code: def test_2x2():
      # Generate correlated data with mean and variance
      np.random.seed(3434)
      x = np.random.normal(size=(1000, 2))
      r = 0.5
      x[:, 1] = r * x[:, 0] + np.sqrt(1 - r**2) * x[:, 1]
      x[:, 0] *= 2
      x[:, 1] *= 3
      x[:, 0] += 1
      x[:, 1] -= 2
  
      # Introduce some missing values
      u = np.random.normal(size=x.shape[0])
      x[u > 1, 0] = np.nan
      u = np.random.normal(size=x.shape[0])
      x[u > 1, 1] = np.nan
  
      bm = BayesGaussMI(x)
  
      # Burn-in
      for k in range(500):
          bm.update()
  
      # Estimate the posterior mean
      mean = 0
      cov = 0
      dmean = 0
      dcov = 0
      for k in range(500):
          bm.update()
          mean += bm.mean
          cov += bm.cov
          dmean += bm.data.mean(0)
          dcov += np.cov(bm.data.T)
      mean /= 500
      cov /= 500
      dmean /= 500
      dcov /= 500
  
      assert_allclose(mean, np.r_[1, -2], 0.1)
      assert_allclose(dmean, np.r_[1, -2], 0.1)
      assert_allclose(cov, np.asarray([[4, 6 * r], [6 * r, 9]]), 0.1)
      assert_allclose(dcov, np.asarray([[4, 6 * r], [6 * r, 9]]), 0.1)
  
file_path: statsmodels/statsmodels/imputation/tests/test_bayes_mi.py
imports: import numpy
  import pandas
  import statsmodels.api
  from statsmodels.imputation.bayes_mi import BayesGaussMI, MI
  from numpy.testing import assert_allclose, assert_equal
label_desc: Use image node_id 2 for calling a global function with example usage: test_2x2() without return types