Many ecologists (and particularly graduate students) have a vague concept of Bayesian statistics, probably based on some cursory information from an intro stats course. Most know that they differ from frequentist methods in the way they approach probability theory and incorporate some kind of “prior knowledge” with likelihood to produce a posterior probability distribution. Although the philosophical differences are not trivial and are the selling point of Bayesian inference for some, I will not delve into them too much. What I will say is that most Bayesian models will use “uninformative priors”, in which case, the result is essentially identical to the maximum likelihood estimates for the same model. So why bother with Bayesian statistics if you can use maximum likelihood and get the same answer?
The real beauty of Bayes lies in hierarchical Bayesian models*. In his 2005 paper in Ecology Letters “Why environmental scientists are becoming Bayesians”, Jim Clark provides some great examples of just how useful these models can be. For starters, they allow for the addition of stochastic factors that help to better describe the variability in our data. This isn’t the kind of variability that will decrease with sampling size but real, biologically meaningful variability that that often has complex relationships and is difficult to quantify – think differences between individuals or ecologically relevant time points. Clark gives the example of the nonlinear relationship between individual tree fecundity and maturation/senescence, which is not well represented by counts of seeds in seed traps.
Figure from Clark (2005) Ecology Letters
Next, hierarchical Bayes differs from classical mixed modeling in that it can more easily deal with complex hierarchical problems. What is a hindrance for more simple models is what allows Bayes to shine with the more complex – it’s reliance on computation power, namely MCMC algorithms. Models that would be too complex to even attempt using traditional methods become possible with hierarchical Bayes.
Finally, ecologists often wish to make predictions based on their models but doing this in a quantitative manner can be a struggle. There seems to be a trade off between simple models that have a lot of predictive power and complex models that do a better job of describing natural processes. This is where hierarchical Bayes truly shines, allowing for predictive power even with complex models. In essence, this is because we’re building the natural structure of the system’s processes into the model. So instead of poking a domino and seeing what happens at the other end of the chain, we are aware of every domino in between and we can predict how each will affect the next.
This is just the tip of the iceberg when it comes to hierarchical Bayes and there is obviously much more to it than I have room for here. There are boundless ways in which the hierarchical Bayesian framework could be useful to ecologists and I think everyone should at least be aware of its utility. Bayesian stats shouldn’t be viewed as an alternative to classical techniques but another, quite useful tool to be pulled from our toolbox in the right situation.
*A special thanks to Tom Miller of Rice University and Kellogg Biological Station’s ELME program for opening my eyes to how useful hierarchical Bayesian models can be!
