Lecture 19 - Bayesian Statistics Flashcards
(15 cards)
What is Bayes theorem?
Thomas Bayes (1702-1761), statistician and philosopher (and minister)
Our perceived probability that something is true depends on the data about it as well as our previous expectations
Bayes formalized that intuition into some equations
It has become hugely important for many aspects of science
Engineering (signal detection theory)
Computing (your spam filter)
Statistics (Bayesian inference)
Human decision-making (are we good Bayesians?)
What is the equation for Bayes theorem?
In notes
The probability that H is true (after seeing the evidence - posterior probability) is the evidence coming from the data (likelihood of these data coming from H) multiplied by the probability of the theory (before seeing the evidence - the prior)
Explain the Bayes Theorem equation
Likelihood is technically just the p(D|H) and the p(D) is the Marginal Likelihood. Together these give the evidence given by the data.
When p(D|H) is big it means that this data becomes more likely to occur when the hypothesis is true. We obviously want that to be a big number to generate a large Posterior probability
When p(D) is big it means that the data are likely to look like this, whether or not the hypothesis is true, so if this is big it means the data aren’t giving much evidence for/against the hypothesis
What are posteriors, likelihoods and priors?
Ultimately, what all that means is: Posterior = likelihood x prior
(Belief = evidence x expectation)
What is a real-world example of Bayes theorem?
Do I need an umbrella today?
Prior: knowledge of the weather in May
Evidence: look out the window
Do I believe the news report that politician X has done something naughty?
Prior: do I think it sounds like the sort of thing they would do?
Evidence: what is the evidence on this occasion?
Do I throw away 100 previous studies of data if I get one study claiming different?
What is a point for debate in Bayes theorem?
One implication of Bayesian theory is that, 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 us less ready to reject things where the evidence doesn’t agree?
Or is it obvious, and human nature – we just ARE Bayesian creatures?
What is the impact of priors?
Graphs in notes
What are Bayesian inferential tests?
Got an idea that intuitively makes sense and some equations that allow us to convert those intuitions into numbers (i.e. statistics)
Could be used to create more useful inference tests
How does Bayesian inferential testing compare to null hypothesis inference testing?
Null hypothesis significance testing (NHST):
Calculate the likelihood of this data when Null is H0
We reject H0 only if there’s really good reason to do so
Failing to reject H0 doesn’t mean it was more likely than H1
Instead, “Bayesians” say:
Calculate the probability of H1 and H0 given the data (with or without informative priors)
Calculate the ratio of those probabilities, the Bayes Factor (BF)…
What is Bayes factor?
Harold Jeffreys (1961)
Usually denoted, K, these are the Bayesian equivalent of a p-value:
K = p(H1) / p(H0)
Small if H0 more likely - e.g. if H0 is 3 times more likely then K = 1/3 = 0.3333
Large is H1 more likely - e.g. if H1 is 3 times more likely then K = 3/1 = 3.0
They operate in both directions (can suggest evidence for the H0, rather than ‘stacking the odds’ in favor of H0)
Bayesians like to point out that BF is continuous. We don’t need a hard boundary (sig/non-sig) (actually, that’s true of p as well, but people like simple categories)
BF categories soon appeared too:
0.0 - 1/3 evidence for H0
1/3 - 3 not much evidence for anything
3 evidence for the theory
What provides evidence for H0?
In NHST H0 is always designed to be Null effect (no difference, no relationship etc.)
In Bayesian statistics H0 is not necessarily the ‘Null’ hypothesis. It can be any other hypothesis (so often referred to as M1/M2 rather than H0/H1)
But often it is the Null effect (are the conditions the same) that we want to know about
Mathematically Bayes Factors can provide support for the Null (e.g. when BF<1/3)
To say 2 things are the same we have to say what level of ‘tolerance’ (which Bayes tests also handle)
Why don’t Bayesians like ‘frequentists’?
Bayesians refer to traditional statistics (and statisticians) as “frequentists” because of the logic of the p value:
They say p only tells you how “frequent” this data would be assuming that H0 is true and you ran many studies
They like to point out this isn’t the same as telling us how likely the original hypotheses are
Jon’s personal point of view:
Data that give a highly significant p also give a significant Bayes Factor
If the parametric t-test and the Bayes test give us the same answer, then does it matter whether it was a p or a BF?
What are credible intervals?
These are the Bayesian equivalent of Confidence Intervals
For confidence intervals, if we ran the experiments many times we would get a new CI each time and 95% of such intervals contain the current mean
That isn’t quite the same as having a confidence interval that is fixed and saying the 95% of means are inside it.
Bayesian “Credible Intervals” aim to give you the “right” thing
How does Bayesian statistics compare to traditional statistics?
Some key differences with traditional stats
Can also incorporate a priori knowledge explicitly in the test (but often used with a “flat” prior)
Balanced test of whether H0/H1 is more likely, rather than assuming H0 unless confident about H1
Can use any hypothesis as H0 (not necessarily the no-difference hypothesis)
Philosophically, a better fit with what stats aims to do
Very popular with enthusiasts
SPSS only added these in version 25 (even though Thomas Bayes died in 1761!!)
Glossary
NHST (Null Hypothesis Statistical Testing): the traditional stats testing we’ve been talking about all year
Frequentist statistics: another (not great) term for NHST
BF (Bayes Factor or K): A bit like p
Credible Intervals or High-density intervals (HDIs): a bit like confidence intervals
Posterior: probability after seeing the data
Prior: probability before seeing the data (expectation)
Likelihood: evidence coming from the data itself
