $i$ in ${1,2,\ldots,18}$
- NOTA: precisione = varianza$^{-1}$ = $s^{-2}$

DEFINIRE il Normal-Normal

sleep_model <- "model{
    # Likelihood model for Y[i]
    for(i in 1:length(Y)) {
        Y[i] ~ dnorm(m, s^(-2))
    }

    # Prior models for m and s
    m ~ dnorm(50, 25^(-2))

}"

$Y_i \sim N(m, s^2)$ per $i$ in ${1,2,\ldots,18}$
- NOTA: precisione = varianza$^{-1}$ = $s^{-2}$
$m \sim N(50, 25^2)$

DEFINIRE il Normal-Normal

sleep_model <- "model{
    # Likelihood model for Y[i]
    for(i in 1:length(Y)) {
        Y[i] ~ dnorm(m, s^(-2))
    }

    # Prior models for m and s
    m ~ dnorm(50, 25^(-2))
    s ~ dunif(0, 200)
}"

$Y_i \sim N(m, s^2)$ per $i$ in ${1,2,\ldots,18}$
- NOTA: precisione = varianza$^{-1}$ = $s^{-2}$
$m \sim N(50, 25^2)$
$s \sim \text{Unif}(0, 200)$

COMPILARE il Normal-Normal

# COMPILA il modello    
sleep_jags <- jags.model(textConnection(sleep_model), 
    data = list(Y = sleep_study$diff_3), 
    inits = list(.RNG.name = "base::Wichmann-Hill",
                 .RNG.seed = 1989))
    sleep_study$diff_3

 [1]  71.8798 -18.0269  33.7877 -36.4096  32.5074  74.9082
 [7]  15.9673 -10.8008  29.1938  33.7556  18.8188  -0.7697
[13]  30.0626 125.1784   5.7331  15.2090  11.9091  41.2199

SIMULARE il Normal-Normal

# COMPILA il modello    
sleep_jags <- jags.model(textConnection(sleep_model), 
    data = list(Y = sleep_study$diff_3),
    inits = list(.RNG.name = "base::Wichmann-Hill", 
                 .RNG.seed = 1989))

# SIMULA la posteriore
sleep_sim <- coda.samples(model = sleep_jags, 
    variable.names = c("m", "s"), 
    n.iter = 10000)

SIMULARE il Normal-Normal

Esercitiamoci!

Modeling bayesiano con RJAGS