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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)?

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  • $\begingroup$ 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 $\endgroup$ – Cam.Davidson.Pilon Aug 3 '14 at 20:01
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    $\begingroup$ @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 $\endgroup$ – Henry Aug 3 '14 at 22:09
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    $\begingroup$ @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. $\endgroup$ – Zen Aug 3 '14 at 22:37
  • $\begingroup$ @Zen yes I was too hasty - you are correct. $\endgroup$ – Cam.Davidson.Pilon Aug 4 '14 at 16:58
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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.

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