Mean-shape parameterization of beta distribution with minimum dispersion

Modeling
1

I have the following working model:

data {
    int<lower=1> N;
    int<lower=2> n_players;
    array[N] int<lower=1, upper=n_players> player_home;
    array[N] int<lower=1, upper=n_players> player_away;
    array[N] int<lower=0> x;
    array[N] int<lower=1> n;
}

parameters {
    vector[n_players - 1] lambda_free;
    vector<lower=0>[n_players] inv_phi;
}

transformed parameters {
    vector[n_players] lambda = append_row(0, lambda_free);

    vector<lower=0, upper=1>[N] mu = inv_logit(lambda[player_home] - lambda[player_away]);
    vector<lower=0>[N] phi         = 2 * inv(inv_phi[player_home] + inv_phi[player_away]);

    vector<lower=0>[N] alpha = mu * phi;
    vector<lower=0>[N] beta  = (1.0 - mu) * phi;
}

model {
    // priors
    lambda_free ~ normal(0, 2);
    inv_phi     ~ std_normal();

    // likelihood
    x ~ beta_binomial(n, alpha, beta);
}

With the help of a harmonic mean and introducing inv_phi I make sure that phi is player/opponent dependent.

Next, I want to make sure that it’s impossible that phi is smaller than 2 because this implies that probabilities near 0 and 1 are more likely than the mean.

In Statistical Rethinking the author uses the following “trick” to make sure that the dispersion is at least 2:

transformed parameters {
    ...
    vector<lower=2>[N] phi = kappa + 2;
}


model {
    kappa ~ exponential(1);
}

How would I implement this trick while keeping the harmonic mean-methodology intact?

Thank you

2

I think this post was superseded by this one: Incomplete beta as prior