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Linear Regression (Linear models for regression)
linear models?
Input data, features, basis functions
Maximum likelihood and least squares
Geometric intuition
Regularised least squares
Multiple outputs
Bias-variance decomposition
The relation between MLE and linear models.
least squares, Lagrange multipliers, multiple
ways of looking at

Why linear models?
– it saves lives
Image and table from wikipedia https://www.pregnancybirthbaby.org.au/apgar-score

Everything should be made as simple as possible, but not simpler. —
Occam’s razor, also spelled Ockham’s razor, also called law of economy or law of
parsimony, principle stated by the Scholastic philosopher William of Ockham (1285–1347/49)
that pluralitas non est ponenda sine necessitate, “plurality should not be posited without
necessity.” The principle gives precedence to simplicity: of two competing theories, the simpler
explanation of an entity is to be preferred. The principle is also expressed as “Entities are not to be
multiplied beyond necessity.”
https://www.britannica.com/topic/Occams-razor

The Lancet, 2003
https://sfar.org/scores2/snap22.php https://www.sciencedirect.com/science/article/abs/pii/S0140673603133971

Linear models and likelihood
Why linear
regression?
But what if ?

Conventions in [Bishop 2006]

Linear models and likelihood
also (3.8)

Computing the likelihood
= … (see next page)

Computing the likelihood

… first rewrite
the error function

stationary
The normal equation:

Obtaining and interpreting MLE results

The normal equation:
Numerical difficulties when
SVD or regularisation will help.
is ill-conditioned.
Cool geometric derivation of the normal equation

Sum-of-squares error

linear models?
Input data, features, basis functions
Maximum likelihood and least squares
Geometric intuition, sequential
Regularised least squares
Multiple outputs
Bias-variance decomposition

Regularised least squares
also (1.4)

Regularisation by

multipliers (appendix E)
The first encounter in SML – we’ll see it again
objective function equality constraint
maximize f(x) subject to g(x)=0
in kernel methods.

Multiple regression outputs

linear models?
Input data, features, basis functions
Maximum likelihood and least squares
Geometric intuition
Regularised least squares
Multiple outputs
Bias-variance decomposition

Bias-variance: the
What are the sources of different dots
What remains constant: learning target, model specification, learning algorithm.
What changes:
Data (subsets), randomness in learning (including but are not limited to initialisations), …
https://www.kdnuggets.com/2016/08/bias-variance-tradeoff-overview.html

Expected square loss
(See Sec 1.5.5)

Generalising squared loss
to Mikowski distances

introduce data D
Q: how does one know to add+subtract E_D[y(x; D)] ?

Unbiased estimators

The bias-variance
decomposition

“explainable models”? Interpretable models
Dawes, .. “The robust beauty of improper linear models in decision making.”
American Psychologist
34 (1979): 571-582.
Ustun, Berk and . “Supersparse linear integer models for optimized medical scoring systems.”
(2015): 349-391.
Rudin, Cynthia. “Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead.”
Machine Learning 102
Nature Machine Intelligence
1 (2019): 206-215.
2022 Squirrel Prize in AI
https://aaai.org/Awards/squirrel-ai-award.php
for Humanity
For pioneering scientific work in the area of interpretable and
transparent AI systems in real-world deployments, the advocacy
for these features in highly sensitive areas such as social justice
and medical diagnosis, and serving as a role model for researchers
and practitioners.
Video: https://www.youtube.com/watch?v=PwLN5irdMT8

super-sparse linear models

Linear models for
linear models?
Input data, features, basis functions
Maximum likelihood and least squares
Geometric intuition
Regularised least squares
Multiple outputs
Bias-variance decomposition
The relation between MLE and linear models.
regression 1
least squares, Lagrange multipliers, multiple
ways of looking at

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