The key point is to show that your analysis does not depend much on the prior in the first place.  

You could try 

  * a conjugate prior (if one exists)
  * a (possibly improper) uniform prior and  
  * a Jeffreys prior (justified by its invariance under re-parameterization) 

and if you get answers that are reasonably close then that should provide some support that your analysis does not depend on arbitrary choice of prior.

From a practical viewpoint, computability and the availability of software may restrict your choice of prior in any case.

See:

  * http://en.wikipedia.org/wiki/Conjugate_prior

  * http://en.wikipedia.org/wiki/Uniform_prior

  * http://en.wikipedia.org/wiki/Jeffreys_prior