Dixons-Coles with over/unders

General
1

Hi,

I’m playing around with the dixons-coles football model[1] and trying to get parameters given not only the score, but the probability of over/under 2.5 goals before the game began. My hope is then to replace the actual score of the game with the probability of the home/away/draw outcomes. I’m struggling to understand why the over/under 2.5 goals isn’t really having any affect on the outputs. Code presented below:

functions {
  real tau(int x, int y, real rho, real mu1, real mu2) {
    real adj;

    if (x == 0 && y == 0)
      adj = 1 - (mu1 * mu2 * rho);
    else if (x == 0 && y == 1)
      adj = 1 + (mu1 * rho);
    else if (x == 1 && y == 0)
      adj = 1 + (mu2 * rho);
    else if (x == 1 && y == 1)
      adj = 1 - rho;
    else
      adj = 1;

    return adj;
  }

  real dixon_coles_log(int[] goals, real rho, real mu1, real mu2) {
    int home;
    int away;
    real prob;

    home = goals[1];
    away = goals[2];

    prob = poisson_lpmf(home | mu1) + poisson_lpmf(away | mu2) + log(tau(home, away, rho, mu1, mu2));
    return prob;
  }
  
  real under_three_log(real under_prob_target,real mu1, real mu2, real rho){
  real under_prob = 0;
  int goals[2];
    
    for(hg in 0:2){
      for(ag in 0:2){
        if( (hg+ag) < 4 ){
          goals[1] = hg;
          goals[2] = ag;
          under_prob = under_prob + exp(  dixon_coles_log(goals,rho, mu1, mu2));
        }
      }
    }
  return (under_prob);
  }
  
}

data {
  //int<lower=0> N;
  //vector[N] y;
  real under_prob_log;
  int home_goals;
  int away_goals;
}

// The parameters accepted by the model. Our model
// accepts two parameters 'mu' and 'sigma'.
parameters {
  real <lower=0> mu1;
  real <lower=0> mu2;
  real rho;
}

// The model to be estimated. We model the output
// 'y' to be normally distributed with mean 'mu'
// and standard deviation 'sigma'.
model {
  //y ~ normal(mu, sigma);
  int score[2];
  
  mu1 ~ normal(0, 1);
  mu2 ~ normal(0, 1);
  rho ~ normal(0, 1);
  
  score[1] = home_goals;
  score[2] = away_goals;
  
  under_prob_log ~ under_three_log(mu1,mu2,rho); 
  score ~ dixon_coles(rho, mu1, mu2);
}

generated quantities{

  real log_link;
  int score[2];
  
  score[1] = home_goals;
  score[2] = away_goals;
  
  log_link = under_three_log(under_prob_log,mu1,mu2,rho); ;
}

with inputs:

dat = list(under_prob_log = 0.3 , home_goals = 1, away_goals = 0)

[1] Predicting Football Results With Statistical Modelling: Dixon-Coles and Time-Weighting - dashee87.github.io

2

Hi and welcome. Sorry for the delay. Thanks for posting the link and the data. Can you post the full model call? And also let folks know what version of Stan and if this is being carried out in R, python, etc.