The Modern Mathematics of Deep Learning

A prior on a hypothesis set

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: 1.2.3 Does it work on new examples?

p is a probability distribution on the hypothesis set. For a finite set, its weights are nonnegative and sum to one.

A prior assigns weights before the training sample is observed.

Source formula example

p

illustration · The example is a finite class; an unrestricted network class can be infinite and have no finite cardinality.

Symbols

$p$: A probability distribution on the hypothesis class.
$\mathcal{F}ₐ$: The network hypothesis class.
$|\mathcal{F}ₐ|$: Its cardinality when finite.

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

hypotheses = ["zero", "identity"]
probabilities = [0.25, 0.75]

Calculated here

Result expression: { hypotheses, probabilities, total: probabilities.reduce((a, b) => a + b, 0), cardinality: hypotheses.length, }

Returns

Example result

hypotheses: array (2 items): [zero, identity]
probabilities: array (2 items): [0.25, 0.75]
total: 1
cardinality: 2

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 unless a source value is specified. The example is a finite class; an unrestricted network class can be infinite and have no finite cardinality.

Source: Inline notation