Probabilistic Operations Research Models
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationProbabilistic Operations Research ModelsPaul BrooksJill HardinDepartment of Statistical Sciences and Operations ResearchVirginia Commonwealth UniversityBNFO 691 December 5, 2006university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationOutline1234Operations Research ModelsAxioms of ProbabilityDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesMarkov ChainsMarkov PropertyBlood Types IISimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event SimulationPaul Brooks, Jill Hardinuniversity-logo
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationOperations Research ModelsOperations ResearchModelsDeterministic ear LinearNonlinearFunctions Functions Functions FunctionsProbabilistic ORDiscrete TimeContinuous TimeDiscrete Continuous Discrete ContinuousSpaceSpaceSpaceSpaceuniversity-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationLinear ProgrammingOperations ResearchModelsDeterministic lesNonlinear LinearNonlinearFunctions Functions FunctionsProbabilistic ORDiscrete TimeDiscreteSpaceContinuous TimeContinuous ul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationInteger ProgrammingOperations ResearchModelsDeterministic lesNonlinear LinearNonlinearFunctions Functions FunctionsProbabilistic ORDiscrete TimeDiscreteSpaceContinuous TimeContinuous ul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationMarkov ChainsOperations ResearchModelsDeterministic lesNonlinear LinearNonlinearFunctions Functions FunctionsProbabilistic ORDiscrete TimeDiscreteSpaceContinuous TimeContinuous ul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDiscrete-Event SimulationOperations ResearchModelsDeterministic lesNonlinear LinearNonlinearFunctions Functions FunctionsProbabilistic ORDiscrete TimeDiscreteSpaceContinuous TimeContinuous ul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDiscrete vs. Continuous ModelsDiscretemeans “space between”countable, e.g., integers, binary numbersattributes, variables, time, spaceContinuousuncountable, e.g., real numbers, intervals of real numbersattributes, variables, time, spaceuniversity-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationLinear vs. Nonlinear ModelsLinearadditivity - every function is the sum of the individualcontributions of activitiesproportionality - the contribution of an activity to a functionis proportional to the level of the activity.Hillier, FS and Lieberman, GJ. Introduction to Operations Research, 6th edition. McGraw-Hill, 1995.NonlinearAdditivity or proportionality (or both) are violateduniversity-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationProbabilistic vs. Deterministic ModelsProbabilisticProbability is used to model behaviors that are uncertain orunknownDeterministicRandomness is not considered; systems are assumed to betotally determined. Sensitivity analysis can help incorporateuncertainty into models.university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesOutline1234Operations Research ModelsAxioms of ProbabilityDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesMarkov ChainsMarkov PropertyBlood Types IISimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event SimulationPaul Brooks, Jill Hardinuniversity-logo
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesWhat is Probability?Simply speaking probabilities are numbers between 0and 1 that reflect the chances of “something”happening.Synonymous with chance, likelihood, odds.Has different interpretations.university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesCards and DiceWhat is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?These are called eventsAre you sure? What assumptions did you make?Were they correct?How can I correct these assumptions?How can I determine a more accurate probability?university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesCards and DiceWhat is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?These are called eventsAre you sure? What assumptions did you make?Were they correct?How can I correct these assumptions?How can I determine a more accurate probability?university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesCards and DiceWhat is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?These are called eventsAre you sure? What assumptions did you make?Were they correct?How can I correct these assumptions?How can I determine a more accurate probability?university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesCards and DiceWhat is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?These are called eventsAre you sure? What assumptions did you make?Were they correct?How can I correct these assumptions?How can I determine a more accurate probability?university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesCards and DiceWhat is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?These are called eventsAre you sure? What assumptions did you make?Were they correct?How can I correct these assumptions?How can I determine a more accurate probability?university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesCards and DiceWhat is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?These are called eventsAre you sure? What assumptions did you make?Were they correct?How can I correct these assumptions?How can I determine a more accurate probability?university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesCards and DiceWhat is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?These are called eventsAre you sure? What assumptions did you make?Were they correct?How can I correct these assumptions?How can I determine a more accurate probability?university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesCards and DiceWhat is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?These are called eventsAre you sure? What assumptions did you make?Were they correct?How can I correct these assumptions?How can I determine a more accurate probability?university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesCards and DiceWhat is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?These are called eventsAre you sure? What assumptions did you make?Were they correct?How can I correct these assumptions?How can I determine a more accurate probability?university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesCards and DiceWhat is the probabilitythat I draw a black card?that I roll a 7?that I roll doubles?These are called eventsAre you sure? What assumptions did you make?Were they correct?How can I correct these assumptions?How can I determine a more accurate probability?university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesOutline1234Operations Research ModelsAxioms of ProbabilityDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesMarkov ChainsMarkov PropertyBlood Types IISimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event SimulationPaul Brooks, Jill Hardinuniversity-logo
