OK, here’s some code for a Normal inverse-Gamma regression analysis using power priors.

R code:

```
N0 <- 1000
true.betas <- c(-1, 1, .5, -.5)
P <- 4
X0 <- matrix(NA, nrow = N0, ncol = P)
for(i in 1:P) X0[, i] <- rnorm(N0)
sy <- 2
y0 <- rnorm(N0, mean = X0%*%true.betas, sd = sy)
summary(lm(y0 ~ -1 + X0))
library(rstan)
rstan_options(auto_write = TRUE)
compiled.model.prior <- stan_model("simple_linear_regression_NIG_prior.stan")
as <- .5
bs <- 2
vb <- 1.5
lm.data <- list(
N_0 = N0,
X_0 = X0,
y_0 = y0,
mu_beta = rep(0, P),
lambda_0 = solve(vb * diag(P)),
alpha0 = as,
beta0 = bs,
a_0 = .65
)
prior.lm <- sampling(compiled.model.prior, data = lm.data)
prior.lm
pairs(prior.lm, pars = "beta")
############
### Posterior
### will draw from the a similar process
N <- 100
X <- matrix(NA, nrow = N, ncol = P)
new.betas <- true.betas + rnorm(P, 0, sd = .2)
for(i in 1:P) X[, i] <- rnorm(N)
y <- rnorm(N, mean = X%*%new.betas, sd = sy)
compiled.model.posterior <- stan_model("simple_linear_regression_NIG_posterior.stan")
lm.data.forposterior <- list(
N_0 = lm.data$N0,
X_0 = lm.data$X0,
y_0 = lm.data$y0,
mu_beta = lm.data$mu_beta,
lambda_0 = lm.data$lambda_0,
alpha0 = lm.data$alpha0,
beta0 = lm.data$beta0,
X = X,
N = N,
y = y,
a_0 = lm.data$a_0
)
posterior.lm <- sampling(compiled.model.posterior, data = lm.data.forposterior)
posterior.lm
pairs(posterior.lm, pars = c("beta"))
```

whence `simple_linear_regression_NIG_prior.stan`

reads as

```
data{
int<lower=0> N0;
int<lower=0> P;
real y0[N0];
matrix[N0, P] X0;
vector[P] mu_beta;
matrix[P, P] lambda_0;
real<lower=0> alpha0;
real<lower=0> beta0;
real<lower=0> a_0;
}
parameters{
vector[P] beta;
real<lower=0> sigma_sq;
}
model{
/*power prior*/
target += a_0 * normal_lpdf(y0 | X0*beta, sqrt(sigma_sq) );
target += multi_normal_lpdf(beta| mu_beta, sigma_sq * lambda_0);
target += inv_gamma_lpdf(sigma_sq | alpha0, beta0);
}
```

and `simple_linear_regression_NIG_posterior.stan`

is

```
data{
int<lower=0> P;
int<lower=0> N0;
real y0[N0];
matrix[N0, P] X0;
vector[P] mu_beta;
matrix[P, P] lambda_0;
real<lower=0> alpha0;
real<lower=0> beta0;
int<lower=0> N;
real y[N];
matrix[N, P] X;
real<lower=0> a_0;
}
parameters{
vector[P] beta;
real<lower=0> sigma_sq;
}
model{
/*power prior*/
target += a_0 * normal_lpdf(y0 | X0*beta, sqrt(sigma_sq) );
target += multi_normal_lpdf(beta| mu_beta, sigma_sq * lambda_0);
target += inv_gamma_lpdf(sigma_sq | alpha0, beta0);
/* Likelihood */
target += normal_lpdf(y | X*beta, sqrt(sigma_sq) );
}
```