====== 11 Bayesian learning ======

Introduction to Bayesian probabilities.

===== Learning outcomes =====
After this practice session, the student
  * is able to formulate a decision task in Bayesian framework;
  * understands all parts of Bayesian decision task formulation (hidden states, observations, decisions, probability distribution over states and observations, loss function, decision strategy, its risk, partial/conditional risk, finding/constructing the optimal strategy);
  * finds optimal strategy for a given decision task.

===== Program =====
  * Q/A
  * Discussion of the bonus quiz from the last week
  * Exercise 1: Probability of illness given a positive test
  * Exercise 2: Bayesian decision task (power outage)
  * Introduction of the bonus quiz for this week


===== Exercise I / Solving together =====
=== Conditional probability, Bayes theorem ===

Although I don't have any symptoms yet, I'm worried about disease X. I'm considering getting tested for it, but haven't yet. I therefore want to study the following phenomena:
  * $X$ - the person is sick (has disease X)
  * $\bar X$ - the person is healthy (does not have disease X)
  * $\ominus$ - the test for disease X came out negative (according to the test, the person does not have disease X)
  * $\oplus$ - the test for disease X came out positive (according to the test, the person has disease X)
For the test I'm considering taking, the manufacturer says it has 90% accuracy for both healthy and sick people.
The prevalence of disease X in the population is 5%.
Try to answer the following questions:
   - What is the probability that a person randomly selected from the population has disease X?
   - What is the probability that I have disease X?
   - How to understand (and denote) test information from the manufacturer?
   - What is the probability that I have disease X if I test positive?
   - What is the probability that I have disease X if I test negative?

> {{page>courses:be5b33kui:internal:quizzes#Conditional probability}}


===== Exercise II / Solving together =====

Introduction to Bayesian probabilities on estimation of gender of people if we know the weight distribution of male and female colleagues {{:courses:be5b33kui:labs:weekly:bayes_intro_2021.pdf|(see pdf)}}.

> {{page>courses:be5b33kui:internal:quizzes#Bayes introduction}}


===== Bonus quiz =====
  * Estimation of the value of old coins based on the weights of the sub-set of a bag of coins.
  * 0.5 points
  * submit your solution to [[https://cw.felk.cvut.cz/brute/|BRUTE]] **lab11quiz** by May 07, midnight
  * format: text file, photo of your solution on paper, pdf - what is convenient for you
  * solution will be discussed on the next lab
  * Students with their family name starting from A to K (included) have to solve and upload {{ :courses:be5b33kui:labs:weekly:BayesDecisionsCoins_A_2025.pdf |subject A}} , while students with family name from L to Z have to solve and upload {{ :courses:be5b33kui:labs:weekly:BayesDecisionsCoins_B_2025.pdf |subject B}}

> {{page>courses:be5b33kui:internal:quizzes#Bayes coins}}


===== Homework =====

  * Start working on the [[courses:be5b33kui:semtasks:05_ml1:start|machine learning]] assignment