The Modern Mathematics of Deep Learning

Independent random sample

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.

Observations and sampling · Describe examples, targets and samples.

The selected rows supply measurements and known answers to training; held-out IDs remain separate.

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 · 1.2.3 Does it work on new examples? · 1.3 The questions ahead

ℙZ is the distribution of a data example Z. Its m-fold product is the joint distribution of m independent examples with that same distribution.

This distinguishes a random sample from one observed sample.

Source formula example

{\mathbbm{P}}_{Z}

illustration · This enumerates the exact joint law for two finite-valued examples; it does not draw random samples.

Symbols

$Z$: A random data example.
$ℙZ$: Its probability law.
$m$: A positive integer sample size.
$S$: A random sample.
$ℙZᵐ$: The product law for m independent examples.

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, 1]
p = [0.25, 0.75]

Calculated here

Result expression: { joint }

joint = values.flatMap((a, i) => values.map((b, j) => ({ sample: [a, b], probability: p[i] * p[j] })), ) Uses: values, p

Returns

Example result

joint: array (4 items): [object (2 items), object (2 items), object (2 items), object (2 items)]

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 enumerates the exact joint law for two finite-valued examples; it does not draw random samples.

Source: Inline notation