How to do positive non-centered parameterization?

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);
        beta[m] = normal_rng(35, 3);
      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?

How about you model log beta?

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?