The Modern Mathematics of Deep Learning

Conditional expectation

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.

Prediction rules and layers · Describe or apply prediction rules, layers and parameters.

The project uses a linear rule with two features. The fitted weights produce scores or normalized mass.

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: 1.2 Learning from examples

The conditional mean of Y given X = x averages possible targets using their conditional probabilities.

For squared loss it is a standard optimal prediction when the relevant conditions hold.

Source formula example

f^{*}(x)={\mathbbm{E}}[Y|X=x]

illustration · This finite example specifies the conditional probabilities directly; general conditional expectations need additional measure-theoretic definitions.

Symbols

$X$: A random input.
$Y$: A real target.
$x$: The input value being conditioned on.
$f*$: The conditional mean prediction.

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

x = 0
targets = [1, 3]
probabilities = [0.25, 0.75]

Calculated here

Result expression: { x, conditionalMean: targets.reduce((s, y, i) => s + y * probabilities[i], 0) }

Returns

Example result

x: 0
conditionalMean: 2.5

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. This finite example specifies the conditional probabilities directly; general conditional expectations need additional measure-theoretic definitions.

Source: Inline notation