bayesian statistics Flashcards
Bayes theorem, posteriors, likelihoods, priors, Bayesian inferential tests, Bayes factor, evidence for H0, frequentists, credible intervals, comparison (25 cards)
who was Thomas Bayes?
1702-1761
statistician, philosopher and minister
formalised that intuition into some equation
what was Bayes theorem?
our perceived probability that something is true depends on the data about it as well as our previous expectations
what aspects of science has Bayes theorem become hugely important for?
engineering (signal detection theory)
computing (spam filter)
statistics (Bayesian inference)
human decision-making (are we good Bayesians?)
what is Bayes formula?
p(hypo|data) =
p(D|H) x p(H)
/
p(D)
what is the posterior probability?
p(H|D)
probability that H is true
after seeing the evidence
what is the likelihood?
p(D|H) / p(D)
evidence coming from the data
the likelihood of these data coming from H
what is the prior?
p(H)
probability of the theory
before seeing the evidence
what does it mean when p(D|H) is big?
data more likely to occur when hypothesis true
want that to be big number to generate a large posterior probability
what does it mean when p(D) is big?
data is likely to look like this whether or not hypothesis is true
if big, data isn’t giving much evidence for/against hypothesis
what is a point of debate for Bayesian theory?
when something is surprising, we intuitively require better evidence to believe it
do you think we should base our stats tests to include our priors (aka beliefs)?
does that make use less ready to reject things where the evidence doesn’t agree?
or is it obvious, and human nature - are we just Bayesian creatures?
what is Null Hypothesis Significance Testing (NHST)?
calculate the likelihood of this data when null is H0
we reject H0 only if there’s a good reason to do so
failing to reject H0 doesn’t mean it was more likely than H1
what are Bayesian inferential tests?
calculate the probability of H1 and H0 given data (with or without informative priors)
calculate the ratio of these probabilities, the Bayes Factor (BF)
who created the Bayes Factor and when?
Harold Jeffreys
1961
how is BF usually donoted?
K
what is K?
Bayesian equivalent of p-value
K = p(H1) / p(H0)
when is K small?
if H0 more likely
when K large?
if H1 more likely
what is BF?
operate in both directions (can suggest evidence for H0 rather than stacking the odds in favour of H1)
what are BF categories?
0.0 - 1/3 = evidence for H0
1/3 - 3 = not much evidence for anything
3 = evidence for theory
what is H0 in NHST?
always designed to be null effect
no difference, no relationship
what is H0 in Bayesian stats?
not necessarily the null hypothesis
can be other hypothesis (often referred to as M1/M2 rather than H0/H1)
often is null effect (are conditions the same) that we want to know about
mathematically Bayes factors can provide support for the null
what do we need to say two things are the same?
what level of tolerance
Bayes tests can handle
what are frequentists?
Bayesians refer to traditional stats or statisticians as “frequentists” because of logic of p value
p only tells you how “frequent” this data would be assuming H0 is true and you ran many studies
like to point out isn’t the same as telling us how likely original hypotheses are
what are credible intervals?
Bayesian equivalent of confidence intervals
