# scipy.stats.bayes_mvs

scipy.stats.bayes_mvs(data, alpha=0.9)

  Bayesian confidence intervals for the mean, var, and std.

  Parameters:
    data array_like

      Input data, if multi-dimensional it is flattened to 1-D by `bayes_mvs`. Requires 2 or more data points.

    alpha float, optional

      Probability that the returned confidence interval contains the true parameter.

  Returns:
    mean_cntr, var_cntr, std_cntr tuple

      The three results are for the mean, variance and standard deviation, respectively. Each result is a tuple of the form:

          (center, (lower, upper))

      with center the mean of the conditional pdf of the value given the data, and (lower, upper) a confidence interval, centered on the median, containing the estimate to a probability `alpha`.

  See also

  `mvsdist`

  Notes

  Each tuple of mean, variance, and standard deviation estimates represent the (center, (lower, upper)) with center the mean of the conditional pdf of the value given the data and (lower, upper) is a confidence interval centered on the median, containing the estimate to a probability `alpha`.

  Converts data to 1-D and assumes all data has the same mean and variance. Uses Jeffrey’s prior for variance and std.

  Equivalent to `tuple((x.mean(), x.interval(alpha)) for x in mvsdist(dat))`

  References

  T.E. Oliphant, “A Bayesian perspective on estimating mean, variance, and standard-deviation from data”, https://scholarsarchive.byu.edu/facpub/278, 2006.

  Examples

  First a basic example to demonstrate the outputs:

      >>> from scipy import stats
      >>> data = [6, 9, 12, 7, 8, 8, 13]
      >>> mean, var, std = stats.bayes_mvs(data)
      >>> mean
      Mean(statistic=9.0, minmax=(7.103650222612533, 10.896349777387467))
      >>> var
      Variance(statistic=10.0, minmax=(3.176724206..., 24.45910382...))
      >>> std
      Std_dev(statistic=2.9724954732045084, minmax=(1.7823367265645143, 4.945614605014631))

  Now we generate some normally distributed random data, and get estimates of mean and standard deviation with 95% confidence intervals for those estimates:

      >>> n_samples = 100000
      >>> data = stats.norm.rvs(size=n_samples)
      >>> res_mean, res_var, res_std = stats.bayes_mvs(data, alpha=0.95)

      >>> import matplotlib.pyplot as plt
      >>> fig = plt.figure()
      >>> ax = fig.add_subplot(111)
      >>> ax.hist(data, bins=100, density=True, label='Histogram of data')
      >>> ax.vlines(res_mean.statistic, 0, 0.5, colors='r', label='Estimated mean')
      >>> ax.axvspan(res_mean.minmax[0],res_mean.minmax[1], facecolor='r',
      ...            alpha=0.2, label=r'Estimated mean (95% limits)')
      >>> ax.vlines(res_std.statistic, 0, 0.5, colors='g', label='Estimated scale')
      >>> ax.axvspan(res_std.minmax[0],res_std.minmax[1], facecolor='g', alpha=0.2,
      ...            label=r'Estimated scale (95% limits)')

      >>> ax.legend(fontsize=10)
      >>> ax.set_xlim([-4, 4])
      >>> ax.set_ylim([0, 0.5])
      >>> plt.show()