Pattern Recognition and Machine Vision

Suppose we have a probability density function for some continuous random variable: e.g. a Gaussian or Normal distribution. p(x) = 1 √ 2πσ 2 exp − (x − µ) 2 2σ 2 (1) Usually when we use this function to calculate the probability of getting the value x in the range x to x to x+δ x, we assume that the parameters σ and µ are known or given. To emphasize this point of view, the probability density is often written as p(x|θ) where θ stands for the parameters {σ , µ}. This is often read as " the probability of x given θ " and regarded as a conditional probability for obtaining x in the range x to x + δ x, given that the parameters σ , µ take the designated values. An alternative view of p(x|θ) is to regard it as a function that, if we have a piece of data, x, tells us how likely the parameter values θ = {µ, σ } are. Note: Viewed this way p(x|θ) is called the likelihood of θ with respect to the sample, x. It is not a probability. For example, it is not correctly normalized.