Classical test of inference for two-group comparisons, like the t-test, often face the problem of small effect size or borderline significance. Moreover, the issue is not so clear when we cannot reject the null.
Here I would like to discuss the following paper from Wetzels et al.: How to quantify support for and against the null hypothesis: a flexible WinBUGS implementation of a default Bayesian t test, Wetzels R, Raaijmakers JG, Jakab E, Wagenmakers EJ. Psychonomic Bulletin & Review (2009) 16 (4): 752-60.
We propose a sampling-based Bayesian t test that allows researchers to quantify the statistical evidence in favor of the null hypothesis. This Savage-Dickey (SD) t test is inspired by the Jeffreys-Zellner-Siow (JZS) t test recently proposed by Rouder, Speckman, Sun, Morey, and Iverson (2009). The SD test retains the key concepts of the JZS test but is applicable to a wider range of statistical problems. The SD test allows researchers to test order restrictions and applies to two-sample situations in which the different groups do not share the same variance.
The paper from Rouder et al., Bayesian t tests for accepting and rejecting the null hypothesis, is also available as PDF.
To my acknowledge, no implementation of the SD test is available in R.