diff --git a/tests/scenarios.py b/tests/scenarios.py
index f8e7cb8..717ab56 100644
--- a/tests/scenarios.py
+++ b/tests/scenarios.py
@@ -41,7 +41,7 @@ def get_multivariate_normal_log_prob(
 ) -> tuple[LogProbFn, tuple[torch.Tensor, torch.Tensor]]:
     mean = torch.randn(dim)
     sqrt_cov = torch.randn(dim, dim)
-    cov = sqrt_cov @ sqrt_cov.T
+    cov = sqrt_cov @ sqrt_cov.T * 0.3 + torch.eye(dim) * 0.7
     chol_cov = torch.linalg.cholesky(cov)  # Lower triangular with positive diagonal
 
     def log_prob(p, batch):
diff --git a/tests/sgmcmc/test_baoa.py b/tests/sgmcmc/test_baoa.py
index 76ba518..468f4f9 100644
--- a/tests/sgmcmc/test_baoa.py
+++ b/tests/sgmcmc/test_baoa.py
@@ -10,7 +10,7 @@ def test_baoa():
     torch.manual_seed(42)
 
     # Set inference parameters (with torch.optim.SGD parameterization)
-    lr = 1e-3
+    lr = 1e-2
     mu = 0.9
     tau = 0.9
 
diff --git a/tests/sgmcmc/test_sghmc.py b/tests/sgmcmc/test_sghmc.py
index 321be53..f0bdc24 100644
--- a/tests/sgmcmc/test_sghmc.py
+++ b/tests/sgmcmc/test_sghmc.py
@@ -10,7 +10,7 @@ def test_sghmc():
     torch.manual_seed(42)
 
     # Set inference parameters (with torch.optim.SGD parameterization)
-    lr = 1e-3
+    lr = 1e-2
     mu = 0.9
     tau = 0.9
 
diff --git a/tests/sgmcmc/test_sgld.py b/tests/sgmcmc/test_sgld.py
index adcc76e..348853e 100644
--- a/tests/sgmcmc/test_sgld.py
+++ b/tests/sgmcmc/test_sgld.py
@@ -10,7 +10,7 @@ def test_sgld():
     torch.manual_seed(42)
 
     # Set inference parameters
-    lr = 1e-3
+    lr = 1e-2
     beta = 0.0
 
     # Run MCMC test on Gaussian
diff --git a/tests/sgmcmc/test_sgnht.py b/tests/sgmcmc/test_sgnht.py
index cc8dd78..74b0d9a 100644
--- a/tests/sgmcmc/test_sgnht.py
+++ b/tests/sgmcmc/test_sgnht.py
@@ -10,7 +10,7 @@ def test_sgnht():
     torch.manual_seed(42)
 
     # Set inference parameters (with torch.optim.SGD parameterization)
-    lr = 1e-3
+    lr = 1e-2
     mu = 0.9
     tau = 0.9
 
diff --git a/tests/sgmcmc/utils.py b/tests/sgmcmc/utils.py
index 520d5cb..4fcddea 100644
--- a/tests/sgmcmc/utils.py
+++ b/tests/sgmcmc/utils.py
@@ -8,9 +8,9 @@
 
 def run_test_sgmcmc_gaussian(
     transform_builder: Callable[[LogProbFn], Transform],
-    dim: int = 3,
-    n_steps: int = 15_000,
-    burnin: int = 5_000,
+    dim: int = 2,
+    n_steps: int = 100_000,
+    burnin: int = 15_000,
     rtol: float = 1e-2,  # Relative reduction of KL for final distribution compared to initial distribution
 ):
     # Load log posterior
@@ -33,6 +33,13 @@ def run_test_sgmcmc_gaussian(
     # Remove burnin
     all_states = all_states[burnin:]
 
+    samples_mean = all_states.params.mean(0)
+    samples_cov = all_states.params.T.cov()
+
+    # Check sufficient statistics
+    assert torch.allclose(samples_mean, mean, atol=1e-1)
+    assert torch.allclose(samples_cov, cov, atol=1e-1)
+
     # Check KL divergence between true and inferred Gaussian
     kl_init = utils.kl_gaussians(torch.zeros(dim), torch.eye(dim) * init_var, mean, cov)
     kl_inferred = utils.kl_gaussians(
@@ -54,7 +61,12 @@ def run_test_sgmcmc_gaussian(
 
     start_num_samples = 500  # Omit first few KLs with high variance due to few samples
     spacing = 100
-    kl_divs_spaced = kl_divs[start_num_samples::spacing]
+
+    kl_divs_min_idx = torch.argmin(kl_divs[2:]) + 2
+    kl_divs_before_min = kl_divs[:kl_divs_min_idx]
+    kl_divs_spaced = kl_divs_before_min[start_num_samples::spacing]
     spaced_decreasing = kl_divs_spaced[:-1] > kl_divs_spaced[1:]
     proportion_decreasing = spaced_decreasing.float().mean()
-    assert proportion_decreasing > 0.8
+    assert proportion_decreasing > 0.5, (
+        f"Proportion decreasing: {proportion_decreasing}"
+    )