In the BAYESAB package, the below example is provided:
A_norm <- rnorm(100, 6, 1.5)
B_norm <- rnorm(100, 5, 2.5)
AB2 <- bayesTest(A_norm, B_norm,
priors = c('mu' = 5, 'lambda' = 1, 'alpha' = 3, 'beta' = 1),
distribution = 'normal')'
from the two distributions A_norm an dB_norm - how is the values for mu, lambda, alpha and beta generated?
Many thanks.
From ?bayesTest (bold-face mine)
• Normal: If your data is well modeled by the normal
distribution, with parameters mu, sigma^2 controlling mean
and variance of the underlying distribution
• Data _can_ be negative if it makes sense for your
experiment
• Uses a conjugate ‘NormalInverseGamma’ distribution for
the parameters *mu* and *sigma^2* in the Normal
Distribution.
• ‘mu’, ‘lambda’, ‘alpha’, and ‘beta’ must be set for prior
distributions over *mu, sigma^2* in accordance with the
parameters of the conjugate prior distributions:
• mu, sigma^2 ~ NormalInverseGamma(mu, lambda, alpha,
beta)
• This is a bivariate distribution (commonly used to model
mean and variance of the normal distribution). You may
want to experiment with both this distribution and the
‘plotNormal’ and ‘plotInvGamma’ outputs separately before
arriving at a suitable set of priors for the Normal and
LogNormal ‘bayesTest’
So to summarise, mu, lambda, alpha, beta denote the (hyper)parameters of the priors on (mu, sigma^2) of the normal distribution. How to choose parameters for your priors is a very broad question, and usually requires domain-specific knowledge. What priors (and in turn which parameter values) to choose is a critical question in Bayesian inference, and you can find a plethora of literature on this.
In this case, you are assuming an Normal-inverse-gamma distribution, where mu is the location parameter, lambda a scaling parameter, and alpha and beta are the parameters of an inverse Gamma distribution that defines the prior on sigma^2. With an understanding of the underlying probability distribution of the prior(s) and taking into account your/any prior knowledge, you choose parameters according to whether you want to consider an informative or non-informative (weak) prior on the parameters mu and sigma^2 of your normal distribution.
