Week 01
Bayesian inference, estimators and posterior summaries | Conjugacy | The beta-binomial model
Embedded Files
The Bayesian approach in machine learning
The Bayesian approach in machine learning
Motivation
Motivation
Classical approach: searching the best model parameters by minimizing the sum of squared distances (error)
BUT for a given finite dataset, many sets of parameters could work
→ different predictions
THUS Bayesian machine learning:
consider ALL potential sets of parameters
compute the probability of those parameters given the data
How does it work?
How does it work?
Based on the Bayes' rule:
Everything is a probability distribution!
Take uncertainty into account + better decision-makingPrior p(w) : prior belief about the parameters before seeing the data
Likelihood p(y|w) : represent the probability our parameters generate the given data
Marginalization p(y) :normalization constant
Posterior p(w|y) : contains all knowledge about the parameters after seeing the data
Posterior, all knowledge? YES!
→ Posterior summaries:
Mean (Bayes estimator)
Credibility interval to evaluate uncertainty
Mode (Maximum A Posteriori estimator)
How to choose the distribution of the prior?
How to choose the distribution of the prior?
Where does prior knowledge come from?
Previous experiments (iterations), experts
Likelihood and prior function forms vary with the situation
[Definition] Conjugacy: when the posterior and the prior are in the same probability distribution family.
Let's study an example of Bayesian conjugate model!
The Beta-Binomial model
The Beta-Binomial model
Motivation
Motivation
For proportion estimations
Example of A/B test, to estimate click-rates
Is A better than B?
What is the probability that the click-rate is below 10%?
How much am I certain about these conclusions?
Key equations
Key equations
We want to estimate the parameter µ, the success probability e.g. click rate.
Uniform / Informative / Weakly priors
About the Prior
→ 2 hyperparameters
About the Likelihood
Binomial distribution (PMF - Probability Mass Function)
About the Posterior
Prior and posterior are conjugate, since they are both Beta distributions!
Effect of the prior distribution: informative VS weekly informative
Vary hyperparameters and compute the posterior distribution with the same likelihood to compare them
About the Predictive Posterior
Obtained by averaging over the uncertainty of the posterior
Estimators & posterior summaries
Estimators & posterior summaries
All information about the parameter µ can be found in the posterior distribution.
Posterior mean as Bayes estimator
Credibility interval as uncertainty
Interval such that the posterior probability of µ belonging to the interval is 0.95
Posterior mode as MAP estimator
(Maximum A Posteriori)
Estimation at highest density, e.g. where the results are the most frequent
Maximum Likelihood estimator: obtained by maximizing the log likelihood (classical approach)
Estimators and posterior summaries' reports & density visualizations when the size of data N increases
Illustration: back to the example of A/B testing
Illustration: back to the example of A/B testing
Posterior histograms for both versions by sampling on posterior distributions + histogram of the difference between A and B
Finally, what is the probability that the click-rate of B is better than A after seeing the data D?
Page updated
Google Sites
Report abuse