maximum likelihood estimation
/ MAK-sih-mum LYKE-lee-hood es-tih-MAY-shun /
Maximum likelihood estimation is a recipe for picking the model settings that make your observed data look least surprising. The logic is detective-like: you have evidence (the data) and several suspect explanations (possible parameter values), and you crown whichever explanation would most plausibly have produced exactly what you saw. In short, choose the parameters under which the data you actually got was the most probable.
Suppose you flip a bent coin 10 times and get 7 heads. What is the coin's hidden bias? MLE answers: the bias that makes '7 heads out of 10' as likely as possible — which works out, sensibly, to 0.7. You write down the probability of your data as a function of the unknown parameter, then turn the dial until that probability peaks. In practice people maximize the logarithm of that probability (the log-likelihood), because turning a product of many small numbers into a sum makes both the math and the computer far happier.
This principle quietly underlies a huge swath of machine learning. Training many models — from logistic regression to neural networks — is maximum likelihood in disguise, and minimizing the popular cross-entropy loss is exactly equivalent to maximizing likelihood. So when a classifier 'learns,' it is often just hunting for the parameters that best explain the training data. The honest limits: MLE can overfit, clinging too tightly to the data it happened to see, and it stays silent on how confident you should be — it hands you a single best guess with no built-in error bars. Add a prior belief and you cross over into the Bayesian world.
Flip a bent coin 10 times and see 7 heads. If the coin's true heads-probability were 0.5, getting 7 heads is possible but not the most likely count. Try 0.6, 0.7, 0.8 and compute how probable '7 of 10' is under each; the probability peaks exactly at 0.7. So the maximum likelihood estimate of the coin's bias is 0.7 — the data's own best guess about itself.
Seven heads in ten flips: the bias that makes that exact outcome most probable is 0.7 — that's the MLE.
Likelihood is not the same as probability: probability asks how likely the data is given fixed parameters, while likelihood flips it to ask which parameters best explain fixed data. Minimizing cross-entropy loss, the standard choice for classifiers, is mathematically the very same thing as maximum likelihood.