Bayesian inference on possibly-non-linear effects In my field, it is occasionally the case that we want to evaluate the degree to which some variable, Y, might be influenced by another variable, X, where X is measured across a range of continuous values, and we are not sure whether to assume that the effect of X on Y forms a linear relationship or some sort of non-linear relationship. Previously, I found generalized additive modelling (GAM) very useful in such cases, because it permitted me to evaluate whether there was a relationship between X and Y without being sure of the precise form of that relationship because GAM uses data-driven methods to find functions that best fit the data (linear or otherwise). I'd typically achieve evaluation by computing likelihood ratios comparing nested models. However, more recently I've been trying to move to the Bayesian statistical framework, and I no longer know what to do with such cases of possibly-non-linear effects. Is there any standard Bayesian approach to this scenario? 
 A: Since GAM are nonparametric models, you will need to look at the literature on Bayesian Nonparametrics in order to find analogous models. This theory might be slightly more difficult to digest, though, given that the priors have to set on infinite dimensional spaces.
If you are brave enough to dig into this area, I would recommend the following book as a reference point to find other relevant references:

Hjort et al. (2010). Bayesian Nonparametrics. Cambridge University Press, Cambridge.

A: "This scenario" possibly refers to one of two things: a test of whether Y is associated with X (with no assumptions of their functional relationship) or a test of whether that relationship is linear. The Bayesian framework is much better for arguing in favor of one model versus another, so this seems adequately suited to the latter problem.
It's worth pointing out that the analogous Bayesian approach to your frequentist LRT inference is valid as well. This involves the basic linear regression model with normal priors on parameters and an inverse gamma prior on the residual variance. Estimation of posterior distributions of parameters are obtained via Gibbs sampling using a non-informative prior on the higher order terms. This allows you to perform Bayesian inference on adjusted higher order effects (quadratic or exponential), as with the likelihood based or asymptotic frequentist theory. This also averts need for Bayesian non-parametrics which can be a contentious issue.
GAMs have an unknown number of priors. You can show that this leads to some hairy issues in high dimensions, i.e. "The curse of dimensionality" which is also a problem with the frequentist GAM. Gibb's sampling, then, is a nightmare, with the highly irregular likelihood function on a highly irregular parameter space.
If I were forced to approach the problem of estimating an X Y association of any functional form in the Bayesian framework, I would take the model selection approach with a finite class of very flexible models, such as smoothing splines, and evaluate the BIC for posterior weights on those models, including the null model. If the "Y only" model is determined to be more predictive of the data generating process given the observed data, I would infer that we have high belief that X is not associated with Y.
