What are Bayes Factors and how can they be used to demonstrate a null finding of "no difference" between two groups? A prominent journal in my field recently updated its advice to authors to state:

Null findings: Authors should only report ‘no difference’ between
  conditions or lack of associations if they can demonstrate this by
  calculating Bayes Factors. A Bayes Factor of less than 0.3 would
  normally be required to be confident that there really is no
  difference or association. Otherwise null findings should be framed as
  ‘the findings were inconclusive as to whether or not a
  difference/association was present’ or some similar wording.

While I understand that in classical NHT, null findings do not demonstrate 'no difference', I am unfamiliar with Bayes Factors and their uses.
What are Bayes Factors? How can they be calculated? How should they be interpreted, including to demonstrate a null finding of "no difference" between two groups?
 A: To quote from a paper by Robert West:

The Bayes Factor (BF) is a ratio showing the degree to which a set of
  data (even just one data point), changes the probability that a given
  hypothesis (H1) is true versus an alternative hypothesis (H0). If the
  BF is 1, then the data support (or fail to support) both hypotheses
  equally and are thus completely uninformative on the matter. If the BF
  is 3 then, given these data, H1 is three times more likely to be
  correct than H0. If the BF is 1/3 then H0 is three times more likely
  to be correct than H1.
It so happens that a BF of 3 is found typically when in classical
  frequentist statistics the P-value would be 0.05 on the same data, and
  so 3 and 1/3 are used commonly as thresholds for statements that there
  is substantial evidence for one or other hypothesis being correct [3].
  For different decisions one may wish to use different thresholds.
Suppose one has undertaken a study of 300 people (150 in each group)
  comparing gradual and abrupt methods for smoking cessation and finds
  an odds ratio favouring abrupt cessation of 1.56 [95% confidence
  interval (CI) = 0.77–3.11, P = 0.22]. Thus, using the classical
  hypothesis testing approach, we would say there was no statistically
  significant difference and researchers often conclude that there is
  ‘no effect’. With a Bayesian approach we can go further. We could
  specify in advance two hypotheses: H0 as the hypothesis that the odds
  ratio in the population is 1 (i.e. there is no difference), and H1 as
  the hypothesis that there is a clinically useful and plausible
  advantage to abrupt cessation of something between 1.1 and 3.0. We can
  calculate a BF to tell us which of these is most likely from the data
  and by how much. In fact, it turns out to be 1.5, which means that H1
  is 50% more likely than H0. The appropriate conclusion, then, is that
  the data favour the hypothesis of a clinically important difference
  relative to none at all, but not strongly. This is more informative
  than simply concluding that there is no statistically significant
  difference and more accurate than asserting that there is ‘no effect’.

Functions to calculate Bayes Factors in R can be found at:
http://www.danny-kaye.co.uk/Docs/Dienes_functions.txt
