Update mcse_sd calculation to not use normality assumption.

CHANGELOG.md

@@ -7,6 +7,7 @@
 ### Maintenance and fixes
 - Make `arviz.data.generate_dims_coords` handle `dims` and `default_dims` consistently ([2395](https://github.com/arviz-devs/arviz/pull/2395))
 - Only emit a warning for custom groups in `InferenceData` when explicitly requested ([2401](https://github.com/arviz-devs/arviz/pull/2401))
+- Update `method="sd"` of `mcse` to not use normality assumption ([2167](https://github.com/arviz-devs/arviz/pull/2167))
 
 ### Documentation
 

arviz/stats/diagnostics.py

@@ -744,8 +744,8 @@ def _ess_sd(ary, relative=False):
     ary = np.asarray(ary)
     if _not_valid(ary, shape_kwargs=dict(min_draws=4, min_chains=1)):
         return np.nan
-    ary = _split_chains(ary)
-    return min(_ess(ary, relative=relative), _ess(ary**2, relative=relative))
+    ary = (ary - ary.mean()) ** 2
+    return _ess(_split_chains(ary), relative=relative)
 
 
 def _ess_quantile(ary, prob, relative=False):
@@ -838,13 +838,15 @@ def _mcse_sd(ary):
     ary = np.asarray(ary)
     if _not_valid(ary, shape_kwargs=dict(min_draws=4, min_chains=1)):
         return np.nan
-    ess = _ess_sd(ary)
+    sims_c2 = (ary - ary.mean()) ** 2
+    ess = _ess_mean(sims_c2)
+    evar = (sims_c2).mean()
+    varvar = ((sims_c2**2).mean() - evar**2) / ess
+    varsd = varvar / evar / 4
     if _numba_flag:
-        sd = float(_sqrt(svar(np.ravel(ary), ddof=1), np.zeros(1)).item())
+        mcse_sd_value = float(_sqrt(np.ravel(varsd), np.zeros(1)))
     else:
-        sd = np.std(ary, ddof=1)
-    fac_mcse_sd = np.sqrt(np.exp(1) * (1 - 1 / ess) ** (ess - 1) - 1)
-    mcse_sd_value = sd * fac_mcse_sd
+        mcse_sd_value = np.sqrt(varsd)
     return mcse_sd_value
 
 
@@ -973,19 +975,21 @@ def _multichain_statistics(ary, focus="mean"):
         # ess mean
         ess_mean_value = _ess_mean(ary)
 
-        # ess sd
-        ess_sd_value = _ess_sd(ary)
-
         # mcse_mean
-        sd = np.std(ary, ddof=1)
-        mcse_mean_value = sd / np.sqrt(ess_mean_value)
+        sims_c2 = (ary - ary.mean()) ** 2
+        sims_c2_sum = sims_c2.sum()
+        var = sims_c2_sum / (sims_c2.size - 1)
+        mcse_mean_value = np.sqrt(var / ess_mean_value)
 
         # ess bulk
         ess_bulk_value = _ess(z_split)
 
         # mcse_sd
-        fac_mcse_sd = np.sqrt(np.exp(1) * (1 - 1 / ess_sd_value) ** (ess_sd_value - 1) - 1)
-        mcse_sd_value = sd * fac_mcse_sd
+        evar = sims_c2_sum / sims_c2.size
+        ess_mean_sims = _ess_mean(sims_c2)
+        varvar = ((sims_c2**2).mean() - evar**2) / ess_mean_sims
+        varsd = varvar / evar / 4
+        mcse_sd_value = np.sqrt(varsd)
 
         return (
             mcse_mean_value,

arviz/tests/base_tests/test_diagnostics.py

@@ -120,10 +120,11 @@ def test_deterministic(self):
         ```
         Reference file:
 
-        Created: 2020-08-31
-        System: Ubuntu 18.04.5 LTS
-        R version 4.0.2 (2020-06-22)
-        posterior 0.1.2
+        Created: 2024-12-20
+        System: Ubuntu 24.04.1 LTS
+        R version 4.4.2 (2024-10-31)
+        posterior version from https://github.com/stan-dev/posterior/pull/388
+            (after release 1.6.0 but before the fixes in the PR were released).
         """
         # download input files
         here = os.path.dirname(os.path.abspath(__file__))