Power prior in R2jags

Question

In a fixed power prior model, the model is set up as:

Suppose that the event follows a Bernoulli distribution with probability p_i, what I want is to raise the Bernoulli likelihood to the power of w in jags:

mod1<-function(){
  for(i in 1 : N) {
    y[i] ~ dbern(p[i])^w #<------------------- 
    logit(p[i]) <- a[group[i]] + t[group[i]]*d[i]
  }
  for (j in 1:3) {
    a[j] ~ dnorm(mu.a, pow(sd.a,-2))
    t[j] ~ dnorm(mu.t, pow(sd.t,-2))
  }
  mu.a~dnorm(0,0.01)
  mu.t~dnorm(0,0.01)
  sd.a ~ dunif(0,10)
  sd.t ~ dunif(0,10)
}

However JAGS does not recognize such code.

As an alternative, I wrote the following STAN code:

mod1 <- '
data {
  int<lower = 1> N;
  real w;
  int y[N];
  vector[N] d;
  int group[N];
  }

parameters {
  vector[3] a;
  vector<lower=0>[3] t;
  real mua;
  real mut;
  real sigmaa;
  real sigmat;
}

model {
  real p[N];
  //prior
  mua~normal(0,10);
  mut~normal(0,10);
  sigmaa ~ uniform(0,10);
  sigmat ~ uniform(0,10);
  for (j in 1:3) {
  a[j] ~ normal(mua, sigmaa); 
  t[j] ~ normal(mut, sigmat);
  }
  
  //likelihood

  for (i in 1:N) {
    p[i]=inv_logit(a[group[i]] + t[group[i]]*d[i]);
    target += bernoulli_lpmf(y[i]|(p[i]))*w;
  }
}
'

which works, but I'd like to use JAGS (R2jags) for this project and was wondering if there is a JAGS equivalent for the above STAN code. THanks!

Here is the sample data if you want to test the code:

y <- rbinom(160, 1, 0.3)
d <- runif(160, 0, 41)
group <- sample(1:3, 160, replace = TRUE)
w <- rep(0.5, 160)
N=160
mydata <- data.frame(y, d, group, x4,N)

Answer 1

I would have thought that using the ones or zeros trick would do the trick. Here's an example with the zeros trick:

Jags Model:

mod1j<-function(){
  C <- 0
  for(i in 1 : N) {
    zeros[i] ~ dpois(phi[i])
    phi[i] <- -LL[i] + C
    LL[i] <- (log(p[i])*y[i] + log(1-p[i])*(1-y[i]))*w[i]
    logit(p[i]) <- a[group[i]] + t[group[i]]*d[i]
  }
  ll <- sum(LL[])
  for (j in 1:3) {
    a[j] ~ dnorm(mu.a, pow(sd.a,-2))
    t[j] ~ dnorm(mu.t, pow(sd.t,-2))
  }
  mu.a~dnorm(0,0.01)
  mu.t~dnorm(0,0.01)
  sd.a ~ dunif(0,10)
  sd.t ~ dunif(0,10)
}

Stan Model

mod1 <- '
data {
  int<lower = 1> N;
  real w[N];
  int y[N];
  real d[N];
  int group[N];
  }

parameters {
  vector[3] a;
  vector[3] t;
  real mua;
  real mut;
  real sigmaa;
  real sigmat;
}

model {
  real p[N];
  //prior
  mua~normal(0,10);
  mut~normal(0,10);
  sigmaa ~ uniform(0,10);
  sigmat ~ uniform(0,10);
  for (j in 1:3) {
  a[j] ~ normal(mua, sigmaa); 
  t[j] ~ normal(mut, sigmat);
  }
  
  //likelihood

  for (i in 1:N) {
    p[i]=inv_logit(a[group[i]] + t[group[i]]*d[i]);
    target += bernoulli_lpmf(y[i]|(p[i]))*w[i];
  }
}
'

Generate Data

y <- rbinom(160, 1, 0.3)
d <- runif(160, 0, 41)
group <- sample(1:3, 160, replace = TRUE)
w <- runif(160, .5, 2)
N=160
mydata <- list(y=y, d=c(scale(d)), group=group, w=w, N=N, zeros=rep(0, N))

Run Models

library(rstan)
smod <- stan(model_code= mod1, data=mydata, chains=2, iter = 25000, warmup=10000)
#> Warning: There were 8607 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: There were 11 transitions after warmup that exceeded the maximum treedepth. Increase max_treedepth above 10. See
#> https://mc-stan.org/misc/warnings.html#maximum-treedepth-exceeded
#> Warning: Examine the pairs() plot to diagnose sampling problems


library(R2jags)
jmod <- jags(mydata, inits=NULL, model.file = mod1j, parameters.to.save = c("a", "t", "ll", "mu.a", "mu.t", "sd.a", "sd.t"), n.chains=2, n.iter=50000, n.burnin=10000, n.thin = 4)

Model Summaries

summary(smod)$summary
#>                mean     se_mean        sd          2.5%         97.5%
#> a[1]     -0.9485418 0.004391921 0.2616977   -1.45203977   -0.42740087
#> a[2]     -0.9229228 0.004768607 0.2498671   -1.39857013   -0.40886121
#> a[3]     -1.3801613 0.006318897 0.2987109   -2.02100962   -0.85652760
#> t[1]     -0.3540932 0.004473947 0.2395895   -0.82275651    0.12607274
#> t[2]     -0.3760353 0.004234657 0.2339069   -0.84264376    0.08154950
#> t[3]     -0.4288344 0.004414164 0.2519173   -0.96762765    0.03517404
#> mua      -1.0691230 0.017084370 0.9030125   -2.82320530    0.82897044
#> mut      -0.3732147 0.014114388 0.7314059   -1.58495700    1.11586195
#> sigmaa    1.0581223 0.036266564 1.3500882    0.04529007    5.25790545
#> sigmat    0.7294629 0.042495890 1.2622050    0.01482299    4.49297131
#> lp__   -115.2883229 0.201683196 5.3385894 -125.86118296 -104.86900546

jmod
#>           mu.vect sd.vect     2.5%    97.5% 
#> a[1]       -0.952   0.260   -1.454   -0.434 
#> a[2]       -0.925   0.250   -1.404   -0.428 
#> a[3]       -1.371   0.301   -1.999   -0.849 
#> t[1]       -0.348   0.237   -0.817    0.119 
#> t[2]       -0.367   0.230   -0.829    0.085 
#> t[3]       -0.423   0.251   -0.966    0.043 
#> mu.a       -1.076   1.010   -3.108    0.868 
#> mu.t       -0.383   0.780   -1.744    1.001 
#> sd.a        1.045   1.405    0.032    5.523 
#> sd.t        0.707   1.211    0.011    4.630 
#> ll       -118.541   1.504 -122.250 -116.472 
#> deviance  237.082   3.008  232.945  244.501 

^{Created on 2023-03-28 with reprex v2.0.2}

The models look pretty comparable. Note, that I took the <lower=0> constraint away from t just to make the JAGS and Stan models look otherwise equivalent. The results look pretty comparable. I also generated unequal weights.