What does Bayesian Comparison of Groups and Posterior Interval say about my Hypothesis? I am comparing the score of two groups: A and B. The score is normally distributed and a two sample t-test yields a p-value >0.05. Therefore I have to reject the Hypothesis that there is significant difference between the mean score of both groups.
However, I also conducted a comparison of groups and posterior confidence intervals according to the Bayesian approach and I need clarification on how to properly interpret the results:
The "mean of the reference group" (A) is 208 and the "difference to the reference group" is 13 (Sigma 29). The 90% posterior confidence interval ranges from -5 to 30, the 75% posterior interval ranges from 0.4 to 26.
-Can I say that participants of group B perform 12 points better on average than participants of group A? 
-Must the hypothesis be rejected as the 90% confidence interval contains zero? Or can it be interpreted as "In 90% of the cases, participants of the experimental group score somewhere between 5 points less and 29 points higher than the control group"
-Can the 75% confidence interval be interpreted as "In 75% of the cases participants of Group B score between 0.4 and 24 points higher than participants of Group A"?
 A: If the mean of group B was larger than the mean of group A then a statement that participants in group B performed better on average than participants of group A is naturally correct. 
However, is the difference in performance likely to reflect an underlying difference in performance ability or just an accident of random sampling? (The samples are random, aren't they?) A relatively large P-value says that the difference is not very extreme relative to the statistical expectations of the model applied. That would be consistent with the accident of random sampling option, but it would also be consistent with the possibility that your sample is small relative to the size needed to expose an effect small relative to the variability. It may also mean that the statistical model is inappropriate.
A P-value greater than 0.05 implies that the null hypothesised values of the parameter of interest (presumably zero in this case) lies within a 95% confidence interval. If zero is not within the 90% confidence interval then the P-value must be greater than 0.1.
The hypothesis that the effect size is zero must be rejected when the observed P-value is greater than the pre-determined critical value (0.05 or 0.1 for you, but I can't tell which) if you are intent on obtaining the long term false positive error rate implied by that critical value. However, you may well not want that. The type of response to or report of data depends on the analytical and experimental purpose, and you have not told us about them.
Now, can a confidence interval be interpreted as you suggest? Not really, but probably well enough for your purposes. See this for the full glory of interpretation of confidence intervals: How to interpret confidence interval of the difference in means in one sample T-test?
A: 
Can the 75% confidence interval be interpreted as "In 75% of the cases participants of Group B score between 0.4 and 24 points higher than participants of Group A"?

No. The credible interval is interpreted as "we have 75% belief that the mean difference in scores between Group A and Group B lies between 0.4 and 24.0".
(Use sig-figs!)
