Scale transformation with error in linear regression


#1

Hi all,

suppose I have a dependent variable y_i that I model through a linear regression and one fixed covariate x_i. Now suppose that I have a scale transformation for the y_i in a way that \tilde{y}_i = y_{i} \beta AND \beta has normal distribution with mean \mu_\beta and standard deviation \sigma_\beta. I am wondering if this model is actually identifiable w.r.t. to \beta and the regression coefficient \alpha (see below)?

data {
    int<lower=1> N;
    vector[N] xs;
    vector[N] ys;
    real mean_alpha;
    real mean_beta;
    real<lower=0> sd_alpha;
    real<lower=0> sd_beta;
}
parameters {
    real alpha;
    real beta;
    real<lower=0> sigma;
}
model {
    vector[N] ys_resc;
    sigma ~ cauchy(0,3);
    alpha ~ normal(mean_alpha, sd_alpha);
    beta  ~ normal(mean_beta, sd_beta);
    ys_resc = ys*beta;
    ys_resc ~ normal(alpha*xs,sigma);
}

#2

Yes, but it may not work particularly well.


#3

Thanks. Could you elaborate a bit more?


#4

Also wouldn’t that be equivalent to having the identity link for the y_i equal to x_i \alpha \beta? How would this product then be identifiable?


#5

Are there multiple values of \alpha and \beta such that the joint PDF of parameters and data is the same for all possible values of the data? No, but that is different than saying there is a stepsize that is small enough to prevent NUTS from diverging.