What is the difference between a Frequentist approach with meta-analysis and a Bayesian approach? Say I am performing an analysis looking at a particular health measure. I am interested in the difference in that measure between patients and controls and whether or not the difference is different from 0. There have been studies in the past looking at my same research question and health measure, but in different samples of patients.
In my Bayesian analysis I would build up a prior distribution based on the previous studies incorporating the mean difference and standard error. 
Please forgive me if this a newbie question as I am newly learning Bayesian stats, but in what ways would the results from my Bayesian analysis differ from the results I would obtain using an inverse variance weighted meta-analysis to combine the mean difference estimates from the prior studies with my current data?
 A: There are ample references on this question in statistical analysis at large, and in meta-analysis.
For instance, have a look here:
Dohoo I, Stryhn H, Sanchez J.Evaluation of underlying risk as a source of heterogeneity in meta-analyses: a simulation study of Bayesian and frequentist implementations of three models. Prev Vet Med. 2007 Sep 14;81(1-3):38-55. Epub 2007 May 2. 
Bennett MM, Crowe BJ, Price KL, Stamey JD, Seaman JW Jr.Comparison of Bayesian and frequentist meta-analytical approaches for analyzing time to event data. J Biopharm Stat. 2013;23(1):129-45. doi: 10.1080/10543406.2013.737210. Hong H, 
Carlin BP, Shamliyan TA, Wyman JF, Ramakrishnan R, Sainfort F, Kane RL. Comparing Bayesian and frequentist approaches for multiple outcome mixed treatment comparisons. Med Decis Making. 2013 Jul;33(5):702-14. doi: 10.1177/0272989X13481110. Epub 2013 Apr 2.
Biggerstaff BJ, Tweedie RL, Mengersen KL. Passive smoking in the workplace: classical and Bayesian meta-analyses. Int Arch Occup Environ Health. 1994;66(4):269-77.
The following passage from the abstract of Biggerstaff et al is particularly interesting: 

...the approximations arising from classical methods appear to be
  non-conservative and should be used with caution. The Bayesian
  methods, which account more explicitly for possible inhomogeneity in
  studies, give slightly lower estimates again of relative risk and
  wider posterior credible intervals, indicating that inference from the
  non-Bayesian approaches might be optimistic.

If you are interested in my personal opinion, Bayesian approaches are typically more flexible but more computationally or theoretically complex. In addition, the frequentist approach is based on the tricky concept of hypothesis testing and type I/II errors, whilst the Bayesian approach enables direct probability statements. Finally, Bayesian analysis forces you to explicitly acknowledge your assumptions. 
Anyway, I would caution against a meta-analysis in which Bayesian and frequentist approaches are quite conflicting.
