Over the years, I got asked about why acoustic score could be a positive number all the time. That occasionally lead to a kind of big confusion from beginner users. So I write this article as a kind of road sign for people.
Acoustic score per frame is essentially the log value of continuous distribution function (cdf). In Sphinx's case, the cdf is a multi-dimensional Gaussian distribution. So Acoustic score per phone will be the log likelihood of the phone HMM. You can extend this definition to word HMM.
For the sign. If you think of a discrete probability distribution, then this acoustic score thingy should always be negative. (Because log of a decimal number is negative.) In the case of a Gaussian distribution though, when the standard deviation is small, it is possible that the value is larger than 1. (Also see this link). So those are the time you will see a positive value.
One thing you might feel disharmonious is the magnitude of the likelihood you see. Bear in mind, Sphinx2 or Sphinx3 are using a very small logbase. We are also talking about a multi-dimensional Gaussian distribution. It makes numerical values become bigger.
My answer on the Sphinx Forum