---
title: Parametric Bayesian TOST
url: https://www.emergentmind.com/topics/parametric-bayesian-tost
type: topic
---

# Parametric Bayesian TOST

Parametric Bayesian Two One-Sided Tests (TOST) integrate Bayesian posterior inference into the classical TOST procedure for equivalence testing, replacing frequentist $p$-values by posterior tail probabilities. This allows conclusions about practical equivalence (rather than simple difference) to be drawn via Bayesian measures of evidence, with posterior probabilities that are valid uniform$(0,1)$ test statistics under the null, supporting both single-hypothesis and multiple-testing contexts. The methodology offers direct control over the tradeoff between type I error, test power, and prior informativeness, and is applicable to a range of parametric models including normal and binomial families [2507.19685], [2107.01271].

## 1. Parametric Model Structure and Equivalence Hypothesis Formulation

The parametric Bayesian TOST operates on data $X = (X_1,\ldots,X_n)^\top$ sampled i.i.d. from a family $\{f(x|\theta): \theta \in \Theta \subseteq \mathbb{R}\}$. The null hypothesis for equivalence is split across two margins $\theta_1^* < \theta_2^*$:

- $H_0$: $\theta \notin (\theta_1^*, \theta_2^*)$
- $H_1$: $\theta \in (\theta_1^*, \theta_2^*)$

This is further decomposed into two one-sided nulls for TOST:
- $H_r$: $\theta \le \theta_1^*$ versus $K_r$: $\theta > \theta_1^*$
- $H_l$: $\theta \ge \theta_2^*$ versus $K_l$: $\theta < \theta_2^*$

A statistic $T(X)$ with the monotone likelihood-ratio property in $\theta$ underpins the construction, supporting the derivation of pivotal quantities [2507.19685]. This setting encompasses scenarios such as comparing two means, two proportions, and equivalence of a mean to a reference value [2107.01271].

## 2. Bayesian Posterior Tail Probabilities and Decision Rules

A continuous prior $\pi(\theta)$ is placed on $\Theta$, yielding posterior $\pi(\theta|x)$ via
\[
\pi(\theta|x) = \frac{f(x|\theta)\pi(\theta)}{\int_\Theta f(x|\theta)\pi(\theta)\,d\theta}.
\]
The Bayesian analogs of one-sided $p$-values are the posterior left and right tail probabilities:
\[
P_r^B(x) = \int_{-\infty}^{\theta_1^*}\pi(\theta|x)d\theta, \qquad P_l^B(x) = \int_{\theta_2^*}^{\infty}\pi(\theta|x)d\theta.
\]

The overall Bayesian “p-value” for equivalence is
\[
P_b(x) = P_r^B(x) + P_l^B(x) = 1 - \Pr(\theta_1^* < \theta < \theta_2^* | x).
\]
Equivalence is declared when both $P_r^B(x) \le \alpha$ and $P_l^B(x) \le \alpha$ for chosen $\alpha$ (often $\alpha=0.05$), precisely parallel to the frequentist TOST procedure [2507.19685].

## 3. Uniformity and Validity of Bayesian TOST Statistics

For monotone likelihood-ratio families and continuous priors, each tail posterior probability is uniformly distributed under its null:
- $U_r = P_r^B(X)$ is uniform$(0,1)$ when $\theta=\theta_1^*$
- $U_l = P_l^B(X)$ is uniform$(0,1)$ when $\theta=\theta_2^*$

As a result, $P_r^B$ and $P_l^B$ serve as valid $p$-values for multiple-testing procedures and FDR contexts. This is foundational to their direct use in the TOST structure and enables plug-in to standard p-value-based procedures, including the Benjamini–Hochberg algorithm [2507.19685].

## 4. Algorithmic Workflow and Implementation

The Bayesian TOST involves the following steps:
1. **Specification**: Define equivalence bounds $\theta_1^*,\theta_2^*$ and the significance level $\alpha$.
2. **Prior Selection**: Choose prior $\pi(\theta)$ over $\Theta$ (e.g., $N(\theta_0, \tau^2)$ for normal models, Beta for binomial cases).
3. **Posterior Calculation**: Compute $\pi(\theta|x)$ given observed data $x$.
4. **Tail Probability Evaluation**:
   \[
   P_r^B(x) = \int_{-\infty}^{\theta_1^*}\pi(\theta|x)d\theta
   \]
   \[
   P_l^B(x) = \int_{\theta_2^*}^{\infty}\pi(\theta|x)d\theta
   \]
5. **Decision Rule**: Declare equivalence if $P_r^B(x) \le \alpha$ and $P_l^B(x) \le \alpha$.

