Deviance as a difference

Peng Yu writes:

On page 180 of BDA2, deviance is defined as D(y,\theta)=-2log p(y|\theta). However, according to GLM 2/e by McCullagh and Nelder, deviance is the different of the log-likelihood of the full model and the base model (times 2) (see the equation on the wiki webpage). The english word ‘deviance’ implies the difference from a standard (in this case,
the base model). I’m wondering what the rationale for your definition of deviance, which consists of only 1 term rather than 2 terms.

My reply:

Deviance is typically computed as a relative quantity; that is, people look at the difference in deviance. So the two definitions are equivalent.