I have a question regarding bayesian statistics. Is it possible to end up with a posterior probability of 1, that a slope is positive?

My likelihood data shows a greatly significant relationship, with a p-value of 0.000715. But does it make sense that I end up with a posterior probability of 1?

And do you know a smart way to calculate the posterior probability of a positive slope?

  • $\begingroup$ Exactly how did you compute this posterior probability? Was it based, perhaps, on a Monte-Carlo method or is the calculation an exact analytical one? $\endgroup$ – whuber Feb 27 '16 at 17:59
  • $\begingroup$ And I suppose we are to assume that this p-value corresponds to a positive point estimate? $\endgroup$ – conjugateprior Feb 27 '16 at 18:01
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    $\begingroup$ That means you have an estimated posterior probability of $1$. As in all simulations, this value is uncertain. Unless the prior assigned zero probability to negative slopes, mathematically there must be some positive posterior probability of a negative slope--but your simulation just hasn't gone on long enough to detect that. $\endgroup$ – whuber Feb 27 '16 at 18:13
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    $\begingroup$ You ended up with the result that, after taking several thousands of samples from the joint posterior, the number of times that particular coefficient was non-positive was 0? $\endgroup$ – conjugateprior Feb 27 '16 at 18:13
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    $\begingroup$ You may run 10⁶ or 10⁹ iterations and still find a probability of one and still be unable to decide whether or not this estimate equal to 1 corresponds to a posterior probability of exactly 1. Simulation cannot help you with this question. $\endgroup$ – Xi'an Feb 27 '16 at 21:01

(I'm assuming that you're doing some sort of linear regression.) The only way you could end up with a posterior probability of 1 that the slope is positive is if either

1) the prior is zero for non-positive slopes, or 2) the likelihood is zero for non-positive slopes.

That's a straightforward consequence of Bayes' Rule.

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