Maximum Likelihood

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Introduction

This is a very widely used mathematical (probabilistic) approach used when trying to build a model from real-world observations.

A key aspect of the approach is that of conditional probability, and we decide to take the real-world observations and say that their probability is in actual fact conditional on certain values that parameters that our notional model take.

Although probabilistic, and made up of probabilities, the likelihood is not itself a formal probability because all its possibilities will not add up to 1. Therefore it is often better to talk of it as a score. This aspect is also a reason why likelihood ratios are popular, because its exact nature disappears in the division.

Formula

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle f(x_1,x_2,\ldots,x_n\mid\theta) = f(x_1\mid \theta)\times f(x_2|\theta) \times \cdots \times f(x_n\mid \theta). }

Now we look at this function from a different perspective by considering the observed values x1, x2, …, xn to be fixed "parameters" of this function, whereas θ will be the function's variable and allowed to vary freely; this same function will be called the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathcal{L}(\theta\,;\,x_1,\ldots,x_n) = f(x_1,x_2,\ldots,x_n\mid\theta) = \prod_{i=1}^n f(x_i\mid\theta). }