Why does the shifted_lognormal() family make stan code such as
target += lognormal_lpdf(Y - ndt | mu, sigma);
How does this subtraction by the shift parameter accomplish shifting the distirbution to the right horizontally?
Further, I am confused about the syntax of response ~ predictors here. Do the predictors here represent the inputs to the mean function or the inputs to the function for the first parameter?
For example, in the following code:
fit3 <- brm(reaction_time ~ 1 + bigram+ trial_num + (1 + bigram | participant_id) , data = experiment_df , family = shifted_lognormal(), prior = c(set_prior("normal(6,0.2)",class = "Intercept"), set_prior("normal(-0.6,0.2)", class = "b", coef = "trial_num"), set_prior("normal(0,1)", class = "b", coef = "bigram"), set_prior("gamma(1,1)", class = "sigma")), control = list(adapt_delta = 0.95), max_treedepth = 20)
Does this mean that the mean function of this shifted lognormal distribution is a function of bigram and trial_num with population and random effects, or instead that the parameter mu is a function of these predictors? And, for the shifted log normal mu is not the mean but exp(mu) is the median.