Advanced Mathematical Methods Part II – Statistics Scientific Method Mel Slater http://www.cs.ucl.ac.uk/staff/m.slater/Teaching/Statistics/ 1
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 2
Is science based on ‘facts’? Thomas Kuhn Decide what this figure represents, and then write down 5 facts about it. Karl Popper 3
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 4
Examples of ‘Hypotheses’ � Continuity Hypothesis (CH): Any infinite set of 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 5
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.oceansonline.com/gaiaho.htm http://www.ecolo.org/lovelock / 6
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 7
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 of consciousness, in the sense that the relevant activity of an appropriate subset of 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 � 8
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 9
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 10
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. 11
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 12
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…. 13
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% � of 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. 14
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. 15
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 ) 16
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 one 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’). 17
Paradigms & Hypotheses � 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. 18
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. 19
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 overthrows the existing paradigm – transition from Newtonian to Einsteinian physics is a classic example. 20
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. 21
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. 22
Objective of this Course � By the end of this course you will be able to competently test hypotheses on 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 23
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