hevruta Introduction Bayesian modelling in the recent decade Lee amp Wagemakers 2013 Some tentative plans Today A general introduction Session 2 Handson introduction into ID: 596851
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Slide1
Bayesian modelling hevruta
IntroductionSlide2
Bayesian modelling in the recent decade
Lee &
Wagemakers
(2013)Slide3
Some tentative plans
Today – A
general introduction
Session 2 – Hands-on introduction into
Bayesian modelling software
Session 3 – Introduction into
Bayesian data analysis
Session 4 – Introduction into
Bayesian modelling
Session 5(optional) – A few advanced issues (e.g. Hierarchical Bayes)
Session 6 onward –
Your part.
How does that sounds?
Meetings frequency? Monthly? Bi-weekly? First Bi-weekly then monthly?Slide4
Bayes theorem and beliefs
E.g. What are the chances you will think it’s an interesting
Hevruta
after today? (B – great
hevruta
! ; D – liked the first meeting)
Great
Hevruta
Sounds boring…
Liking the first meeting
Liking the first meeting
Not liking it
Not liking it
8
4
Given that you liked the first meeting -> the chances you’d think it’s a great
Hevruta
are now 0.67
You can also say that one will be 67% certain of his belief that this is a great
Hevruta
Likelihood
Prior
DataSlide5
Bayes theorem and beliefs
E.g. What are the chances you will think it’s an interesting
Hevruta
after today? (B – great
hevruta
! ; D – liked the first meeting)
Great
Hevruta
Sounds boring…
Liked the first meeting
Liked the first meeting
Didn’t like it
Didn’t like it
8
4
Given that you liked the first meeting -> the chances you’d think it’s a great
Hevruta
are now 0.67
You can also say that one will be 67% certain of his belief that this is a great
Hevruta
Beliefs as distributions of credibilitySlide6
Bayes theorem and beliefs
Let’s say that your opinion regarding the
Hevruta
is a continuous parameter:
Very boring!
That’s great!
-100
100
PriorSlide7
Bayes theorem and beliefs
Let’s say that your opinion regarding the
Hevruta
is a continuous parameter:
Very boring!
That’s great!
-100
100
Prior
LikelihoodSlide8
Bayes theorem and beliefs
Let’s say that your opinion regarding the
Hevruta
is a continuous parameter:
Very boring!
That’s great!
-100
100
Prior
Likelihood
Posterior
The posterior weights the prior and likelihood in accordance with their precisionSlide9
Examples of applications to perceptionSlide10
Examples of applications to perceptionSlide11
Non-Informative priors
Very boring!
That’s great!
-100
100
PriorSlide12
Non-Informative priors
Very boring!
That’s great!
-100
100
Prior
LikelihoodSlide13
Non-Informative priors
Very boring!
That’s great!
-100
100
Prior
Likelihood
PosteriorSlide14
Why go Bayesian?
Modelling:
A coherent mathematical framework for optimal learning, perceptual processes, and modification of beliefs.
Can be useful to find conditions in which people diverge from the optimal predictionsSlide15
Bullock, J. G. (2007)Slide16
Why go Bayesian?
Modelling:
A coherent mathematical framework for optimal learning, perceptual processes, and modification of beliefs.
Can be useful to find conditions in which people diverge from the optimal predictions
Statistics:
Many problems with null hypothesis significance testing
(e.g. multiple comparisons, interpretation of p-values, etc.)
A coherent framework for study replication and accumulation of evidence –
“today’s posterior is tomorrow’s prior”
A much more flexible framework of data analysis (e.g.
almost no assumptions are required)
Very suitable for hierarchical statistical modellingSlide17
Data analysis example
We want to say something about the height of a population.
We have the following sample:Slide18
Data analysis example
The first question is -
what is the data generating process?
.
We assume a normal process so that:
(data variance is assumed to be known for simplicity)
Slide19
Data analysis example
The second question is
what kind of prior do we want to assume
Slide20
Data analysis example
For a
weakly informative prior
we can choose a prior that represents
what we know about heights in general, but allows for high variation. E.g.
Slide21
Data analysis example
Note that the green curve represents a distribution representing the credibility of different
. It is not the data distribution
.
The posterior mean is 192.11, while the sample mean is 192.31, Thus, the posterior seems to correspond well with the sample
Slide22
How does it actually happen?
Kruschke
, 2015Slide23
How does it actually happen?
For each value of
- its probability given the data is calculated
E.g.
For
,
and
(log likelihood = -348)
For
,
and
(log likelihood = -144)
Finally, each value of
to create a probability distribution (summing up to 1)
Slide24
The product of Bayesian data analysis
In its core, Bayesian data analysis
does not lead to a yes/no conclusion regarding parameters of interest.
Rather it provides a belief probability distribution, which In most cases can be used to conclude on
what is the most likely state of the parameter, and how uncertain this state is.
Bayesian model comparison
(i.e. Bayes factors), is a more familiar approach to compare the relative probability of two models. Although often thought of as a dichotomous decision process, it is not.
It only says how much more likely is model A in comparison to model B.
Slide25
Some good recent books on Bayesian modelling
Kruschke
, J. (2014).
Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan
. Academic Press.
A very comprehensive hands-on introductory book to Bayesian data analysisSlide26
Some good recent books on Bayesian modelling
Lee &
Wagenmakers
(2013). Bayesian cognitive modelling
A good introductory book into Bayesian modelling.Slide27
Some good recent books on Bayesian modelling
Gelman
, A., et al. (2013).
Bayesian Data Analysis.
CRC press.
A
more advanced,
mathematically oriented tutorial.