The same steps apply in discrete models (e.g., binomial data), replacing integrals with summations as appropriate [2507.19685]. For explicit “two-interval” Bayesian tests (2IT), one computes posterior probability $P_E = \Pr(\theta \in [\theta_1^*,\theta_2^*] | x)$ and applies high threshold criteria (e.g., $P_E \ge 0.95$), paralleling but not identical to the TOST tail-probability approach [2107.01271].

## 5. Power, Conservativeness, and Prior Specification

The power function for the Bayesian TOST depends on prior choice:
- **Binomial-Beta Models**: Priors with $p,q<1$ yield less conservative posterior p-values; as $p,q$ increase, conservativeness increases and power drops.
- **Normal Models**: For prior $N(\theta_*,\tau^2)$, $\tau^2 \to \infty$ recovers the frequentist case ($P_b \to P_f$). Very small $\tau^2$ gives an overly informative, extremely conservative test. Choosing moderate $\tau^2$ (by empirical Bayes or elicitation) balances the trade-off.

Closed-form expressions for $P_b(x)$ and $P_f(x)$ allow explicit comparison. For instance, in the normal-mean model with known $\sigma$,
\[
P_b(x) = 1-\Phi\left(\frac{n\tau(\bar x-\theta_1^*)}{\sigma\sqrt{\sigma^2+n\tau^2}}\right)
+\Phi\left(\frac{n\tau(\bar x-\theta_2^*)}{\sigma\sqrt{\sigma^2+n\tau^2}}\right),
\]
\[
P_f(x) = 1-\left\{\Phi\left(\frac{\sqrt n(\bar x-\theta_1^*)}{\sigma}\right) + \Phi\left(\frac{\sqrt n(\bar x-\theta_2^*)}{\sigma}\right)\right\}.
\]
Power ($\beta_B(\theta)$, $\beta_F(\theta)$) is computed by integrating these under the sampling law. Often, suitably chosen priors achieve greater power near the center of the equivalence region [2507.19685].

## 6. Correlation Structure and Multiple Testing

In the normal model, it is shown that $\operatorname{Cov}\{P_b(X),P_f(X)\} = 0 \implies \rho\{P_b,P_f\}=0$ (Proposition 8 of [2507.19685]). This independence ensures separate inferential roles for the Bayesian and frequentist procedures and supports valid FDR procedures.

Simulations for both single-hypothesis and multiple-testing (up to $k=1\,000$ hypotheses) demonstrate that
- Type I error control is near the nominal level,
- Power increases with sample size, relaxed equivalence margins, and prior variance,
- Bayesian posterior p-value–based FDR is competitive with standard $p$-value FDR as prior variance grows.

Under dependence or in FDR settings, collections of $P_b$ values can be submitted directly to algorithms such as Benjamini–Hochberg, as they retain the uniform null distribution [2507.19685].

## 7. Extensions, Comparison, and Practical Considerations

The parametric Bayesian TOST generalizes to the Bayesian two-interval test (2IT) framework, which replaces p-values with posterior probabilities of interval hypotheses, applicable to superiority, non-inferiority, or equivalence [2107.01271]. The Bayesian TOST can realize direct sample size determination via expected posterior probabilities and permits sequential and optional stopping without adjusting type I error, thanks to the likelihood principle.

Differences between Bayesian TOST and fully posterior-interval tests are detailed in their respective treatments: the TOST approach employs tail probabilities as pivotal quantities, matching the frequentist conceptual structure; the 2IT computes posterior mass in the equivalence interval versus its complement and adopts threshold-based decision rules that may directly represent the probability of equivalence.

Across approaches, by tuning the informativeness of the prior, practitioners can interpolate between fully uninformative, classical TOST-like behavior and potentially more powerful, informative analyses when robust prior knowledge is available. The methodology encompasses a wide range of standard parametric models and is computationally straightforward in settings with conjugate priors.

**References:**
- "A Comparison of the Bayesian Posterior Probability and the Frequentist $p$-Value in Testing Equivalence Hypotheses" [2507.19685].
- "Bayesian two-interval test" [2107.01271].

Source: https://www.emergentmind.com/topics/parametric-bayesian-tost