# Log likelihood

## Introduction

When evidence is entered in a Bayesian network or Dynamic Bayesian network, the Probability (likelihood)
of that evidence, denoted `P(e)`

can be calculated.

The Probability of evidence `P(e)`

indicates how likely it is that
the network could have generated that data. The lower the value, the less likely.

## Log value

The Log Likelihood `Log(P(e))`

is also reported, as the `P(e)`

can often report zero, due to underflow caused by the repeated multiplication of small probability values, using floating point arithmetic.

##### NOTE

Bayes Server uses sophisticated algorithms to detect very small log-likelihood values

An example of zero likelihood:

##### NOTE

Log Likelihood values are often used to detect unusual data, known as Anomaly detection.

## Range of values

The likelihood `P(e)`

in networks with only discrete nodes lies in the range [0, 1], therefore the log-likelihood lies in the range [-Infinity, 0]. For networks that contain one or more continuous nodes (with or without discrete nodes) the likelihood (pdf) lies in the range [0, +Infinity], therefore the log-likelihood lies in the range [-Infinity, +Infinity].

## Log-likelihood -> Probability

While log-likelihood values from the same model can be easily compared, the absolute value of a log-likelihood is somewhat arbitrary and model dependent. The `HistogramDensity`

class in the API can be used to build a distribution of Log-Likelihood values for a model which can then be used to convert log-likelihood values to a value in the range [0,1].

##### NOTE

This techniques is often used in anomaly detection applications when we wish to report the health of a system as a single meaningful value.

## Log-likelihood (Most probable)

When Most probable explanation (MPE) is used, the log-likelihood/likelihood is the same value you would get if you were to calculate it without **Most Probable explanation** on, but having evidence set according to the most probable configuration.