The Modern Mathematics of Deep Learning

Expectation and variance

Penguin learning project

The project uses 342 complete Palmer penguin records: 240 training-pool and 102 held-out rows. The selected training rows determine feature scaling and model parameters. Source CSV

This project trains weights with softmax, unlike the homepage lab, which uses nearest species averages. Same penguin measurements; different classification methods. Predictions can differ.

Probability and expectation · Describe random variables and their averages under a law.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

Consider this concept alongside the project; its source assumptions may differ. These observations do not prove a theorem.

Input: 12 × 2; Held-out evaluation: 102 · #58 · 40.6 mm, 193 mm. Known answer: Adélie; Prediction: Adélie · [≈0.7383, ≈0.2444, ≈0.01737].

Used in these source sections: 1.1 A note on symbols · 1.2 Learning from examples

Expectation is the probability-weighted average. Variance is the expected squared distance from that average.

These quantities describe the center and spread of a distribution.

Source formula example

{\mathbbm{E}}[X]

illustration · This invented finite example illustrates the notation; it does not establish a general theorem.

Symbols

$X$: A real random variable with finite second moment.
$𝔼[X]$: Its expectation.
$𝕍[X]$: Its variance.

Source formula example

This source example uses its own inputs and assumptions. Editing it does not change the project.

Edit a value or the code, then run the example. This is a finite demonstration, not a proof of the paper’s theorem.

This example’s data flow

Initial values in code

values = [0, 2]
probabilities = [0.25, 0.75]

Calculated here

Result expression: { mean, variance }

mean = values.reduce((s, x, i) => s + x * probabilities[i], 0) Uses: values, probabilities
variance = values.reduce((s, x, i) => s + (x - mean) ** 2 * probabilities[i], 0) Uses: values, mean, probabilities

Returns

Example result

mean: 1.5
variance: 0.75

JavaScript

Read numbers from input and return the result. JavaScript basics.

Result

Enable JavaScript to run this example.

Technical statement

The numerical values in this example are invented. This invented finite example illustrates the notation; it does not establish a general theorem.

Source: Inline notation