BAYES_MVS_CI

This function wraps SciPy’s Bayesian mean-variance-standard-deviation estimator to produce equal-tailed credible intervals for three population parameters: mean, variance, and standard deviation.

Given observed data and confidence level \alpha, it returns posterior centers and interval bounds based on Jeffreys prior assumptions used by the underlying method.

For each parameter \theta, the reported interval is an equal-tailed posterior interval:

P(\theta_{L} \leq \theta \leq \theta_{U} \mid \text{data}) = \alpha

Excel Usage

=BAYES_MVS_CI(data, alpha)
  • data (list[list], required): 2D array of sample observations (unitless).
  • alpha (float, optional, default: 0.9): Credible interval probability level (0 to 1).

Returns (list[list]): 2D table with parameter name, posterior center, and lower/upper credible bounds.

Example 1: Credible intervals for balanced sample values

Inputs:

data alpha
6 9 12 0.9
7 8 13

Excel formula:

=BAYES_MVS_CI({6,9,12;7,8,13}, 0.9)

Expected output:

Metric Center Lower Upper
Mean 9.16667 6.87407 11.4593
Variance 12.9444 3.50782 33.9015
Std Dev 3.31475 1.87292 5.8225
Example 2: Credible intervals with decimal-valued sample

Inputs:

data alpha
1.5 2 2.5 0.95
3 3.5 4

Excel formula:

=BAYES_MVS_CI({1.5,2,2.5;3,3.5,4}, 0.95)

Expected output:

Metric Center Lower Upper
Mean 2.75 1.76834 3.73166
Variance 1.45833 0.340931 5.2634
Std Dev 1.1126 0.583893 2.29421
Example 3: Single-row sample array is accepted

Inputs:

data alpha
10 11 9 12 0.8

Excel formula:

=BAYES_MVS_CI({10,11,9,12}, 0.8)

Expected output:

Metric Center Lower Upper
Mean 10.5 9.44284 11.5572
Variance 5 0.799822 8.55616
Std Dev 1.78412 0.894328 2.92509
Example 4: Scalar sample input is normalized to 2D

Inputs:

data alpha
4 5 6 0.9

Excel formula:

=BAYES_MVS_CI({4,5,6}, 0.9)

Expected output:

Metric Center Lower Upper
Mean 5 3.31415 6.68585
Variance Infinity 0.333808 19.4957
Std Dev 1.77245 0.577761 4.4154

Python Code

from scipy.stats import bayes_mvs as scipy_bayes_mvs

def bayes_mvs_ci(data, alpha=0.9):
    """
    Compute Bayesian credible intervals for mean, variance, and standard deviation from sample data.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.bayes_mvs.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        data (list[list]): 2D array of sample observations (unitless).
        alpha (float, optional): Credible interval probability level (0 to 1). Default is 0.9.

    Returns:
        list[list]: 2D table with parameter name, posterior center, and lower/upper credible bounds.
    """
    try:
        def to2d(x):
            return [[x]] if not isinstance(x, list) else x

        data = to2d(data)

        if not isinstance(data, list) or not all(isinstance(row, list) for row in data):
            return "Error: Invalid input - data must be a 2D list"

        if not (0 < alpha < 1):
            return "Error: alpha must be between 0 and 1"

        flat = []
        for row in data:
            for value in row:
                try:
                    flat.append(float(value))
                except (TypeError, ValueError):
                    continue

        if len(flat) < 2:
            return "Error: data must contain at least two numeric values"

        mean_res, var_res, std_res = scipy_bayes_mvs(flat, alpha=alpha)

        return [
            ["Metric", "Center", "Lower", "Upper"],
            ["Mean", float(mean_res.statistic), float(mean_res.minmax[0]), float(mean_res.minmax[1])],
            ["Variance", float(var_res.statistic), float(var_res.minmax[0]), float(var_res.minmax[1])],
            ["Std Dev", float(std_res.statistic), float(std_res.minmax[0]), float(std_res.minmax[1])]
        ]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

2D array of sample observations (unitless).
Credible interval probability level (0 to 1).