## How to determine distributions for uncertain priors?

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### How to determine distributions for uncertain priors?

Hi

I am doing a bayesian efficient design now. However, I don't know how to determine distributions for uncertain priors in my design. For example, for what type of priors I can assign a normal distribution or uniform distribution? In addition, if I assume one prior follows a normal distribution, how can I determine the mean and standard deviation for this distribution?

Kind regards,
Shan
Shan

Posts: 7
Joined: Mon Dec 09, 2019 11:41 pm

### Re: How to determine distributions for uncertain priors?

Priors should always be set carefully as they have a very large impact on the efficiency of the design.

Bayesian (uncertain) priors are generally set based on parameter estimates from a pilot study. You collect data from 10% of your sample and estimate parameters beta and standard errors se. Then you use b[(n,beta,se]) as normally distributed random prior, where the se is used for the level of uncertainty.

If you have no pilot data but do know the approximate range of the parameter values, say lower<beta<upper, then you can use b[(u,lower,upper)] as a uniformly distributed random prior.

Michiel
Michiel Bliemer

Posts: 955
Joined: Tue Mar 31, 2009 4:13 pm

### Re: How to determine distributions for uncertain priors?

[quote="Michiel Bliemer"]
Hi Michiel,
Many thanks for your detailed reply. Now, I understand that if assuming uncertain priors follow normal distributions, I should use estimated betas in my pilot survey as means and standard errors as standard deviations. I still have two questions regarding to bayesian efficient designs.
1. Many of the estimated coefficients in my pilot survey are not statistically significant. Should I only treat these priors as uncertain priors, or I should treat all priors as uncertain priors?
2. Signs for 3 out of 17 coefficients are unexpected in my pilot survey. If assuming these priors also follow normal distributions, can I assign a very small value , (which has an expected sign) as mean and still use the standard error in pilot survey as standard deviation in bayesian efficient design?
Best,
Shan
Shan

Posts: 7
Joined: Mon Dec 09, 2019 11:41 pm

### Re: How to determine distributions for uncertain priors?

1. You generally would treat all prior as uncertain. However, this would become an issue if your model has many parameters (e.g. 17), you generally want to only include up to 10 Bayesian priors and keep the others as fixed priors since it is difficult to get stable results doing random draws over a 17-dimensional distribution space. I would make priors of attributes that have the largest contribution to utility (i.e. beta*X) Bayesian while keeping the others fixed.

2. I think this would be OK, especially if these attributes only have a small contribution to utility (and therefore the efficient design will not be as sensitive to these priors). If these priors correspond to very important attributes (with a high contribution to utility) you may need to be more careful and perhaps increase the standard deviation of the priors.

Michiel
Michiel Bliemer

Posts: 955
Joined: Tue Mar 31, 2009 4:13 pm

### Re: How to determine distributions for uncertain priors?

[quote="Michiel Bliemer"]
Thank you for your reply. Now I know what to do next.
Best,
Shan
Shan

Posts: 7
Joined: Mon Dec 09, 2019 11:41 pm