# Distributions of parameters

What are the traditional distributions for assigning probabilities to model parameters? For instance, assume that we have a binomial distribution: $$y \sim Bin(n,\theta)$$ Then we can distribute $\theta$ parameter with a Beta distribution and parameters $\alpha$ and $\beta$ equal to 1. $$\theta \sim Beta(1,1)$$ I consider that if we would like to derive the distribution of the mean of a Gaussian probability distribution, then we could use the central limit theorem and assume that: $$\mu \sim N(0,1)$$ But what about other distributions? For instance, what is the distribution of $\lambda$ in the Weibull distribution? Or what is the distribution of $k$ for Chi-squared distribution? I'm not sure where to find papers on this topic because in most cases the usage of techniques involving parameters distributions seems like more of an intuitive methodology than some kind of rule.

Update: I'm asking this question for the Bayesian parameters estimation and Bayesian model selection, where I calculate posterior with: $$P(\theta|y) = P(y|\theta)P(\theta)$$ So I need to find the prior $P(\theta)$, that's what I was interested in.

• This is about conjugate priors for the exponential family of probability distributions (which includes many distributions): people.eecs.berkeley.edu/~jordan/courses/260-spring10/… – Eskapp Dec 15 '16 at 20:47
• @Eskapp thanks a lot for the link, will definitely work on this paper. – olejnik_ Dec 15 '16 at 20:52
• Wikipedia has a useful table for conjugate priors en.wikipedia.org/wiki/Conjugate_prior – Jon Dec 15 '16 at 21:31
• The beta is a conjugate prior to the binomial. But nowhere in the question do you mention Bayesian methods. Maybe one can infer it since you are talking about distributions (presumably posterior distributions) for parameters. But it is puzzling to me when you talk about the mean of a normal distribution where you invoke the central limit theorem but don't mention any prior. – Michael R. Chernick Dec 15 '16 at 22:09
• So $\alpha$ and $\beta$ are your prior parameters. Those do not come directly from your data. There are a few ways to estimate those prior parameters using your observed data set, but they're generally supposed to come from "prior" information. – Jon Dec 18 '16 at 19:34

• @MichaelChernick for now I'm working on the Bayesian approaches, I mean Bayesian model selection and Bayesian parameter estimation, where I calculate posterior $P(\theta | y) = P(y | \theta)P(\theta)$. The distribution I need is prior, $P(\theta)$. I already understood that I need to work with conjugate priors, thank you for this advice. The $N(0,1)$ distribution was meant to be the prior. – olejnik_ Dec 16 '16 at 13:08