Choosing a suitable prior distribution π(θ) is a crucial step in Bayesian inference, as it encodes the information about the parameter of interest before observing the data. The choice of prior should be based on prior knowledge or beliefs about the parameter, if available, or it should be chosen in a way that avoids introducing biases into the inference.


Here are some correct ways to estimate a suitable prior π(θ):


Expert opinion: If there are subject matter experts who have knowledge about the parameter, their opinion can be used to inform the prior. This can be done through elicitation techniques, such as asking experts to specify their beliefs in the form of probability distributions.


Historical data: If there are previous studies or data on the same or similar parameter, the prior can be based on the results of those studies. For example, if we are interested in estimating the effect of a new drug, we can use the prior based on the effect sizes observed in previous studies.


Uninformative priors: If there is no prior knowledge or information available, uninformative priors can be used. These are prior distributions that are chosen to have minimal influence on the posterior distribution. Examples of uninformative priors include the uniform distribution or Jeffreys prior.


Bayesian model selection: If there are multiple models that can be used to describe the data, Bayesian model selection can be used to choose the prior. This involves computing the marginal likelihood of the data under each model and using these values to weight the priors.


Empirical Bayes: If there is a large amount of data available, empirical Bayes methods can be used to estimate the prior from the data itself. This involves using the data to estimate the parameters of the prior distribution, which can then be used as the prior for subsequent analyses.