Bayesian Regression and Its Advantages
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Dr Oliver Perra
Full resource: https://www.ncrm.ac.uk/resources/online/all/?id=20843
Summary
•
Bayes vs. Frequentist approach
•
Advantages of a bayes approach
Bayesian approach
•
A different view of hypothesis testing from traditional
approach:
•
Allows to include previous knowledge on the phenomena
•
Incorporates every piece of new information (however small)
to update our knowledge in light of new evidence.
Conventional Methods (Frequentist)
•
Parameters (e.g.: population mean) are unknown but
fixed.
•
A
counter-factual
approach:
o
Null hypothesis
o
Estimate probability of observed difference occurring from
null hypothesis if we
replicated the experiment several
times
.
Bayesian Analysis
•
Parameters are
uncertain:
o
Described by probability distributions.
•
A-priori assumptions about the distribution of a
parameter are updated considering new data:
o
Estimate
posterior
distributions
that
take into
account new information.
o
Sample from posterior
distribution to summarise
parameter plausible values and their uncertainty.
Bayesian analysis
•
First element: A prior
p
= proportion who will vote Red
H
= number of respondents who indicate Red preference
N
= total number of respondents
Prior
DATA
:
R=3
N=5
R/N = 0.60
Bayesian analysis
•
Second element: Likelihood function.
p
= proportion who will vote for Red candidate
R
= number of respondents who intend to vote for Red
N
= total number of respondents
My prior assumption
DATA
:
R=3
N=5
R/N = 0.60
Prior
Bayesian analysis
•
Bayesian Rule
Posterior
DATA
:
R=3
N=5
R/N = 0.60
Posterior:
Posterior
p
= proportion who will vote for Red candidate
R
= number of respondents who intend to vote for Red
N
= total number of respondents
Prior
Bayesian analysis
Prior
New Data
:
R
=29
N
=50
R/N
= 0.58
Posterior
p
= proportion who will vote for Red candidate
R
= number of respondents who intend to vote for Red
N
= total number of respondents
Sampling to summarise
90% of prob. of Red between 49.40% and 66.26%
Posterior
p
= proportion who will vote for Red candidate
R
= number of respondents who intend to vote for Red
N
= total number of respondents
Bayesian Analysis
•
Bayesian analysis “takes a question in the form of a
model and uses logic to produce an answer in the
form of probability distributions” (McElreath, 2018:
Page 10)
www.ncrm.ac.uk
Bayesian regression offers a unique approach to hypothesis testing by incorporating prior knowledge and updating beliefs with new evidence. Contrasting with frequentist methods, Bayesian analysis considers parameters as uncertain and describes them using probability distributions. This methodology allows for a more comprehensive interpretation of data, emphasizing the posterior distribution to summarize parameter values and uncertainty.
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Presentation Transcript
An Introduction to Bayesian Regression Part #1 Dr Oliver Perra Full resource: https://www.ncrm.ac.uk/resources/online/all/?id=20843
Summary Bayes vs. Frequentist approach Advantages of a bayes approach
Bayesian approach A different view of hypothesis testing from traditional approach: Allows to include previous knowledge on the phenomena Incorporates every piece of new information (however small) to update our knowledge in light of new evidence.
Conventional Methods (Frequentist) Parameters (e.g.: population mean) are unknown but fixed. A counter-factual approach: o Null hypothesis o Estimate probability of observed difference occurring from null hypothesis if we replicated the experiment several times.
Bayesian Analysis Parameters are uncertain: o Described by probability distributions. A-priori assumptions about the distribution of a parameter are updated considering new data: o Estimate posterior distributionsthattake into account new information. o Sample from posterior distribution to summarise parameter plausible values and their uncertainty.
Bayesian analysis First element: A prior Prior DATA: R=3 N=5 R/N = 0.60 p = proportion who will vote Red H = number of respondents who indicate Red preference N = total number of respondents
Bayesian analysis Second element: Likelihood function. Prior My prior assumption DATA: R=3 N=5 R/N = 0.60 Likelihood func.: R ~ Binomial (N, p) p = proportion who will vote for Red candidate R = number of respondents who intend to vote for Red N = total number of respondents
Bayesian analysis Bayesian Rule Posterior Posterior Prior DATA: R=3 N=5 R/N = 0.60 Likelihood func.: R ~ Binomial (N, p) Posterior: Pr(p | R, B) = ?? ?,? ? ??(?) ??( ?,?) p = proportion who will vote for Red candidate R = number of respondents who intend to vote for Red N = total number of respondents
Bayesian analysis New Data: R=29 N=50 R/N = 0.58 Prior Posterior Likelihood func.: R ~ Binomial (N, p) Pr(p | R, B) = ?? ?,? ? ??(?) ??( ?,?) p = proportion who will vote for Red candidate R = number of respondents who intend to vote for Red N = total number of respondents
Sampling to summarise Posterior p = proportion who will vote for Red candidate R = number of respondents who intend to vote for Red N = total number of respondents 90% of prob. of Red between 49.40% and 66.26%
Bayesian Analysis Bayesian analysis takes a question in the form of a model and uses logic to produce an answer in the form of probability distributions (McElreath, 2018: Page 10)
