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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 ...
The log-odds function of probabilities is often used in state estimation algorithms [11] because of its numerical advantages in the case of small probabilities. Instead of multiplying very small floating point numbers, log-odds probabilities can just be summed up to calculate the (log-odds) joint probability. [12] [13]
A graph showing how the log odds ratio relates to the underlying probabilities of the outcome X occurring in two groups, denoted A and B. The log odds ratio shown here is based on the odds for the event occurring in group B relative to the odds for the event occurring in group A.
Any of the probabilities could have been selected to be so defined. This special value of n is termed the "pivot index", and the log-odds (t n) are expressed in terms of the pivot probability and are again expressed as a linear combination of the explanatory variables:
In probability theory, odds provide a measure of the probability of a particular outcome. Odds are commonly used in gambling and statistics.For example for an event that is 40% probable, one could say that the odds are "2 in 5", "2 to 3 in favor", or "3 to 2 against".
Interpreting negative log-probability as information content or surprisal, the support (log-likelihood) of a model, given an event, is the negative of the surprisal of the event, given the model: a model is supported by an event to the extent that the event is unsurprising, given the model.
Beta(0,0), the beta distribution for α=0, β=0 (uniform distribution on log-odds scale). The logarithmic prior on the positive reals (uniform distribution on log scale). [citation needed] These functions, interpreted as uniform distributions, can also be interpreted as the likelihood function in the absence of data, but are not proper priors.
The name Log5 is due to Bill James [1] but the method of using odds ratios in this way dates back much farther. This is in effect a logistic rating model and is therefore equivalent to the Bradley–Terry model used for paired comparisons , the Elo rating system used in chess and the Rasch model used in the analysis of categorical data.