Maths for ML / Notation

\sum_{i=1}^{n} x_i

summation

Sums and productsBeginner

Say it “sum from i equals 1 to n of x i”

Add up the expression for each value of the index — it's a for-loop that accumulates.

Example

\sum_{i=1}^{3} i^2 = 1+4+9 = 14

In code

sum(x[i] for i in range(n))  # or x.sum()

x.sum()

Where you’ll see it in ML

Losses averaged over a dataset, weighted sums inside every neuron.

Don’t confuse it with

$\prod_{i=1}^{n} x_i$ product Multiply the expression together for every value of the index.
$\sigma,\ \sigma^2,\ \sigma(z)$ sigma Lowercase σ is the standard deviation (σ² the variance) — or, as a function σ(z), the sigmoid. Capital Σ is a sum or a covariance matrix.
$\int_a^b f(x)\,dx$ integral Continuous version of a sum: total area under f between a and b.
$\mathbb{E}_{x\sim p}[f(x)]$ expectation The average value you'd get if you sampled many times — weighted by probability.
$\boldsymbol{\Sigma} = \operatorname{Cov}(\mathbf{x})$ covariance matrix Matrix whose entries say how pairs of features vary together. Same symbol as sum — context (bold, no limits) tells you which.
$\operatorname{softmax}(\mathbf{z})_i = \frac{e^{z_i}}{\sum_j e^{z_j}}$ softmax Turns a vector of any scores into positive numbers that sum to 1 — a probability distribution.

Lessons that use it

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