Modeling a continuous [0, 1] variable

Modeling
1

Let’s say I have prob which is a number between 0 and 1 and some covariates X. I want to model this like

prob ~ inv_logit(N(alpha + beta X, …))

The below code (where I flip logit and normal) works but I find it unsatisfactory as the sigma could push the normal() bit outside the 0, 1 range.

Is there a better way of modeling this?

data {
  int N;
  matrix[N, 2] X;
  real<lower=0, upper=1> prob[N];
}

parameters {
  vector[2] beta;
  real alpha;
  real<lower=0> sigma;
}

model {
  beta ~ normal(0, 1);
  alpha ~ normal(0, 1);
  prob ~ normal(inv_logit(alpha + X * beta), sigma);
}
2

Hey there! You could use the Beta (proportion) distribution, which is a parameterization of the Beta distribution with location an scale (so it works well for beta regression).

You could also try to model \text{logit(prob)} \sim N(\alpha + X\beta, \sigma), although you would need to include the Jacobian correction for the logit transformation of prob and probably tight priors for \sigma. This is sort of like wrapping the inv_logit function around the whole left hand side/normal distribution, so that the outcome is guaranteed to fall in [0,1].

Cheers,
Max

2 Likes
3

Thanks Max, I have not seen stan examples where a function would be on the LHS, do you have an example?

In the beta prop would my Xbeta be the mu or kappa?

4

Ok I think I figured the beta prop out


data {
  int N;
  matrix[N, 2] X;
  real<lower=0, upper=1> prob[N];
}

parameters {
  vector[2] beta;
  real alpha;
  real<lower=0> kappa;
}

transformed parameters {
  vector<lower=0, upper=1>[N] mu;
  mu = inv_logit(alpha + X * beta);  
}

model {
  beta ~ normal(0, 1);
  alpha ~ normal(0, 1);
  kappa ~ gamma(1, 1);
  prob ~ beta_proportion(mu, kappa);
}
1 Like
5

Looks good! Does its work?

That should “just work”. Something like this…

logit(prob) ~ normal(alpha + X*beta, sigma);
target += .... // Jacobian correction

…but I’d go with the Beta regression anyways.

Cheers,
Max