Sensitivity analysis on the priors?

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I'm not clear on how to perform sensitivity analysis on the priors. Many sites have different answers. One site indicates to perform three non-informative, weakly informative and known priors. Another suggest running the model with different priors.

Here are my questions:

I want to estimate this parameter A, in which A = mean1 + mean2 . So, the priors for mean1 and mean2 are formulated (coding) as the following :

1. mean1 ~ N(u1, tau1) mean2 ~ N(mean1, tau2)

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where tau1 ~ gamma(alpha1,beta1) tau2 ~ gamma(alpha2,beta2).With the above formulation, how do I perform the sensitivity analysis ?

• Should I define a range of u1, tau1, and tau2 ? If so, what is the systematic procedures ? It seems to me there are various combinations: varying u1 and fixing tau1 and tau2; or fix u1 and tau2, varying tau1; or fix u1 and tau1, varying tau2. How do we know what are the correct priors to use ?
• How do I incorporate weakly informative priors to the above formulation ?
• What if the sensitivity test returns different estimated mean1 and mean2 by varying the priors ? How do we make conclusion ?
  

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