Before you can read Maximum likelihood estimation (MLE) comfortably, you must be able to look at every squiggle on the page and hear a plain sentence. This note builds each squiggle from nothing, in the order they depend on each other. Never take a symbol on faith — if you meet it below and it isn't defined yet, that's a bug; tell me.
Picture: dots sitting on a horizontal number line; each dot carries a tag x1,x2,… and a dashed box drawn around all of them is the whole dataset.
Figure 1 — Six orange dots on a horizontal value-axis, tagged x1 through x6; a plum dashed rectangle encloses them all and is labelled X={x1,…,xn},n=6. If the image fails: imagine a ruler with six beads on it, lassoed together by a loop marked "X".
Why the topic needs it: MLE is a method for explaining data. Data is nothing but a bag of these dots, so this is the atom everything is built from.
So if you flipped a coin 10 times, n=10 and X holds all 10 results.
Picture: the dashed box in Figure 1 — the box is X, the number of dots inside is n.
Why the topic needs it: MLE scores a guess by how well it explains the entire bag X, not one dot. The count n later becomes the number of factors we multiply and the number of terms we add.
Picture: two dials that don't touch, each spitting out dots — no wire connects them.
Why the topic needs it: Without i.i.d. the joint probability of the whole dataset would be a tangled mess. i.i.d. lets us write it as one clean product (Section 8).
Two brand-new symbols live here: p(⋅) and θ. Take them one at a time.
Picture: a bell curve whose shape/position is controlled by the dial θ; the height of the curve above a point x is p(x∣θ) — how strongly the machine "votes" for that value.
Figure 2 — A teal bell curve labelled p(x∣θ). A vertical orange dashed line rises from a point x on the axis up to the curve; the dot where it meets the curve is annotated "height = how strongly the dial votes for this x". A plum note near the summit says "θ sets the shape/position". If the image fails: picture a hill; the taller the hill is directly above your data point, the happier that dial is with that point.
Why the topic needs it: This height is the currency of MLE. A good dial setting makes the curve tall exactly where your data dots landed.
Figure 3 — Two curves over the same dial-axis θ: a teal "likelihood (scaled)" and a plum "log-likelihood (scaled)". Both peak at the same orange dashed vertical line marked θ=0.7. If the image fails: two differently-shaped hills whose summits stand at the exact same spot on the ground.
Figure 4 — A teal log-likelihood hill ℓ(θ). An orange flat bar sits on the summit (slope = 0); a plum uphill tangent on the left is marked "slope > 0" and a plum downhill tangent on the right "slope < 0". If the image fails: a hill where the tangent stick tilts up on the way up, tilts down on the way down, and lies perfectly flat at the very top.
Read top-to-bottom: dots become a dataset, i.i.d. lets us multiply heights into a likelihood, the natural log tidies it into a sum, the flat-slope (or zero-gradient) condition plus a boundary check finds its peak, and argmax hands back θ^MLE.
Why the topic needs it: Keeping truth (θ) and guess (θ^) visually separate is exactly what lets us later ask honest questions like "is our guess biased?" (Bias-Variance Tradeoff) or "how jumpy is it?" (Fisher Information, Cramér-Rao Bound).
Once these atoms are solid, the parent page and its neighbours reuse them: Method of Moments is an alternative recipe for the same "find the dial" goal; EM Algorithm runs MLE when some data is hidden; Loss Functions in ML shows that minimising a loss is often maximising a likelihood in disguise; and Likelihood Ratio Test compares two dial-settings using the very L(θ∣X) built in §8. Prefer Hindi? See the Hinglish version.
