# Bayesian statistics: Can a posterior probability be exactly 1?

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?

• 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? – whuber Feb 27 '16 at 17:59
• And I suppose we are to assume that this p-value corresponds to a positive point estimate? – conjugateprior Feb 27 '16 at 18:01
• 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. – whuber Feb 27 '16 at 18:13
• 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? – conjugateprior Feb 27 '16 at 18:13
• 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. – Xi'an Feb 27 '16 at 21:01