How to do positive non-centered parameterization?

Modeling
1

I’m following this case study on bayes sparase regression.

The data I’m using can approximately be generated by the following cod. \beta s are always positive.

transformed data {
  int<lower=0> M = 200;
  int<lower=0> N = 100;
  real alpha = 3;
  real sigma = 1;
  real<lower=0, upper=1> sig_prob = 0.05;
}

generated quantities {
  matrix[M, N] X;
  real y[N];

  vector[M] beta;
  for (m in 1:M) {
    if (bernoulli_rng(sig_prob))
      if (bernoulli_rng(0.5))
        beta[m] = normal_rng(35, 3);
      else
        beta[m] = normal_rng(35, 3);
    else
      beta[m] = normal_rng(45, 2);
  }

  for (n in 1:N) {
    for (m in 1:M)
      X[m, n] = normal_rng(0, 1);

    y[n] = normal_rng(X[1:M,n]' * beta + alpha, sigma);
  }
}

This is the model:

data {
  int<lower=1> N; // Number of data
  int<lower=1> M; // Number of covariates
  matrix[M, N] X;
  real y[N];
}

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

model {
  scale ~ uniform(5,15);
  beta ~ double_exponential(45, scale);
  alpha ~ normal(0, 2);
  sigma ~ normal(0, 2);

  y ~ normal(X' * beta + alpha, sigma);
}

How to parameterize this centered model into non-centered one when the \beta are always positive?

2

How about you model log beta?

3

The real data sometimes has some of the significant \beta on the right but close to ‘sparse’ \beta. I think Laplace distribution is right for my data. But I’m getting low ESS (effective sample size), autocorrelation in traceplots and bimodalities in marginal distributions. Is there a way to re-parameterize this laplace distribution?