In probability theory and computer science, a log probability is simply a logarithm of a probability. The use of log probabilities means representing probabilities on a logarithmic scale ${\displaystyle (-\infty ,0]}$, instead of the standard ${\displaystyle [0,1]}$ unit interval.

Since the probabilities of independent events multiply, and logarithms convert multiplication to addition, log probabilities of independent events add. Log probabilities are thus practical for computations, and have an intuitive interpretation in terms of information theory: the negative expected value of the log probabilities is the information entropy of an event. Similarly, likelihoods are often transformed to the log scale, and the corresponding log-likelihood can be interpreted as the degree to which an event supports a statistical model. The log probability is widely used in implementations of computations with probability, and is studied as a concept in its own right in some applications of information theory, such as natural language processing.

Representing probabilities in this way has several practical advantages:

The logarithm function is not defined for zero, so log probabilities can only represent non-zero probabilities. Since the logarithm of a number in ${\displaystyle (0,1)}$ interval is negative, often the negative log probabilities are used. In that case the log probabilities in the following formulas would be inverted.

Any base can be selected for the logarithm.

In this section we would name probabilities in logarithmic space ${\displaystyle x'}$ and ${\displaystyle y'}$ for short:

The product of probabilities ${\displaystyle x\cdot y}$ corresponds to addition in logarithmic space.

The sum of probabilities ${\displaystyle x+y}$ is a bit more involved to compute in logarithmic space, requiring the computation of one exponent and one logarithm.