"Fully Bayesian" vs "Bayesian" I have been learning about Bayesian statistics, and I often have read in articles 

"we adopt a Bayesian approach"

or something similar. I also noticed, less often: 

"we adopt a fully Bayesian approach"

(my emphasis). Is there any difference between these approaches in any practical or theoretical sense ? FWIW, I am using the package MCMCglmm in R in case that is relevant.
 A: "Bayesian" really means "approximate Bayesian".
"Fully Bayesian" also means "approximate Bayesian", but with less approximation.
Edit: Clarification.
The fully Bayesian approach would be, for a given model and data, to calculate the posterior probability using the Bayes rule 
$$
p(\theta \mid \text{Data}) \propto p(\text{Data} \mid \theta)p(\theta) \>.$$ 
Except for very simple models, this has too large computational complexity, and approximations are necessary. More accurate approximations, such as using MCMC with Gibbs sampling for all parameters $\theta$, are sometimes called "Fully Bayesian". Less accurate approximations, such as using point estimation for some parameters, cannot be called "Fully Bayesian". Some approximate inference methods are in-between, like Variational Bayes or Expectation Propagation, and they are sometimes (rarely) also called "Fully Bayesian".
A: As a practical example:
I do some Bayesian modeling using splines.  A common problem with splines is knot selection.  One popular possibility is to use a Reversible Jump Markov Chain Monte Carlo (RJMCMC) scheme where one proposes to add, delete, or move a knot during each iteration.  The coefficients for the splines are the Least Square estimates.
Free Knot Splines
In my opinion this makes it only 'partially Bayesian' because for a 'fully Bayesian' approach priors would need to be placed on these coefficients (and new coefficients proposed during each iteration), but then the Least Squares estimates do not work for the RJMCMC scheme, and things become much more difficult.
A: I would add a characterization that has not been mentioned so far.  A fully Bayesian approach "fully" propagates the uncertainty in all the unknown quantities through the Bayes theorem.  On the other hand, Pseudo-Bayes approaches such as empirical Bayes do not propagate all the uncertainties.  For example, when estimating posterior predictive quantities, a fully Bayesian approach would make use of the posterior density of the unknown model parameters to obtain the predictive distribution for the target parameter.  An EB approach would not account for the uncertainty in all the unknowns - for example, some of the hyper-parameters may be set to particular values, thus underestimating the overall uncertainty.  
A: The terminology "fully Bayesian approach" is nothing but a way to indicate that one moves from a "partially" Bayesian approach to a "true" Bayesian approach, depending on the context. Or to distinguish a "pseudo-Bayesian" approach from a "strictly" Bayesian approach.
For example one author writes: "Unlike the majority of other authors interested who typically used an Empirical Bayes approach for RVM, we adopt a fully Bayesian approach" beacuse the empirical Bayes approach is a "pseudo-Bayesian" approach. 
There are others pseudo-Bayesian approaches, such as the Bayesian-frequentist predictive distribution (a distribution whose quantiles match the bounds of the frequentist prediction intervals). 
In this page several R packages for Bayesian inference are presented. The MCMCglmm is presented as a "fully Bayesian approach" because the user has to choose the prior distribution, contrary to the other packages. 
Another possible meaning of "fully Bayesian" is when one performs a Bayesian inference derived from the Bayesian decision theory framework, that is, derived from a loss function, because Bayesian decision theory is a solid foundational framework for Bayesian inference. 
A: I think the terminology is used to distinguish between the Bayesian approach and the empirical Bayes approach.  Full Bayes uses a specified prior whereas empirical Bayes allows the prior to be estimated through use of data.
A: I would use "fully Bayesian" to mean that any nuissance parameters had been marginalised from the analysis, rather than optimised (e.g. MAP estimates).  For example a Gaussian process model, with hyper-parameters tuned to maximise the marginal likelihood would be Bayesian, but only partially so, whereas if the hyper-parameters defining the covariance function were integrated out using a hyper-prior, that would be fully Bayesian.
