How to gently introduce epidemiologists/public health coworkers to advanced predictive modeling? Coming from a social science and epidemiology background, my coworkers were trained on  least squares regression, logistic regression, and survival analysis. They like to see 95% confidence intervals and p-values with the parameter coefficents, and are distrustful of more current predictive tools such as neural networks, CART, bagging & boosting, as well as penalized regression techniques.
 A: I'm going to weigh in as an Epidemiologist.

I can see inertia setting in as researchers & professionals in the health care field move into middle management and beyond and are out of touch with new developments in statistics. 

First, I would strongly advise you not to assume this is simply inertia, either in the form of the discipline not wanting to adopt new techniques, or your coworkers falling out of touch with new developments in statistics. You can go to academic epidemiology conferences where new and very methodologically sophisticated work is being done, and still not necessarily find much on predictive modeling.
The hint is in the name. Predictive modeling.
Epidemiology, as a field, is not particularly interested in prediction for it's own sake. Instead, it's focus is on developing etiological explanations for observed disease patterns in a population. The two are related, but distinct, and this often leads to something of a philosophical distrust of more modern classification and prediction techniques that purely attempt to maximize the predictive impact of a model. At the extreme end of this is the people who are of the opinion that variable selection should be performed primarily with the use of something like a directed acyclic graph, which could be considered the opposite of where predictive modeling is heading. This is largely why the methodological developments in epidemiology have been concentrated in causal inference and systems models in the recent past - both are built off etiological and causal arguments, rather than prediction.
This results in it not being part of their background, not being something they encounter much in the literature, and to be perfectly frank, a high likelihood that their exposure to it has been via people who don't actually understand the problems they are trying to solve.
This, in the comments, is a perfect example:

That throws some people - the fact we're purposefully introducing bias into penalized regression to improve predictive accuracy

Very nearly every epidemiologist I know, if you made them pick, would pick a reduction in bias over an increase in accuracy.
That is not to say that it never gets brought up. There are times when predictive models do get used - often in clinical cases where the prediction of this particular patient's outcome is of considerable interest, or outbreak detection, where these techniques are useful because we don't know what's coming and can't make etiological arguments. Or when prediction really is the goal - for example, in many exposure estimation models. They're just somewhat niche in the field.
