Advanced Mathematical Methods Part II Statistics Scientific Method - - PowerPoint PPT Presentation

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Part II – Statistics Scientific Method

Mel Slater

http://www.cs.ucl.ac.uk/staff/m.slater/Teaching/Statistics/

Advanced Mathematical Methods

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Outline

Is science based on ‘facts’? The norms of scientific method What is a hypothesis? How is a hypothesis tested? Paradigm based science What happens when ‘facts’ don’t fit? Scientific revolutions Probability and statistics

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Is science based on ‘facts’?

Karl Popper Thomas Kuhn Decide what this figure represents, and then write down 5 facts about it.

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Norms of Scientific Method

Establish a ‘hypothesis’ Carry out some procedure that may

be used to test the hypothesis

• E.g. gather ‘data’

Hypothesis may be…

• Rejected
• Confirmed
• Neither rejected nor confirmed

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Examples of ‘Hypotheses’

Continuity Hypothesis (CH): Any infinite set

• f real numbers is either countably infinite or

has the same cardinality as the entire set of real numbers (G. Cantor, 1874) http://www.ii.com/math/ch/#overview

• CH been shown to be consistent with

standard set theory

• Not-CH has also been shown to be

consistent with standard set theory

Still an open question in mathematics

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Examples of ‘Hypotheses’

The Gaia Hypothesis: Earth functions as a

single organism that maintains conditions necessary for its survival.

Formulated by James Lovelock

http://www.ecolo.org/lovelock/ http://www.oceansonline.com/gaiaho.htm

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Examples of ‘Hypotheses’

Life on Mars: The planet Mars

contains living organisms.

(Martian meteorites discovered

with fossils that appear to give evidence of life).

http://www.space.com/scienceastronomy/solarsystem/mars-meteorite990930.html

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Examples of ‘Hypotheses’

Neural Correlates of Consciousness: …

An alternative hypothesis is that there are special sets of “consciousness” neurons distributed throughout cortex and associated systems. Such neurons represent the ultimate neuronal correlate

• f consciousness, in the sense that the

relevant activity of an appropriate subset

• f them is both necessary and sufficient

to give rise to an appropriate conscious experience or percept (Crick and Koch 1998).

• http://www.klab.caltech.edu/~koch/Elsevier-NCC.html

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Examples of ‘Hypotheses’

George Bush Election Victory: A

majority of people who voted in Florida voted for Kerry.

• http://www.michaelmoore.com/words/index.php?id=284

http://news.bbc.co.uk/1/hi/entertainment/arts/1896944.stm

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Examples of ‘Hypotheses’

Extra Sensory Perception: It is possible to

transfer information from the mind of one person to another without any (known) physical means of transmission.

Experiments were carried out by J. Rhine in

the 1930s on telepathy.

http://www.williamjames.com/Science/ESP.htm http://www.rhine.org/index.html

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Testing hypotheses?

Continuity

• A type of set with cardinality greater than countable but less than

the continuum might be constructed.

Gaia

• What type of evidence would be relevant?

Life on Mars

• Living organisms or evidence relating to this might be found

Neural Correlates of Consciousness

• A set of neurons identified that when knocked out always results

in ‘non-consciousness’

George Bush Election Victory

• A 100% survey might be conducted

Extra Sensory Perception

• Some examples of information transfer in the absence of all

possible physical means.

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Falsifiability

Karl Popper developed the notion

that a hypothesis must be falsifiable if it is a valid hypothesis.

‘Falsifiable’ means that it must be

possible to obtain evidence that could lead to its rejection.

http://cla.calpoly.edu/~fotoole/321.1/popper.html

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Null Hypothesis

A hypothesis is best framed in ‘null form’. A ‘null hypothesis’ in itself can never be

proven – it can only be rejected.

It is usually of the form that would take

infinite time to be able to ‘prove’.

Let’s consider each of our hypotheses in

this form….

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Null Hypotheses

• Continuity- There is no set with cardinality greater than countable

but less than the continuum.

• A type of set with cardinality greater than countable but less than the

continuum might be constructed.

