My data lie in the unit interval (proportion). I assume them to be drawn from a beta distribution with parameters a and b. Are there recommendations for the choice of the priors for a and b ?
Thank you!
If you want an uninformative prior, you may consider using Jeffrey's Prior.
Kathryn Blackmond Laskey, Bayesian Inference and Decision Theory, Unit 7: Hierarchical Bayesian Models recommends that you transform the hyperparameters of the beta distribution. Dr Laskey proposes this transformation because the two hyperparameters of the beta distribution are so closely correlated that the Gibbs sampling is inefficient. Her transformation, and the only transformation that others consider is U=alpha/(alpha+beta) and V=alpha+beta. U is the mean of the beta distribution and V is a rough measure of the precision of the beta distribution with larger values implying more precision. She suggests a uniform distribution for U and a gamma(1,20) distribution for V. She uses the rat tumour data from Gelman et al.
By Beta Parameters, I assume you are talking about Beta Distributions.
The parameters of a beta distribution is usually the observation count +1. The prior in absence of any other information can be treated as uniformly distributed. This kind of prior is called a non-informative prior.
The question is still little vague, I can furnish more insight if I get the details
You would normally use the number of successes and the number of samples to define the prior parameters. So for no samples you would have a uniform prior. As your number of samples grows, your prior will start to contract around the observed success/samples ratio, become more symmetric, and start to look more like a Gaussian distribution. Anyway, the Wikipedia page on this is pretty good.
Let’s write the Beta distribution as in equation 1,
where x is a r.v. with values in [0,1] (e.g., your vector of proportions) and B(a,b) is the Beta function with parameters a,b.
The Beta distribution is fit for a prior modelling of the unknown distribution of the variable x. In this sense, a and b can be seen as the hyper-parameters of the prior distribution Beta. A relative simple way to estimate the hyper-parameters is the method of moments. Firstly, we calculate the sample mean (M) and the sample variance (V) over the vector x, and then we take the estimates of a and b as in equations 2 and 3
If you parametrize the Beta distribution by the population mean m and precision parameter v, a good prior for m can be again a Beta distribution because m lies in the interval (0,1). For v you can choose any prior distribution defined over the positive reals (for instance a Gamma distribution) or a degenerate uniform (non-informative) prior. If you have covariates and you are interested in Bayesian Beta regression models I suggest you to have a look at the following reference:
Branscum, A.J., Johnson, W.O. & Thurmond, M.C. (2007). Bayesian beta regression: applications to household expenditure data and genetic distance between foot-and-mouth disease viruses. Australian and New Zealand Journal of Statistics,49, 287–301.
One approach is to use inverse gamma distribution with appropriate parameters according to what is described here:
http://www.stat.missouri.edu/~dsun/8640/Lab8.IntroWinBUGS.pdf
The idea is to set the inverse of the parameter to have a gamma prior distribution since
gamma distribution only takes positive values.
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Probability
- Probability Theory