DIRICHLET

Overview

The DIRICHLET function computes properties and samples from the Dirichlet distribution, a multivariate generalization of the Beta distribution. The Dirichlet is commonly used in Bayesian statistics, machine learning, and probability for modeling proportions that sum to 1, such as probabilities across multiple categories. The probability density function (PDF) for a $d$ -dimensional Dirichlet is:

f(x_1, \ldots, x_d; \alpha_1, \ldots, \alpha_d) = \frac{1}{B(\alpha)} \prod_{i=1}^d x_i^{\alpha_i - 1}

where $B(\alpha)$ is the multivariate Beta function, $x_i \geq 0$ , $\sum x_i = 1$ , and $\alpha_i > 0$ .

Supported methods include PDF, log-PDF, mean, variance, covariance, entropy, and random sampling. For more details, see the scipy.stats.dirichlet documentation .

This wrapper simplifies the function to only require the most common parameters (x, alpha, and method), and excludes random seed and sample size options. This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=DIRICHLET(x, alpha, [method])

x (2D list, required for ‘pdf’ and ‘logpdf’): Table of $d$ columns, each row is a point at which to evaluate the function. Each row must sum to 1 and have non-negative values.
alpha (2D column vector, required): Table with one column and $d$ rows, specifying the concentration parameters. All values must be positive.
method (string, optional, default=pdf): Which property to compute. One of pdf, logpdf, mean, var, cov, entropy, rvs.

The function returns a 2D list of results for each input point, or an error message (string) if the input is invalid. For pdf and logpdf, each input row returns a single value. For mean, var, and rvs, returns a column vector. For cov, returns a $d \times d$ matrix. For entropy, returns a single value.

Examples

Example 1: PDF Calculation

Inputs:

x		alpha
0.2	0.8	1.0	2.0

Excel formula:


=DIRICHLET({0.2,0.8}, {1.0;2.0})

Expected output:

Result
1.600

Example 2: Log-PDF Calculation

Inputs:

x		alpha
0.2	0.8	1.0	2.0

Excel formula:


=DIRICHLET({0.2,0.8}, {1.0;2.0}, "logpdf")

Expected output:

Result
0.470

Example 3: Mean Calculation

Inputs:

alpha
1.0	2.0

Excel formula:


=DIRICHLET(, {1.0;2.0}, "mean")

Expected output:

Result
0.333
0.667

Example 4: Covariance Matrix

Inputs:

alpha
1.0	2.0

Excel formula:


=DIRICHLET(, {1.0;2.0}, "cov")

Expected output:


0.056	-0.056
-0.056	0.056

Python Code


from scipy.stats import dirichlet as scipy_dirichlet
from typing import List, Optional, Union
 
def dirichlet(x: Optional[List[List[float]]] = None, alpha: List[List[float]] = None, method: str = 'pdf') -> Union[List[List[Optional[float]]], str]:
    """
    Computes the PDF, log-PDF, mean, variance, covariance, entropy, or draws random samples from a Dirichlet distribution.
 
    Args:
        x: 2D list of float values. Points at which to evaluate the function (required for 'pdf' and 'logpdf' methods).
        alpha: 2D list of float values (column vector). Concentration parameters of the distribution. Required.
        method: Which method to compute (str): 'pdf', 'logpdf', 'mean', 'var', 'cov', 'entropy', 'rvs'. Default is 'pdf'.
 
    Returns:
        2D list of results for each input point, or an error message (str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate alpha
    if not isinstance(alpha, list) or len(alpha) < 1 or not all(isinstance(row, list) and len(row) == 1 for row in alpha):
        return "Invalid input: alpha must be a 2D list (column vector) with at least one row."
    try:
        alpha_vec = [float(row[0]) for row in alpha]
    except Exception:
        return "Invalid input: alpha must contain numeric values."
    if any(a <= 0 for a in alpha_vec):
        return "Invalid input: all alpha values must be positive."
    d = len(alpha_vec)
    # Validate method
    valid_methods = {'pdf', 'logpdf', 'mean', 'var', 'cov', 'entropy', 'rvs'}
    if method not in valid_methods:
        return f"Invalid method: {method}. Must be one of {sorted(valid_methods)}."
    dist = scipy_dirichlet(alpha_vec)
    # PDF and logPDF require x
    if method in {'pdf', 'logpdf'}:
        if x is None or not isinstance(x, list) or len(x) < 1 or not all(isinstance(row, list) and len(row) == d for row in x):
            return f"Invalid input: x must be a 2D list with each row of length {d}."
        results = []
        for row in x:
            try:
                vals = [float(v) for v in row]
                if any(v < 0 or v > 1 for v in vals) or abs(sum(vals) - 1.0) > 1e-8:
                    results.append(None)
                    continue
                if method == 'pdf':
                    res = dist.pdf(vals)
                else:
                    res = dist.logpdf(vals)
                # Disallow nan/inf
                if res is None or not isinstance(res, (int, float)) or res != res or abs(res) == float('inf'):
                    results.append(None)
                else:
                    results.append(float(res))
            except Exception:
                results.append(None)
        # Always return 2D list of shape (n, 1)
        return [[r] for r in results]
    # Mean, var, cov, entropy
    if method == 'mean':
        res = dist.mean()
        return [[float(v)] for v in res]
    if method == 'var':
        res = dist.var()
        return [[float(v)] for v in res]
    if method == 'cov':
        cov = dist.cov()
        # Return as 2D list
        return [[float(cov[i, j]) for j in range(d)] for i in range(d)]
    if method == 'entropy':
        res = dist.entropy()
        return [[float(res)]]
    if method == 'rvs':
        # Draw one sample
        try:
            sample = dist.rvs(size=1)
            # sample is shape (1, d)
            return [[float(v)] for v in sample[0]]
        except Exception as e:
            return f"scipy.dirichlet rvs error: {e}"
    return "Unknown error."

Example Workbook

Link to Workbook