Fix posterior dimensionality wording and tidy Stan code

- info.json: posterior is T-dimensional over (s, nu), not (T+1)
- stochastic_volatility.stan: remove empty functions block and stale comments

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

Author: colinpochart2

posterior_database/models/info/stochastic_volatility.info.json

@@ -2,7 +2,7 @@
   "name": "stochastic_volatility",
   "keywords": ["time_series", "stochastic_volatility", "marginalized"],
   "title": "Stochastic Volatility Model with Marginalized Precision",
-  "description": "Stochastic volatility model from Hoffman & Gelman (2014, NUTS paper, p. 1614-1615). Daily log-returns follow a Student-t distribution with time-varying scale s_i. The log-scale follows a Gaussian random walk with precision tau, which is marginalized analytically. Resulting posterior is (T+1)-dimensional over (s, nu).",
+  "description": "Stochastic volatility model from Hoffman & Gelman (2014, NUTS paper, p. 1614-1615). Daily log-returns follow a Student-t distribution with time-varying scale s_i. The log-scale follows a Gaussian random walk with precision tau, which is marginalized analytically. Resulting posterior is T-dimensional over (s, nu), where T is the number of data points.",
   "urls": "https://www.jmlr.org/papers/v15/hoffman14a.html",
   "model_implementations": {
     "stan": {

posterior_database/models/stan/stochastic_volatility.stan

@@ -1,14 +1,10 @@
-functions {
-
-}
-
 data {
     int<lower=1>T; //number of days
     vector<lower=0>[T] y; //index values of sp500
 }
 
 transformed data {
-    vector[T-1] log_returns;     // log y_i - log y_{i-1} for i = 2, ..., T
+    vector[T-1] log_returns;
     for (i in 2:T) {
         log_returns[i-1] = log(y[i]) - log(y[i-1]);
   }
@@ -31,9 +27,7 @@ model {
     for (i in 2:T) {
         target += student_t_lpdf(log_returns[i-1] | nu, 0, s[i]);
         }
-
-    // Term from marginalizing out tau analytically:
-    //   target += -((T + 1) / 2.0) * log(0.01 + 0.5 * sum((log s_i - log s_{i-1})^2))
+   
     {
         real sum_sq = 0;
         for (i in 2:T) {