# Selecting informative priors

I am questioning myself on how to chose the priors for a bayesian analysis in Rstudio. I'm trying to investigate the chances of relapse in a set of patients.

These patients are all affected by a variant of the disease that, by itself, predisposes them to relapse more than the "regular" variant of the pathology.

I know from bigger studies in literature that the "regular" group has a ~3% chances of relapse, with respect to the ~11% of the variant I'm considering.

If I wanted to predict the chances of relapse, based on some covariates, could I possibly include this prior knowledge in the model? How would I do It, practically?

Thanks

I'm not sure if it's directly applicable to your situation, but here is a brief discussion on choosing an informative prior in a beta-binomial context. I'm not sure how much experience you have with Bayesian inference, so my apologies if this is way too simple.

Let $$\theta$$ be the relapse probability, which you believe to be around 11%. It seems natural to use a beta distribution, which has support $$(0,1)$$ and is conjugate with binomial data.

One beta distribution that is "more or less" concentrated around $$0.11$$ is $$\theta \sim \mathsf{Beta}(\alpha=3, \beta=24).$$ It has mean $$3/27 = 0.111$$ and it puts 95% of its probability in $$(0.024, 0.251).$$ [Computations in R.]

qbeta(c(.025, .975), 3, 24)
[1] 0.02445808 0.25130292


Then if you observe 7 relapses among 50 patients in the class being studied, your posterior distribution is $$p(\theta|x) \propto \mathsf{Beta}(3+7 = 10,\; 24+43 = 67),$$ which has mean 0.149, median 0.127 and gives the 95% posterior probability interval $$(.065, 0.213).$$

[For reference: An Agresti-Coull frequentist 95% CI for the same data is $$(0.067, 0.266).$$ A Bayesian interval estimate based on the Jeffreys noninformative prior is $$\mathsf{Beta(.5,.5)}$$ is $$(0.065, 0.255).]$$

qbeta(.5, 10, 67)
[1] 0.1266618
qbeta(c(.025, .975), 10, 67)
[1] 0.06494157 0.21293324


If you want your prior to be more influential, then try priors with shape parameters in the same ratio, but both larger--perhaps $$\mathsf{Beta}(6, 48),$$ which puts about 95% of its probability in the narrower prior interval $$(0.042, 0.207).$$ And so on.

qbeta(c(.025, .975), 6, 48)
[1] 0.04269639 0.20658533


Of course, with a strong prior such as $$\mathsf{Beta}(60, 480),$$ we would hardly pay attention to data based on only 50 patients. Posterior interval $$(0.089,0.140).$$

qbeta(c(.025,.975), 60+7,480+43)
[1] 0.08925147 0.14034708

• Many thanks for sharing your experience. I’m a novice to Bayesian approach, so every bit matters. I appreciate the intuitive and practical approach of your answer. I thought of beta distribution as my prior, because I understood it is the “most representative” of a binomial outcome. Choosing a bd concentrated around 0.11 is interesting...would this be considered a weakly informative prior? Or a non-informative prior with some “tweaking”? Pardon the inaccuracies. Have a great day! Jun 21, 2019 at 7:53
• twin.sci-hub.se/5543/c6ec6c36460340f21be9ff0628e69fe9/… According to this paper we might be talking of conditional mean priors. Jun 21, 2019 at 19:28
• Choice of a prior centered near 11% seems natural in view of your 3rd paragraph. The details of how much probability the prior puts near 11% depend on how strongly you believe what you say there. In practical applications, the choice of prior is often somehow based on past data on the situation at hand or on informed hunches of the relevance of 'similar' situations. // Uninformative priors are fine in the face of ignorance, but with them Bayesian results look a lot like frequentist ones. Jun 21, 2019 at 20:55
• @BruceET well explained +2. Should be chosen as answer. Dec 4, 2021 at 13:28

Based on your description, I do not think you have usable informative prior.

"These patients are all affected by a variant of the disease that, by itself, predisposes them to relapse more than the "regular" variant of the pathology." It seems there is no reliable scientific evidence to support this statement. Also it is a relative index and has no probability distribution to describe its reliability.

"I know from bigger studies in literature that the "regular" group has a ~3% chances of relapse, with respect to the ~11% of the variant I'm considering." Even the statement is true, it has no relation with your objective of "investigate the chances of relapse in a set of patients".

• The statement regarding the incidence of relapse in the two groups is taken from the most populated study I came across in literature, which I assumed as my “expert opinion”. It is a recognised and shared claim in the field, so I thought it could somehow influence my prior. I understand that this parameter is not appropriate and I should refer to a distribution of probabilities instead. Thanks! Jun 21, 2019 at 2:05