Uninformative Priors, Informative Priors
Two more types of priors are uninformative and informative priors.
Like we mentioned in the previous post on priors, all priors contain some information.
Although Uninformative priors are often used in attempt to produce an “objective analysis”, there is no such thing! So this kind of prior is used when there is an analysis at stake for being as objective as possible.
One pitfall of uninformative priors is that they can often be unbounded and therefore: not make practical sense and not be proper probability distributions.
An example of this kind of prior could be one that is for a parameter that is defined by the age at which Parkinson’s is developed in an adult. Since people ca be at least 0 years of age, that could be a lower bound, but what about an upper bound? If there is no obvious or previous knowledge to go off of, we can use positive infinity.
This doesn’t make practical sense, because life expectancies tend to not go above a certain age. This kind of prior wouldn’t make a proper probability distribution either, because the a non-zero constant value for a probability density results in an infinite total probability (integrate).
Sometimes it is essential to include a significant amount of information in a prior:
If we want to go about constructing an informative prior, one way is through moment matching, which involves using previous data to construct parameters that fit for a new prior distribution.
For example, suppose I had a set of 100 data points.
If I wanted to construct a prior from this raw dataset, I could first graph it and look at its shape and other properties. Then, I can determine a probability distribution that best describes the data.
Then, from this data, I can extract the relevant parameter values belonging to the matching probability distribution.
Eliciting Priors (Informative Prior type)
These are a special kind of informative prior that hinges on the use of “expert opinion” versus older data.
This prior type allows us to use “subjective views” to craft an informative prior.
How do we do this?
Generally, this method requires asking experts to provide values of interest ,such as percentages or ratios of a certain condition, and then compare them against each other.
We can then take these values that experts provide and take the average of the parameters of interest to create a distribution. We could also use all of the provided expert opinions to construct linear regression parameters.
An example of this could be to interview a panel of 100 epidemiologists to provide estimates for the parameters of the SIR model (commonly used for spread of disease) specifically for SARS-CoV-2 in the past year. Then we can do two things:
- Take the average of the 100 individual responses to provide us with a prior.
- OR We can make a linear regression model using all 100 observations at once to construct a prior.
Main Takeaway: All priors contain some information. Uninformative priors run the risk of being impractical and creating improper probability distributions. Informative priors can come from either previously collected data/analyses or come from multiple expert opinions (eliciting priors).
Research done from: “Priors” A Student’s Guide to Bayesian Statistics, by Ben Lambert, SAGE, 2018.