Bayesian t-test assumptions I know that the traditional independent t-test assumes homoscedasticity (i.e., equal variances across groups) and normality of the residuals.
They are usually checked by using levene's test for homogeneity of variances, and the shapiro-wilk test and qqplots for the normality assumption.
Which statistical assumptions do I have to check with the bayesian independent t test? How may I check them in R with coda and rjags or winbugs?
 A: The traditional t test assumes normality and homogeneity of variance because those assumptions make derivation of the sampling distribution of t mathematically tractable, and frequentists want the sampling distribution so they can compute a p value. 
But if your data aren't well described by equal-variance normal distributions, then the best-fitting parameter values don't really tell you much, and the p value might be way off too. (You can fit a hat to your foot, and find the best fitting hat size and the probability that a null hat would produce sizes as big as your foot, but that information about hats is not a very useful description of your foot.)
Bayesian approaches to comparing groups also must make model assumptions, of course. If the Bayesian model assumes equal-variance normal distributions, it too will not be a very good description of heterogeneous non-normal data. Now to your question: How do you check whether the model is a good description? This is usually called a posterior predictive check, because you check whether the data appear to be well described by the model when set at the posterior parameter values. Of course, in the Bayesian world, you don't conduct the test with p values (IMHO, see this article [Kruschke, J. K. (2013). Posterior predictive checks can and should be Bayesian: Comment on Gelman and Shalizi, ‘Philosophy and the practice of Bayesian statistics’. British Journal of Mathematical and Statistical Psychology, 66, 45-56. doi:10.1111/j.2044-8317.2012.02063.x].
Fortunately, Bayesian models implemented in software like JAGS or Stan are very flexible, so if your data don't match the assumptions of traditional models, you instead use a model that better describes your data. In the case of two groups, it's easy to assume heavy-tailed distributions with unequal variances. For an extensive description with accompanying software in JAGS, see this page.
Finally, Bayesian "null hypothesis testing" is done by computing a Bayes factor for model comparison. The models in the model comparison must make assumptions about normality or not, homeogeneous variance or not, etc. In any case, the Bayes factor by itself tells you only which of the models is least bad, not whether any of the models is any good. You still need to do a posterior predictive check that involves looking at the detailed predictions of the winning model.
A: Heteroskedasticity is a different type of problem in Bayesian thinking than in Frequentist thinking.  The Frequentist is just concerned with the sampling distribution of some statistic, in this case, the distribution of the sample mean.  The t-test is the sampling distribution of $\bar{x}$ when $\sigma^2$ is fixed but unknown.  There are an infinite number of problems that can be solved with a t-test, while the same issue would be solved by many different Bayesian models.
If a person using Bayesian methods knew that there was homoskedasticity or was willing to behave as if there were, then the likelihood function would just be the normal distribution.  On the other hand, if there was a suspicion of heteroskedasticity then the person using a Bayesian method, they have to model the form of the heteroskedasticity.  The person using Frequentist methods only cares that it exists, but does not care how it comes about.  The person using Bayesian methods has to model why it exists and how it exists.  As there is not a unique solution as to the cause it could vary with the mean, it could vary with time, it could vary with a specific variable.
The person using a Bayesian method would construct two models, one that is homoskedastic and one that is not and run them jointly.  Let's imagine that the real concern is whether $\mu\ge{5}$ is true.  Then two things are true.  The first is that while the researcher wants an accurate model, the researcher also does not care at all what the variance is or if it changes.  It is a nuisance parameter and nothing more.  The second is that the researcher may behave differently if there is a clear answer to the question than if there is no clear answer.
For the first one, Bayesian methods use marginalization to remove the impact of having multiple models and having parameters you do not care about.  To have an example of this, let's assume that the follow parameters are being estimated:


*

*$\mu$ for the center of location, regardless of model

*$\sigma^2$ for the variance of the homoskedastic model

*$\gamma^2$ as one component of the variance of the heteroskedastic model

*$\lambda$ as the factor triggering heteroskedasticity, which may be a function itself

*$\mathcal{M}_1$ to designate the homoskedastic model as a parameter

*$\mathcal{M}_2$ to designate the heteroskedastic model as a parameter.


It is important to note that Bayesian methods treat separate models as parameters themselves with probabilities to be estimated.
The posterior density function before marginalization would be $$\Pr(\mu;\sigma^2;\gamma^2;\lambda;\mathcal{M}_1;\mathcal{M}_2|X),$$ where $X$ is the data.  This would be a five dimension density function. since $\Pr(\mathcal{M}_1)+Pr(\mathcal{M}_2)=1$.  When you marginalize you would remove anything you are not concerned with.  After marginalization, you would end up with:  $$\Pr(\mu\ge{5}|X)=\int_5^\infty\sum_{i=1}^2\int_0^\infty\int_0^\infty\int_0^\infty\Pr(\mu;\sigma^2;\gamma^2;\lambda;\mathcal{M}_i|X)\mathrm{d}\sigma\mathrm{d}\gamma\mathrm{d}\lambda\mathrm{d}\mu.$$
You need to state assumptions about the true value of $\mu,\sigma,\gamma,$ and the relative probabilities that $\mathcal{M}_1$ is true relative to $\mathcal{M}_2$.  You also need to state the form of the likelihood function so that it is clear how the heteroskedasticity is generated in the first place.  Any statistical package using Bayesian methods could model this.  You may have to do some manual coding.
The second part mentioned above is that you may be interested in not only the hypothesis but secondarily with whether heteroskedasticity is present or not.  In that case, you would marginalize out the center of location and the variance terms.  Imagine that the probability that the data was homoskedastic was 99.99%, then you may just drop the results of the other model and treat it as non-existent.  In that case, you would marginalize out the variance.  On the other hand, imagine that there was a 54% chance the variance was fixed and a 46% chance that it varied in some manner.  In that case, you definitely would want to do the procedure above to properly weight the risks that you have the wrong model.
