08-05-2013, 12:47 PM
Bayes Theorem
Bayes Theorem.ppt (Size: 169.5 KB / Downloads: 43)
Written Work
Only two works published during his life
Divine Benevolence (1731)
Introduction to the Doctrine of Fluxions (1736)
However, he never published his mathematical works
He focused in the areas of probability and statistics
“The probability of any event is the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the chance of the thing expected upon it’s happening.”
Publishing of Bayes Theorem
Richard Price examined Bayes’ work after his death
Responsible for the communication to the Royal Society on Baye’s work
An Essay Toward Solving a Problem in the Doctrine of Chances
An Essay Toward Solving a Problem in the Doctrine of Chances
The essay demonstrates the estimation of future occurrences of an event, given information of the history of the event.
The probability of future occurrences of an event can be solved using this formula,
A generalization of this formula constructs the present form of Bayes Theorem.
Pierre Simon Laplace
French mathematician
Responsible for current form of Bayes Theorem
Bayes found the probability that x is between two values given a number of successes and failures
Laplace found an expression for the probability of a number of future successes and future failures given the number of successes and failures.
Applications
Classification scale in classroom
A simple classification divides a population in two categories
Mastering
Non-mastering
A test would determine which category a student falls in. The Probabilities can be collected and the result would show mastered skills and others that need attention.
Diagnostic testing
Tests identify if a person has a particular disease or not
Tests are not always 100% positive
Bayes Theorem estimates if a person tests positive for a disease if he or she truly has the disease
A cancer is found in 1 in every 2000 people. If a person has the disease there is a 90% the test will result positive. If a person does not have the disease, the test will result in a false positive 1% of the time.
Introduction to Bayesian Statistical Inference
Statistical inference is used to draw conclusions from known data in samples to populations for which data is unknown.
Inference problems are often found in decision problems
There are two different ways to approach probability
Frequentist
Subjective, or Bayesian
Frequentist – can be applied to events found through a random process
Frequentists work on problems associated with collective and not individual events.
Bayesian – has no one correct probability
Depends on prior and observed data
Finding the probability of an American man’s height is over 1.75 meters, the result would depend on if we know anything about the man.