Hypothesis testing I

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Hypothesis testing I
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Hypotheses

• What is an hypothesis? A claim, a set of claims,
  a theory, a model.
• Distinguish
• Context of discovery
• Context of justification hypothesis testing
•  

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Hypothetico-deductive method

• Put forward an hypothesis H
• Deduce an observable consequence K from H.
• 3. Test whether K holds or not.

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Asymmetry

• Two possibilities
• K false (H false).
• K true (H ??)
• If K false, then H false.
• But NOT
• If K true, then H true.

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Observable consequence

• If K is true we can know that K.
• If K is false we can know that K is false.
• K should be directly observable (?)

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Relationship between hypotheses and observable
consequence

• Scientific hypotheses in practice never have any
  observable consequences.
• They rely on auxiliary hypotheses
  (hjälphypoteser)
• Example 
• Hypothesis (H) the temperature in the water is
  25 degrees celsius.
• Observable consequence (K) when the thermometre
  is submerged in the water it will show 25 degrees
  (-.5)
• Auxilliary hypothesis (Ah) The thermometre
  works it displays the temperature of the water
  (-.5 degrees).

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Auxilliary hypotheses

• H-D method gets the form
• Put forward an hypothesis H and auxilliary
  hypothesis Ah.
• Deduce an observable consequence K from H Ah
• Test whether K.
• Two possibilities
• K false (hence, either H or Ah false The result
  can always be explained away).
• K true (HAh gets support, but isnt verified)

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Poppers demarcationcriterium

• A scientific theory must be falsifiable.
• Problem with auxilliary hypothesis

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Background assumptions

• In practice (typically) observable consequence K
  is derived from H Ah implicit assumptions
• Example
• Ptolemaic solarsystem. Retrograde movement. Tycho
  Brahe.
• Example
• Models

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Quine/Duhem-thesis

• An observation tests our theories as a whole. We
  cannot test our theories claim by claim.
• Only partially true. If we devise a multitude of
  different kinds of experiments with the
  hypothesis H as a common factor and all the tests
  fails, chances are there is something wrong with
  H.
• But to identify what has gone wrong (which
  hypothesis/auxilliary hypothesis that is false)
  requires judgement an evaluation.

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Ad hoc-hypotheses

• An ad hoc hypothesis is an auxilliary hypothesis
  introduced to save the main hypothesis in the
  light of data.
• Situation we have observed that K, and we have
  our favourite theory H -- how can we derive K
  from H? What auxilliary hypothesis Ah is needed?
• To call an auxilliary hypothesis ad hoc is to
  judge it as unfit.
•  
•  

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Exempel

• Immanuel Velikovsky Worlds in collision the
  world has been subject, at various stages in its
  history, to cosmic disasters produced by near
  collisions with massive comets. One of these
  comets (that went on to become Venus) passed by
  during the Israelites captivity in Egypt and
  caused the parting of the red sea.
•  
• testable consequence every group of people
  would have noticed this happening, and if they
  were literate, would have recorded it. However
  many literate communities failed to note anything
  and Velikovsky attributed this to collective
  amnesia caused by the fact that the experience
  was so horrendous and traumatic.

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The scientist is encouraged by Natures YES, but
not discouraged by its NO (Lakatos)
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How avoid ad hoc hypotheses?

• Two criteria 
• Main criterium H Ah should have observable
  consequences over and above the observable
  consequence that caused the problem.
• Secondary criterium some of these consequences
  should already be verified.

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Support

• Not a matter of logic an evaluation.
• The hypothesis Every R is a B is falsified by
  one R that isnt a B.
• The hypothesis Every R is a B gets support from
  an R that is a B.

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The paradox of the Raven

• A white swan supports the hypothesis that all
  ravens are black
• If H and H are equivalent, then H gets support
  from K if and only if H gets support from K.
• Every R is a B is equivalent to Every non-B is
  a non-R
• A white swan supports the hypothesis that
  everything that isnt black, isnt a raven.
• So a white swan supports the hypothesis that all
  ravens are black.

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Support Repeating Experiments

• Example test Newtons theory of gravity by
  dropping a stone from the leaning tower of Pisa
  over and over and over again
• Each new test gives less and less support.
  Paradox?
• One answer What one is really testing is a
  restricted version of the hypothesis Newtons
  theory as restricted to Pisa (but isnt every
  test a restriction of the hypothesis)  

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Support Good Data

• Good Data
• A precise prediction of a phenomenon that we
  would not expect if the hypotheses is false.
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•  

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• Neptune was found when astronomers John Couch
  Adams and Urbain Jean Joseph Leverrier calculated
  the position of a new planet which they thought
  was altering the orbit of Uranus. When Neptune
  was first observed by German astronimer Johann
  Gottfried Galle in 1846. Even after the discovery
  of Neptune the orbits of Uranus and Neptune were
  still diferent from what astronomers thought they
  should be. This led to the search for yet another
  planet.
• The astronomers were surprised when they
  discovered Pluto, as they were expecting a planet
  much larger than the earth. But Pluto is only
  3000km in diameter, smaller than what was
  previously thought to be the smallest planet in
  our solar system, Mercury. With the discovery of
  Pluto, both Uranus and Neptune are following
  their predicted path.

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Support Data too good to be true

• Data too good to be true
• Data supports the hypothesis too well, with
  respect to the measurment errors one could
  expect.
• Example  
• Mendels results as support for his theory of
  genetic inheritance. (??? Fisher, contested by
  Seidenfeld)

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• An hypothesis H has strong support
• If there is a varied collection of supporting
  instances K.
• At least some of these cannot be explained by
  alternative hypotheses.
• Some of the supporting instances have been
  acquired after the hypothesis was formulated.
• H is simple and can be integrated with other
  theories (coherence)
• H has a high explanatory value and can be used
  to predict new phenomena.  
• NOTA BENE support not absolute, we compare a
  theory with other theories (intersubjektivitet)