An informed prior has a distribution that adds information to the Bayesian inference. It is either the result of a previous statistical analysis of other data that gave you information about the parameter (example), or it has been constructed from an expert's estimate of the parameter.

Informed priors can be modeled in various ways. A conjugate prior will be an informed prior if the parameter values create a distribution with a shape that is different from an uninformed prior. For example, a Beta(1,1,1) distribution is usually considered an uninformed prior when estimating a binomial probability because it assigns equal weight to all values of p between 0 and 1. Thus, a Beta(4,2,1) distribution, for example, is an informed prior because its shape is different: it peaks at 0.75, as shown in the figure below:

Informed priors can be constructed graphically to describe an expert's estimate, but it is a good idea to check whether there are data underlying the opinion that could be used in a statistical analysis.

An informed prior can also come out of a logical argument. For example, if 100 people are in a room, we might estimate the number of people who are female to be Binomial(0.5,100) before collecting any information about the group (example).