What Is a Sufficient Statistic for the Beta Distribution?

Assume $X_1,\ldots,X_n$ are independent and identically distributed as $\operatorname{Beta}(\alpha,\beta)$, with both shape parameters unknown and positive. The data contain information about $\alpha$ and $\beta$ through two summaries: the sum of the logarithms of the observations and the sum of the logarithms of their complements.

This is a result about the usual two-parameter Beta model. If a shape parameter is known, or the parameters are constrained to vary together, the sufficient statistic can be smaller.

Derive the statistic from the likelihood

The density of one observation is

$$f(x\mid\alpha,\beta)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1},\qquad 0<x<1.$$

For an iid sample, multiply the individual densities:

$$\begin{aligned} L(\alpha,\beta;\mathbf{x}) &=\left[\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\right]^n \prod_{i=1}^n x_i^{\alpha-1}(1-x_i)^{\beta-1}\\ &=\left[\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\right]^n \exp\left((\alpha-1)\sum_{i=1}^n\log x_i+(\beta-1)\sum_{i=1}^n\log(1-x_i)\right). \end{aligned}$$

The sample appears in the likelihood only through the two sums. By the factorization theorem, a sufficient statistic is

$$T(\mathbf X)=\left(\sum_{i=1}^n\log X_i,\;\sum_{i=1}^n\log(1-X_i)\right).$$

Equivalently, you can report the two products $\prod_i X_i$ and $\prod_i(1-X_i)$, since taking logarithms is a one-to-one transformation on positive values. The log-sum form is usually more convenient for calculations.

Why it is minimal sufficient

For this full two-parameter family, the likelihood-ratio criterion also shows that the pair is minimal sufficient. Compare two possible samples $\mathbf{x}$ and $\mathbf{y}$. Their likelihood ratio is independent of $(\alpha,\beta)$ exactly when

$$\sum_i\log x_i=\sum_i\log y_i \quad\text{and}\quad \sum_i\log(1-x_i)=\sum_i\log(1-y_i).$$

Thus the pair retains precisely the sample information needed to distinguish likelihoods as the two parameters vary. “Minimal” means that every sufficient statistic for this model must determine this pair; it does not mean the pair is the only possible way to represent the information.

Worked example

Take the sample $0.2, 0.4, 0.9$. Its two statistic components are

$$\begin{aligned} \sum_i\log x_i&=\log(0.2\cdot0.4\cdot0.9)=\log(0.072)\approx-2.6311,\\ \sum_i\log(1-x_i)&=\log(0.8\cdot0.6\cdot0.1)=\log(0.048)\approx-3.0366. \end{aligned}$$

So the sufficient statistic for this sample is approximately $(-2.6311,-3.0366)$. For another sample of the same size, calculate these two sums; if both match, the samples give the same likelihood dependence on $\alpha$ and $\beta$.

The sample mean is not enough in general. For instance, $(0.2,0.8)$ and $(0.4,0.6)$ both have mean $0.5$, but their products are $0.16$ and $0.24$. Therefore their sums of log observations differ, and their likelihoods vary differently with $\alpha$.

If a parameter is known or constrained

The statistic depends on which parameters are unknown:

Model	Sufficient statistic
Both $\alpha$ and $\beta$ unknown	$\left(\sum_i\log X_i,\sum_i\log(1-X_i)\right)$
$\beta$ known, $\alpha$ unknown	$\sum_i\log X_i$
$\alpha$ known, $\beta$ unknown	$\sum_i\log(1-X_i)$
$\alpha=\beta=\theta$ unknown	$\sum_i\log[X_i(1-X_i)]$

These follow by retaining only the terms in the likelihood that involve the free parameter or parameters. For example, when $\beta$ is fixed, the term involving $\sum_i\log(1-X_i)$ does not vary with $\alpha$, so it is not needed to form a sufficient statistic for $\alpha$.

Common mistakes

• Using only the sample mean. The likelihood depends on products of observations and complements, not just their sum. The mean is generally not sufficient for the two shape parameters.
• Keeping only one log-sum when both parameters are unknown. Each shape parameter has its own data-dependent term, so both sums are needed.
• Treating the answer as universal across Beta models. Known parameters and restrictions such as $\alpha=\beta$ change which summaries are needed.
• Forgetting the model assumptions. This result assumes an iid sample with values in $(0,1)$ from a Beta distribution. A different sampling model can have a different sufficient statistic.