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## A gentle introduction to the mathematics of biosurveillance: Bayes Rule in addition to Baye

McHugh, Josh, Contributing Editor has reference to this Academic Journal, PHwiki organized this Journal A gentle introduction to the mathematics of biosurveillance: Bayes Rule in addition to Bayes Classifiers Associate Member The RODS Lab University of Pittburgh Carnegie Mellon University http://rods.health.pitt.edu Andrew W. Moore Professor The Auton Lab School of Computer Science Carnegie Mellon University http://www.autonlab.org awm@cs.cmu.edu 412-268-7599 Note to other teachers in addition to users of these slides. Andrew would be delighted if you found this source material useful in giving your own lectures. Feel free to use these slides verbatim, or to modify them to fit your own needs. PowerPoint originals are available. If you make use of a significant portion of these slides in your own lecture, please include this message, or the following link to the source repository of Andrews tutorials: http://www.cs.cmu.edu/~awm/tutorials . Comments in addition to corrections gratefully received. What were going to do We will review the concept of reasoning with uncertainty Also known as probability This is a fundamental building block Its really going to be worth it What were going to do We will review the concept of reasoning with uncertainty Also known as probability This is a fundamental building block Its really going to be worth it (No I mean it it really is going to be worth it!)

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Discrete R in addition to om Variables A is a Boolean-valued r in addition to om variable if A denotes an event, in addition to there is some degree of uncertainty as to whether A occurs. Examples A = The next patient you examine is suffering from inhalational anthrax A = The next patient you examine has a cough A = There is an active terrorist cell in your city Probabilities We write P(A) as the fraction of possible worlds in which A is true We could at this point spend 2 hours on the philosophy of this. But we wont. Visualizing A Event space of all possible worlds Its area is 1 Worlds in which A is False Worlds in which A is true P(A) = Area of reddish oval

The Axioms Of Probability The Axioms Of Probability 0 <= P(A) <= 1 P(True) = 1 P(False) = 0 P(A or B) = P(A) + P(B) - P(A in addition to B) The area of A cant get any smaller than 0 And a zero area would mean no world could ever have A true Interpreting the axioms 0 <= P(A) <= 1 P(True) = 1 P(False) = 0 P(A or B) = P(A) + P(B) - P(A in addition to B) The area of A cant get any bigger than 1 And an area of 1 would mean all worlds will have A true Interpreting the axioms 0 <= P(A) <= 1 P(True) = 1 P(False) = 0 P(A or B) = P(A) + P(B) - P(A in addition to B) Interpreting the axioms 0 <= P(A) <= 1 P(True) = 1 P(False) = 0 P(A or B) = P(A) + P(B) - P(A in addition to B) A B P(A or B) B P(A in addition to B) Simple addition in addition to subtraction These Axioms are Not to be Trifled With There have been attempts to do different methodologies as long as uncertainty Fuzzy Logic Three-valued logic Dempster-Shafer Non-monotonic reasoning But the axioms of probability are the only system with this property: If you gamble using them you cant be unfairly exploited by an opponent using some other system [di Finetti 1931] Another important theorem 0 <= P(A) <= 1, P(True) = 1, P(False) = 0 P(A or B) = P(A) + P(B) - P(A in addition to B) From these we can prove: P(A) = P(A in addition to B) + P(A in addition to not B) A B Conditional Probability P(AB) = Fraction of worlds in which B is true that also have A true F H H = Have a headache F = Coming down with Flu P(H) = 1/10 P(F) = 1/40 P(HF) = 1/2 Headaches are rare in addition to flu is rarer, but if youre coming down with flu theres a 50-50 chance youll have a headache. Conditional Probability H = Have a headache F = Coming down with Flu P(H) = 1/10 P(F) = 1/40 P(HF) = 1/2 P(HF) = Fraction of flu-inflicted worlds in which you have a headache = worlds with flu in addition to headache ----- worlds with flu = Area of H in addition to F region ----- Area of F region = P(H in addition to F) --- P(F) Definition of Conditional Probability P(A in addition to B) P(AB) = --- P(B) Corollary: The Chain Rule P(A in addition to B) = P(AB) P(B) Probabilistic Inference H = Have a headache F = Coming down with Flu P(H) = 1/10 P(F) = 1/40 P(HF) = 1/2 One day you wake up with a headache. You think: Drat! 50% of flus are associated with headaches so I must have a 50-50 chance of coming down with flu Is this reasoning good Probabilistic Inference H = Have a headache F = Coming down with Flu P(H) = 1/10 P(F) = 1/40 P(HF) = 1/2 P(F in addition to H) = P(FH) = Probabilistic Inference H = Have a headache F = Coming down with Flu P(H) = 1/10 P(F) = 1/40 P(HF) = 1/2 What we just did P(A ^ B) P(AB) P(B) P(BA) = --- = --- P(A) P(A) This is Bayes Rule Bayes, Thomas (1763) An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53:370-418 Bad Hygiene Good Hygiene You are a health official, deciding whether to investigate a restaurant You lose a dollar if you get it wrong. You win a dollar if you get it right Half of all restaurants have bad hygiene In a bad restaurant, ¾ of the menus are smudged In a good restaurant, 1/3 of the menus are smudged You are allowed to see a r in addition to omly chosen menu Bayesian Diagnosis

Bayesian Diagnosis Bayesian Diagnosis Bayesian Diagnosis

Bayesian Diagnosis Bayesian Diagnosis Bayesian Diagnosis

Discussion What new data sources should we apply algorithms to EG Self-reporting What are related surveillance problems to which these kinds of algorithms can be applied Where are the gaps in the current algorithms world Are there other spatial problems out there Could new or pre-existing algorithms help in the period after an outbreak is detected Other comments about favorite tools of the trade.

## McHugh, Josh Contributing Editor

McHugh, Josh is from United States and they belong to Wired Magazine and they are from San Francisco, United States got related to this Particular Journal. and McHugh, Josh deal with the subjects like Computers; Internet

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