Lecture 5 Flashcards
(86 cards)
What is a Bayesian Inference
the outcome of a learning process that is governed by relative predictive success
Bayesian learning cycle
- prior knowledge
- prediction
- data
- prediction error
- knowledge update
If your predictive updating factor is larger than 1…
then your beliefs, in a particular value of data, increases.
If it is lower than 1.. decrease
So your beliefs increases…
if your prediction for the data is better when you condition on that particular data
Do you want surprises in statistics
nope, probably means that the data isn’t really good
theta (0met streepje)
gaat altijd over een unknown proportion. van een hele populatie bijvoorbeeld
A more complex model…
does FIT the data better, but only in terms of FITTING it. But it is very poorly at prediction!!!!!!
Suppose bf01= 3, What is the correct interpretation
- the observed data are 3 times more likely under the H0 than under H1. or “H0 outpredicts H1 by a factor of 3” - its about the evidence coming from the data
NOT: after seeing the data, H0 is now 3 times more likely than H1. This is correct ONLY when H0 and H1 are equally likely a priori
a rough conceptualisation of the bayesian learning cycle
Prior knowledge
prediction
data
prediction error
knowledge update
and prior knowledge again
What does Bayes’ rule compute in terms of beliefs?
It computes how we update our posterior beliefs about unknown parameters (θ) using prior beliefs and observed data.
Write out Bayes’ Rule using θ and data.
P(θ∣data)= P(data∣θ)⋅P(θ)
——————-
P(data)
What does 𝑃 (𝜃∣data) represent?
The posterior belief about the parameter θ after seeing the data.
What does P(θ) represent?
The prior belief about θ before seeing any data.
What does P(data∣θ) represent?
The likelihood – the probability of observing the data if θ were true.
What does P(data) represent?
The evidence – the overall probability of the data under all possible θ.
What is the informal interpretation of Bayes’ Rule?
Posterior = Prior × Predictive updating factor
What does the ratio
P(θ∣data)
————-
P(θ) represent?
The change in support for θ after seeing data (how much belief in θ increases/decreases).
What does the ratio
P(data∣θ)
———–
P(data)
represent?
The predictive success of θ – how well it explains the data relative to alternatives.
What equality connects the two ratios above?
P(θ)
=
P(data∣θ)
————–
P(data)
What does this equation tell us about belief updating?
The change in belief (support) is proportional to the predictive success of θ.
What is the interpretation of “the extent to which an account of the world out-predicts another”?
It is how much better θ explains the data than competing hypotheses, and this drives belief change.
What does it mean if data are very surprising under a model?
It suggests the model is not a good fit — it loses credibility.
What is meant by “surprise lost is credibility gained”?
A model that reduces surprise (predicts well) gains support in Bayesian inference.
What is the parameter θ in the binary outcome example?
The unknown proportion (e.g., of cat people), which lies between 0 and 1.