chapter 6.3 Flashcards
classical statistics
so called because it is more or less standard - at least presently.
Problems with p-values
scientists primary interest in statistical hypothesis testing is to figure out which hypotheses are true. But a p-value doesnt indicate how probable the hypothesis itself is, that is, how likely the hypothesis is to be true. It only indicates how probable the observation is given the null
second problem with classical statistics
because this procedure doenst find the likelyhood of the null hypothesis, it doesnt take into account any information we might have in favor of - or against - the truth of the null hypothesis.
For example, in our tea tasting example, it may be the right decision not to reject the null hypothesis even though your friend guesses correctly so often. What we know about how tasting works and about what properties a cup of tea can and cant have suggest this tasting feat should be nearly impossible
Third problem of classical statistics
the probability of the observation given the null hypothesis, the p-value, doesnt directly relate to the alternative hypothesis at all. This only tellss you something about the relationship between the null hypothesis and the observed data. And yet, the alternative hypothesis, the bold and speculative conjecture, is what scientists are truly interested in knowing about. How likely is the alternative hypothesis to be true? this is the million dollar question in hyp testing and the observed data
Bayesian statistics aims
to determine when ab observation counts as evidence for one hypothesis and against a competing hypothesis and how that observation should change our degree of belief that each of these competing hypothesis and how that observation should change our degree of belief that each of these competing hypotheses is true.
Bayesian statisticians do not use p-values and significance testing, and they do not attempt to reject null hypotheses. For bayesians, an observation counts as evidence for a hypothesis when it raises the probability of the hypothesis.
Bayes’s theorem in its simplest formulation
Pr(HIO) = Pr(OIH)*Pr(H)/Pr(O)
The rational degree of belief in a hypothesis H after observation O is just the conditional probability of H given O. Remember that this, Pr(HIO), is what we criticized classical statistics for not being able to provide.
Pr(H) prior probability of the hypothesis;
Pr(HIO) posterior probability of the hypothesis.
This is because Pr(H) is our rational degree of belief before making the observation, that is, prior to the obs, while Pr(HIO) is our rational degree of belief posterior to making the observation
Taking into account the prior probability of a hypothesis enables us to hold implausible hypotheses to a higher standard of evidence than plausible hypotheses are held to.
When is the hypothesis confirmed bayesian approach
If Pr(HIO) > Pr(H), then we say that the observation O confirms hypothesis H
That is, an observation confirms a hypothesis if the probability of the ypothesis, a rational degree of belief that the hypothesis in question is true, goes up once the observation has been made. So, comparing the prior and posterior probabilities shows us whether an observation fonfirms or disconfirms a hypothesis and by how much. A big increase in prob implies a large degree of confirmation and a small increase implies a small degree of confirmation; etc etc.
