# Prior boundaries that are not constant but random variables

I know one can use the Bound method to impost constant constraints on the priors. What I need to do is to impost a constraint relating the prior random variables. I am wondering is this possible?

I used to do it this way:

``````with pm.Model():
bound = pm.Uniform('b',-1,0)
mu = pm.Normal('mu',0.,1.)
b_mu = pm.Deterministic('b_mu', tt.switch(tt.le(mu, bound), bound, mu))
trace = pm.sample()

plt.scatter(trace['bound'], trace['b_mu'])
``````

where `b_mu` would be constrained by `bound`. However, I am not completely sure doing so is correct as it makes the prior improper. Maybe an additional constraint with `pm.Potential` is needed.

It is working!

I am not an expert in Bayesian but I have read a recent paper (https://arxiv.org/abs/1712.03549) that if the prior is improper, sometimes the MCMC sampling can still behave as if it is proper (it shouldn’t). Is there a way to check using PyMC3? I think PPC could not detect this?

Thanks a lot!

What I am concerned about is subtle bias. However, I do not know the precise detail of how it will affect the estimation etc. I guess it depends on how the bounded prior is implemented in the model. Maybe the mathematicians can weight in on this @aseyboldt, @colcarroll, @AustinRochford, @fonnesbeck etc.

So that shouldn’t be an improper distribution. You can write down the integral if you’re patient, or you can point out that your example is equivalent to

``````with pm.Model():
bound = pm.Uniform('b',-1,0)
mu = pm.Normal('mu',0.,1.)
b_mu = pm.Deterministic('b_mu', tt.max([bound, mu]))
trace = pm.sample(10000)
``````

and so the pdf of `b_mu` is bounded above by the sum of `bound` and `mu`, so `∫ b_mu < ∫ bound + mu = 2 < ∞`.

The same argument should hold for the `min`/`max` of any (proper) RVs!

Also, that’s a cool pdf!

1 Like

It is reassuring to know that what I have been doing is not wrong - I always thought of this kind of a hack