# Name for the Bayesian posterior probability that a regression coefficient is larger than zero

I have the following logistic regression: $$\text{logit} (y) = \beta_0 + \beta_1\, x$$ from which I can estimate the following posterior probability (using a Bayesian approach): $$P(\beta_1>0\,|\,\text{Data}).$$ Is there a particular name for that probability (something like Bayesian one-sided p-value)?

• Bayesian one-sided p-value is the best choice. Alternatively, you can give human-readable context (I'm assuming this is for a report or article?): the posterior probability the coefficient is greater than 0 is... - no need for potentially misleading names – Cam.Davidson.Pilon Aug 3 '14 at 20:01
• @Cam.Davidson.Pilon: I think the posterior probability the coefficient is greater than 0 is... is a better choice than anything to do with p-values – Henry Aug 3 '14 at 22:09
• @Cam.Davidson.Pilon: I think that a Bayesian p-value is defined differently. It's not the posterior probability of an event related to the parameters of the model. It's the (predictive) posterior probability that some statistic $T(Z)$ of a "future" observation $Z$ is more extreme than $T(\text{Data})$. This is the definition in Gelman et al. – Zen Aug 3 '14 at 22:37
• @Zen yes I was too hasty - you are correct. – Cam.Davidson.Pilon Aug 4 '14 at 16:58

## 1 Answer

As @Zen referenced in comments, 'Bayesian $p$-value' has a specific definition, distinct from your quantity of interest. As defined in Bayesian Data Analysis:

The Bayesian $p$-value is defined as the probability that the replicated data could be more extreme than the observed data as measured by the test quantity [...] where the probability is taken over the posterior distribution of $\theta$ and the posterior predictive distribution of $y_{rep}$[.]

$$p_B = Pr(T(y^{rep}, \theta) \ge T(y, \theta \vert y)) = \iint \mathbb{I}_{T(y^{rep}, \theta)\ge T(y,\theta)}p(y^{rep}\vert \theta)p(\theta \vert y)dy^{rep}d\theta$$

$\mathbb{I}$ the indicator function, $T$ some test statistic.

All to say that a $p$-value reference is not what you're looking for: I'd follow others commenters' advice and stick to "the posterior probability that $\beta$ is greater than zero," or whatever similar language is natural in context.