lower truncations off by one

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Bob Carpenter

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Sep 4, 2016, 8:47:19 AM9/4/16
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Michael and Ben are both right.

Ben: The lower truncations in our generated
code for discrete distributions normalize in such a way
to make the lower bounds exclusive and the upper bounds
inclusive. The lower-bound checks make the lower
bound inclusive.

Michael: The CDFs and CCDFs are doing the right thing,
with CDF being Pr[X <= x] and CCDF being Pr[X > x],
so that CDF + CCDF = 1.

I revised the issue and included the currently generated
code for for lower bounds, upper bounds,
and both:

https://github.com/stan-dev/stan/issues/2054

This one's going to be a hassle to fix because we only need
to make the change for discrete distributions, so I'm going to need
to test for discreteness. I think the easiest thing to do is going
to be to adjust for the lower bound using the pmf.

I'll try to fix this later tonight (Sunday 4 September).

This will cause sampling errors in cases where the discrete
distribution parameters are Stan model parameters (e.g, poisson(lambda)
where lambda is declared as a parameter).

- Bob

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