lecture 10 Flashcards
The problems with NHST
1) does not tell us what is probably true
(scientists and ordinary people are interested in testing hypotheses because they want to know which hypotheses are true)
But NHST and p-values do not indicate how probable a hypothesis is
At best, NHST and p-values help us find out what we should not believe
2) Neglects prior knowledge
Scientists and ordinary people often have a fairly good idea of how the world works before testing
BUT NHST does not allow us to take account of any information we might have for/against the truth of a hypothesis prior to making observations
3) Inconclusive when we cannot reject Null
Scientists and ordinary people are interested in evidence that could also confirm the null (default) hypothesis, not just disconfirm it
BUT NHST does not give us evidence for the Null hypothesis when we fail to reject it
Why not?
If H, the O
O
Therefore H
is deductively invalid
Bayesian statistics steps
Formulate competing hypotheses and assign a probability to them
Gather data about random variable of interest, and evaluate to what degree observed data (dis)confirm hypotheses
Inductively update probabilities of hypotheses given the evidence
Basic ideas Baesian stats
Statistical version of inductive method
Observed data provide evidence for or against hypotheses
Degrees of belief in truth of hypotheses should change based on the evidence (subjectivist interpretation probability)
Evidence for as probability raising
An observation counts as evidence for (against) a hypothesis if it raises (lowers) the prbability of the hypothesis
Example
P(Matteo healthy | Neg result) > P(Matteo healthy)
P(Matteo healthy | Pos result) < P(Matteo healthy)
How should degrees of belief change in the ligh of the evidence
P(HIO) = P(OIH) * P(H)/P(O)
Conditional probability of observation O given that hypothesis H is true P(OIH)
Prior probability of hypothesis H P(H)
Posterior probability of hypothesis H given observed evidence O P(HIO)
Evidence for as proability raising
P(HIO) > P(H) - observation O confirms H
P(HIO) <P(H) - observation O disconfirms H
Bayesian hypothesis testing 1
calculate posterior probabilities for a set of competing hypothesis under test
Compare posterior probabilities to eacho ther to identify which hypothesis is more likely, given observations
new degree of belief in hypothesis H given the evidence should be equal to the posterior probability of H: Pnew(H) = P(HIO)
Bayesian hypothesis testing 2
Bayes factor
ratio of probabilty of the observations given one hypothesis H1 to the probabilty of the observation given a second hypothesis H2
P(OIH1)/P(OIH2)
Advantages of bayesian stats vs NHST
Indicates what we should believe is probably true (or false) based on the evidence (e.g. whether I should get worried based on the positive test)
Allows us to account of our prior knowledge of hte world (e.g. information about base rates of the disease)
Enables us to determine by how much the observed data confirm or disconfirm competing hypothesis
E.g. evidence bears on the hypothesis that I have the disease and that I dont have the disease
Problems with bayesian stats
1) choice of prior probabilities
If no objective criteria to determine common priors, then different subjective degrees of belief in different hypotheses are warranted
if different prior beliefs are warranted, then this may lead to diverging conclusions and disagreement
Not always the right approach to hypothesis testing
in some field like archaeology, abductive reasoning seems more apt for inquiry than bayesian stats, since probabilistic infomration may be absent, but explanatory considerations present
Bayesian replies to disadvantage
we do often have shared background knowledge to determine common priors
Different choices of priors may be good
Different assumptions become transparent
Enables rational disagreement
Enables convergence of initially different beliefs if we keep updating on shared evidence
No need to define bayesian stats as the only approach to hyp testing
Use bayesian framework when we have easily available probabilitsic information
