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What happens when you don't have an idea of the parameters distribution? What approach should we use?

Most of the time we aim to undersatnd if a certain variable has any influence over the presence/absence of a certain species, and the variable is accepted or not according to the variable importance. This means that most of the times we are not thinking on the expetcted distribution a parameter should have.

Is it correct to assume that all parameters follow a normal distribution, when all i know is that b1,b2,b3 and b4 should vary between -2 and 2, and b0 can vary between -5 and 5 ?

model {
    # N observations
    for (i in 1:N) {
        species[i] ~ dbern(p[i])
        logit(p[i]) <- b0 + b1*var1[i] + b2*var2[i] + 
            b3*var3[i] + b4*var4[i]
    }
    # Priors
    b0     ~ dnorm(0,10)
    b1   ~ dnorm(0,10)
    b2 ~ dnorm(0,10)
    b3  ~ dnorm(0,10)
    b4  ~ dnorm(0,10)
}
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If you do not have a prior, you cannot use Bayesian inference. And hence MCMC methodology, –  Xi'an Oct 16 '12 at 16:52

3 Answers 3

up vote 6 down vote accepted

Parameters in linear predictor are t-distributed. When the number of records goes to infinity, it converges to normal distribution. So yes, normally it is considered correct to assume normal distribution of parameters.

Anyways, in bayesian statistics, you need not to assume parameter distribution. Normally you specify so called uninformative priors. For each case, different uninformative priors are recommended. In this case, people often use something like (you can tweak the values of course):

dunif(-100000, 100000)

or

dnorm(0, 1/10^10)

The second one is preferred, because it is not limited to particular values. With uninformative priors, you have take no risk. You can of course limit them to particular interval, but be careful.

So, you specify uninformative prior and the parameter distribution will come out itself! No need to make any assumptions about it.

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Unfortunately, this is not exactly true: the bounds in the uniform prior above can influence the result, esp. when testing hypotheses. This is a drawback of Winbugs in my opinion. –  Xi'an Oct 16 '12 at 16:54
    
@Xi'an - of course, that's what I say. That's why I prefer the "flat normal" in this case - i.e. the second option. Possibly with tweaking the second parameter. –  Curious Oct 16 '12 at 16:55
1  
Hmmm, this is not a flat prior at all... –  Xi'an Oct 16 '12 at 19:10
    
You are free to use dnorm(0, 1/10^10) or whatever –  Curious Oct 16 '12 at 20:33

Unfortunately, harmless seeming priors can be very dangerous (and have even fooled some seasoned Bayesians).

This recent paper, provides a nice introduction along with plotting methods to visualize the prior and posterior (usually marginal priors/posterior for the parameter(s) of interest).

Hidden Dangers of Specifying Noninformative Priors. John W. Seaman III, John W. Seaman Jr. & James D. Stamey The American StatisticianVolume 66, Issue 2, May 2012, pages 77-84. http://amstat.tandfonline.com/doi/full/10.1080/00031305.2012.695938

Such plots in my opinion should be obligatory in any actual Bayesian analysis, even if the analyst does not need them – what is happening in a Bayesian analysis should be made clear for most readers.

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1  
good link, it's a pitty that it is not available freely. –  Curious Oct 16 '12 at 13:57

Sensitivity analysis is usually a good way to go: try different priors and see how your results change with them. If they are robust, you'll probably be able to convince many people about your results. Otherwise, you'll probably want to somehow quantify how the priors change the results.

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