As long as you are using the posterior distribution to generate predictions, then you can calculate whatever error metric you would like. For example, if you have a matrix named "predictions" that consist of a sample of predicted values for each observation (columns=observations, rows=predicted values from posterior), then the "Bayesian RMSE" calculated in R from simulated data would look something like this:

n <- 50
m <- 100 
y <- sample(n)

# simulate errors 
errors <- matrix(rnorm(n*m,0,1), nrow=m, ncol=n)

# 100 predictions (rows) for each y (cols)
predictions <- t(y + t(errors))

# rmse function
rmse <- function(y,yhat){sqrt(mean((yhat-y)^2))}

# calculate posterior of rmse values
rmse_dist <- apply(predictions,1,rmse,y=y)

# summarize distribution
summary(rmse_dist)

Where the result is a vector of RMSE values equal to the number of predicted values sampled from the posterior predictions.

You can calculate whatever error metric you would like as long as you are using the posterior distribution to generate predictions. For example, if you have a matrix named "predictions" that consist of a sample of predicted values for each observation (columns=observations, rows=predicted values from posterior), then the "Bayesian RMSE" calculated in R from simulated data would look something like this:

n <- 50
m <- 100 
y <- sample(n)

# simulate errors 
errors <- matrix(rnorm(n*m,0,1), nrow=m, ncol=n)

# 100 predictions (rows) for each y (cols)
predictions <- t(y + t(errors))

# rmse function
rmse <- function(y,yhat){sqrt(mean((yhat-y)^2))}

# calculate posterior of rmse values
rmse_dist <- apply(predictions,1,rmse,y=y)

# summarize distribution
summary(rmse_dist)

Where the result is a vector of RMSE values equal to the number of predicted values sampled from the posterior predictions.