Maths for ML / Notation

\mathbb{E}_{x\sim p}[f(x)]

expectation

Probability and statisticsBeginner

Say it “expected value of f of x, x drawn from p”

The average value you'd get if you sampled many times — weighted by probability.

Example

\mathbb{E}[\text{die roll}] = 3.5

In code

np.mean(samples)

samples.mean()

Where you’ll see it in ML

The training loss is an expectation over the data distribution, estimated by a batch average.

Don’t confuse it with

$\bar{x}$ bar Usually the mean (average) of x. Occasionally a complex conjugate.
$\sum_{i=1}^{n} x_i$ summation Add up the expression for each value of the index — it's a for-loop that accumulates.
$\int_a^b f(x)\,dx$ integral Continuous version of a sum: total area under f between a and b.
$\mu$ mu (mean) Conventionally the mean of a distribution.

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