Update phi parameterisation

NEWS.md

@@ -16,6 +16,10 @@
 * Changed all instances of arguments that refer to the maximum of a distribution to reflect the maximum. Previously this did, in some instance, refer to the length of the PMF. By @sbfnk in #468.
 * Fixed a bug in the bounds of delays when setting initial conditions. By @sbfnk in #474.
 
+## Model changes
+
+* Updated the parameterisation of the dispersion term `phi` to be `phi = 1 / sqrt_phi ^ 2` rather than the previous parameterisation `phi = 1 / sqrt(sqrt_phi)` based on the suggested prior [here](https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations#story-when-the-generic-prior-fails-the-case-of-the-negative-binomial) and the performance benefits seen in the `epinowcast` package (see [here](https://github.com/epinowcast/epinowcast/blob/8eff560d1fd8305f5fb26c21324b2bfca1f002b4/inst/stan/epinowcast.stan#L314)). By @seabbs in # and reviewed by @sbfnk.
+
 # EpiNow2 1.4.0
 
 This release contains some bug fixes, minor new features, and the initial stages of some broader improvement to future handling of delay distributions.

inst/stan/functions/observation_model.stan

@@ -55,13 +55,12 @@ void report_lp(array[] int cases, vector reports,
                array[] real rep_phi, real phi_mean, real phi_sd,
                int model_type, real weight) {
   if (model_type) {
-    real sqrt_phi; // the reciprocal overdispersion parameter (phi)
+    real dispersion = 1 / pow(rep_phi[model_type], 2); 
     rep_phi[model_type] ~ normal(phi_mean, phi_sd) T[0,];
-    sqrt_phi = 1 / sqrt(rep_phi[model_type]);
     if (weight == 1) {
-      cases ~ neg_binomial_2(reports, sqrt_phi);
+      cases ~ neg_binomial_2(reports, dispersion);
     } else {
-      target += neg_binomial_2_lpmf(cases | reports, sqrt_phi) * weight;
+      target += neg_binomial_2_lpmf(cases | reports, dispersion) * weight;
     }
   } else {
     if (weight == 1) {
@@ -83,9 +82,9 @@ vector report_log_lik(array[] int cases, vector reports,
       log_lik[i] = poisson_lpmf(cases[i] | reports[i]) * weight;
     }
   } else {
-    real sqrt_phi = 1 / sqrt(rep_phi[model_type]);
+    real dispersion = 1 / pow(rep_phi[model_type], 2);
     for (i in 1:t) {
-      log_lik[i] = neg_binomial_2_lpmf(cases[i] | reports[i], sqrt_phi) * weight;
+      log_lik[i] = neg_binomial_2_lpmf(cases[i] | reports[i], dispersion) * weight;
     }
   }
   return(log_lik);
@@ -94,20 +93,20 @@ vector report_log_lik(array[] int cases, vector reports,
 array[] int report_rng(vector reports, array[] real rep_phi, int model_type) {
   int t = num_elements(reports);
   array[t] int sampled_reports;
-  real sqrt_phi = 1e5;
+  real dispersion = 1e5;
   if (model_type) {
-    sqrt_phi = 1 / sqrt(rep_phi[model_type]);
+    dispersion = 1 / pow(rep_phi[model_type], 2);
   }
 
   for (s in 1:t) {
     if (reports[s] < 1e-8) {
       sampled_reports[s] = 0;
     } else {
       // defer to poisson if phi is large, to avoid overflow
-      if (sqrt_phi > 1e4) {
+      if (dispersion > 1e4) {
         sampled_reports[s] = poisson_rng(reports[s] > 1e8 ? 1e8 : reports[s]);
       } else {
-        sampled_reports[s] = neg_binomial_2_rng(reports[s] > 1e8 ? 1e8 : reports[s], sqrt_phi);
+        sampled_reports[s] = neg_binomial_2_rng(reports[s] > 1e8 ? 1e8 : reports[s], dispersion);
       }
     }
   }