• Gaia – The Earth is not a living organism.
• What type of evidence would be relevant?
• Life on Mars – There is no life on Mars
• Living organisms or evidence relating to this might be found
• Neural Correlates of Consciousness – There is no isolatable

specific set of neurons responsible for consciousness.

• A set of neurons identified that when knocked out always results in ‘non-

consciousness’

• George Bush Election Victory – George Bush won less than 50%
• f the votes cast.
• A 100% survey might be conducted
• Extra Sensory Perception – There is no way that information can

be transmitted from one brain to another without (known) physical means.

• Some examples of information transfer in the absence of all possible

physical means.

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Grand Hypotheses

Most of the hypotheses we have

considered are ‘grand’ – imply big changes in our understanding of the universe.

They challenge an orthodoxy. They are controversial. They are (except for first) easily

understandable by the general public.

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Hypotheses of ‘Normal Science’

Everyday science is small scale puzzle solving

activity (Kuhn).

Hypotheses are highly specific, non-controversial,

non-understandable except by specialists.

E.g. Kolmogorov Refined Similarity Hypothesis for

Isotropic Turbulence (www.nagare.or.jp/fdr/abstracts/15/p337_348.pdf)

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Examples of Hypotheses in VE

Locomotion: Moving through a VE by ‘walking in place’ on the

average results in a greater reported sense of presence than using a wand-based point-and-click method.

Conversational Eye Gaze: An avatar that displays natural eye-

gaze behaviour during a conversation will have greater believability that

• ne which does not.

Anxiety in Fear of

Public Speaking: People who have FOPS will exhibit anxiety when speaking in front of a virtual audience (under ‘certain conditions’).

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Normal science operates within a paradigmatic

framework (Kuhn)

A paradigm is a collection of shared beliefs,

ideas, theories, methods, problems, approaches, ways of thinking, …. typically on a grand scale, e.g.

• Galileo & astronomy
• Newtonian Physics
• Einstein & relativity

Normal science is puzzle

solving within a paradigm

Hypotheses are relative

to a paradigm.

Paradigms & Hypotheses

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Idealised Scientific Method

Simplified…

• State a hypothesis – typically in null

form – must be absolutely clear what it means, under which conditions it is supposed to hold

• Gather data – construct an experiment

– collect observations

• The data may lead to rejection of the

null hypothesis or not.

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Paradigms & Revolutions

Experimental results, data, observations typically exist that

do not fit into the paradigm

Check these examples yourself:

• Immanuel Velikovsky – challenge to orthodoxy of ancient

history and cosmology

• J. B. Rhine – interesting evidence for ‘telepathy’
• Barry Marshal & John Warren – discovered bacteria H. Pylori –

and that it was a significant cause of stomach ulcers – countered the prevailing paradigm in medicine.

When data contradicts a paradigm, it is the data that is

typically rejected, and the discoverer is often vilified and attacked by established scientists.

Sometimes a new paradigm will emerge in a ‘revolution’ that

• verthrows the existing paradigm – transition from

Newtonian to Einsteinian physics is a classic example.

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Normative Science and History

The norms of scientific method

• Hypothesis generation
• Data gathering
• Hypothesis non-rejection/rejection

An ‘ideology’ – a set of rules that are supposed to

be followed

In practice this happens only at a very small scale –

data that threatens the paradigm is usually suppressed in some way

But adherence to norms of scientific method

constitutes an idea to follow in practice

(If not your papers won’t be published anyway!) ‘Scientific truth’ is a social convention.

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Probability and Statistics

Probability theory provides a calculus of

uncertainty

Statistics is a set of methods that allow

inferences to be made from data to probability statements about hypotheses.

• A probability may be assigned to a

hypothesis.

• Data observed in relation to the hypothesis
• A new probability for the hypothesis can be

calculated as a result.

Find out about Thomas Bayes and Karl

Pearson.

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Objective of this Course

By the end of this course you will be

able to competently test hypotheses

• n the basis of multivariate data

gathered from real experiments.

This requires a working

understanding of probability and distribution theory

The fundamental ideas of Baysian

and classical statistical testing.

The use of the GLIM statistical

language for Generalised Linear Models, and also MATLAB.

… and an appreciation of the

rigours and realities of scientific method!

http://www.cartoonstock.com/directory/s/schrodinger.asp