Understanding Science 2. Bayes’ Theorem © Colin Frayn,

Recap Assumptions of science a)Underlying laws b)Accurate senses c)Occam’s Razor Absolute proof –Can be achieved with mathematical claims –Difficult or impossible for scientific laws Spectrum of certainty –Science moves theories on the spectrum Scientific Theories –Empirical models –Well tested, predictive, falsifiable © Colin Frayn,

Clarifications “False” does not imply “completely wrong” –E.g. Newtonian Physics vs. Relativity –E.g. the Flat Earth theory, the Spherical Earth theory Carl Sagan’s Dragon –Can we show it doesn’t exist? –Should we bother? Predictive laws versus specific statements –“There are no dragons” © Colin Frayn,

Introduction New evidence arrives –What does that do? –Moving around the spectrum of certainty Prior knowledge –Did you see Elvis? When could we call something a “fact”? –A scientific fact is “near enough”! © Colin Frayn,

Examples On trial for murder –DNA testing –Very accurate –…but a very large population A rare disease –Rare disease or rare misdiagnosis –Intuition doesn’t help © Colin Frayn,

Organic Gravity – An Example Organic gravity ”Gravity only acts on organic things” Vs. Newtonian gravity “Gravity acts identically on every type of object” Test 1 – drop an apple –Both theories are equal Test 2 – drop a stone –Newtonian gravity wins © Colin Frayn,

In More Detail Let’s look at what we just did Test 1 didn’t really help –It didn’t differentiate –It provided equal support to each Test 2 solved the issue –Distinguished between the proposals –Provided support to Newtonian theory © Colin Frayn,

Equal Support What do we do when we cannot distinguish between two possibilities? Look at the prior probability of each Example: Diagnosing a rare disease 1.The patient has a rare disease 2.The test was wrong © Colin Frayn,

Putting it all together... © Colin Frayn, Probability of a Hypothesis given the Evidence Probability of a Hypothesis given the Evidence P ( H | E ) Depends on... 1.The support that E gives to H 2.The prior probability of H

Finally, Bayes’ Theorem © Colin Frayn, P (H | E) = P (E | H) * P (H) P (E) Posterior Support Prior

Evidential Support “How much does evidence E support hypothesis H?” –P(E|H)/P(E) Eating garlic scares away vampires –Given that I don’t see any vampires P(E) = 1 –Vampires don’t exist! P(E|H) is also 1 –So test is useless –That is, it has no differentiating power © Colin Frayn,

Non-discriminating Evidence © Colin Frayn, P ( | ) = P ( | ) * P ( ) P ( ) 1 Posterior probability is equal to the prior i.e. We’ve learned nothing whatsoever

Priors “What is the chance that our hypothesis might be true ignoring the new evidence?” –P(H) A “flat prior” means “no preference” –P(H) is the same for all hypotheses The “status quo” –E.g. “Elvis is alive” –… or any other conspiracy theory –… or and pseudoscientific claim © Colin Frayn,

Organic Gravity Revisited Dropping an apple gave no preference –P(H) = 0.5 for both © Colin Frayn, P(Newtonian | Stone Falls) = P(Stone Falls | Newtonian) * P(Newtonian) P(Stone Falls) P(Organic | Stone Falls) = P(Stone Falls | Organic) * P(Organic) P(Stone Falls)

Assumptions Assumption of completeness –Don’t have to make this assumption –Though we do need some way to calculate P(E) Assumption that the evidence was accurate –Can factor this into P(E|H) Assumption that you understand your models –Do you really know P(E|H)? © Colin Frayn,

Summary Bayes theorem allows us to update hypotheses in response to evidence It evaluates the support that evidence gives for a hypothesis It underlies all of science © Colin Frayn,