The Modern Mathematics of Deep Learning

A Bayes-optimal predictor

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: 1.2.2 Can the network express the answer?

f* is a predictor that minimizes population risk over the allowed measurable predictors.

Its risk gives a reference for approximation and learning error.

Source formula example

f^{*}

illustration · The example uses squared loss and a constant input, for which the conditional mean is optimal; other losses have other Bayes rules.

Symbols

$f*$: A Bayes-optimal predictor.
$ℛ$: Population risk.

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

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

Calculated here

Result expression: { targets, probabilities, fStar, risk }

fStar = targets.reduce((s, y, i) => s + y * probabilities[i], 0) Uses: targets, probabilities
risk = targets.reduce((s, y, i) => s + probabilities[i] * (fStar - y) ** 2, 0) Uses: targets, probabilities, fStar

Returns

Example result

targets: array (2 items): [0, 2]
probabilities: array (2 items): [0.25, 0.75]
fStar: 1.5
risk: 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 are invented. The example uses squared loss and a constant input, for which the conditional mean is optimal; other losses have other Bayes rules.

Source: Inline notation