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityClassical/AnalyticalTheoretically determined probabilitiesProbability of rolling a 3 on a fair (normally marked) die: 1/6Probability of drawing a black card in a standard deck: 1/2Advantagesprobabilities are accurateno experimentation requiredobjectiveDisadvantage: only possible to compute under the best ofcircumstances (e.g., we know that the die is fair)university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityClassical/AnalyticalTheoretically determined probabilitiesProbability of rolling a 3 on a fair (normally marked) die: 1/6Probability of drawing a black card in a standard deck: 1/2Advantagesprobabilities are accurateno experimentation requiredobjectiveDisadvantage: only possible to compute under the best ofcircumstances (e.g., we know that the die is fair)university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityClassical/AnalyticalTheoretically determined probabilitiesProbability of rolling a 3 on a fair (normally marked) die: 1/6Probability of drawing a black card in a standard deck: 1/2Advantagesprobabilities are accurateno experimentation requiredobjectiveDisadvantage: only possible to compute under the best ofcircumstances (e.g., we know that the die is fair)university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityClassical/AnalyticalTheoretically determined probabilitiesProbability of rolling a 3 on a fair (normally marked) die: 1/6Probability of drawing a black card in a standard deck: 1/2Advantagesprobabilities are accurateno experimentation requiredobjectiveDisadvantage: only possible to compute under the best ofcircumstances (e.g., we know that the die is fair)university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityClassical/AnalyticalTheoretically determined probabilitiesProbability of rolling a 3 on a fair (normally marked) die: 1/6Probability of drawing a black card in a standard deck: 1/2Advantagesprobabilities are accurateno experimentation requiredobjectiveDisadvantage: only possible to compute under the best ofcircumstances (e.g., we know that the die is fair)university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityRelative Frequency/EmpiricalObserved proportion of successful events10 cards selected, 6 of them black probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years probability of having at least 60 inchesof snow this year is 0.95.Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjectiveDisadvantage: situation must be replicableuniversity-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityRelative Frequency/EmpiricalObserved proportion of successful events10 cards selected, 6 of them black probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years probability of having at least 60 inchesof snow this year is 0.95.Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjectiveDisadvantage: situation must be replicableuniversity-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityRelative Frequency/EmpiricalObserved proportion of successful events10 cards selected, 6 of them black probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years probability of having at least 60 inchesof snow this year is 0.95.Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjectiveDisadvantage: situation must be replicableuniversity-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityRelative Frequency/EmpiricalObserved proportion of successful events10 cards selected, 6 of them black probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years probability of having at least 60 inchesof snow this year is 0.95.Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjectiveDisadvantage: situation must be replicableuniversity-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityRelative Frequency/EmpiricalObserved proportion of successful events10 cards selected, 6 of them black probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years probability of having at least 60 inchesof snow this year is 0.95.Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjectiveDisadvantage: situation must be replicableuniversity-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityRelative Frequency/EmpiricalObserved proportion of successful events10 cards selected, 6 of them black probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years probability of having at least 60 inchesof snow this year is 0.95.Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjectiveDisadvantage: situation must be replicableuniversity-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityRelative Frequency/EmpiricalObserved proportion of successful events10 cards selected, 6 of them black probability ofselecting a black card is 0.6Minnesota has had snowfall of at least 60 inches in 95 ofthe last 100 years probability of having at least 60 inchesof snow this year is 0.95.Advantagescan collect empirical data to estimate probabilities whenthey can’t be determined analyticallyobjectiveDisadvantage: situation must be replicableuniversity-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityPersonal/Subjective“What do you think are the odds?”What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?Relies on expert information (definition of “expert” is fluid).Advantage:always applicable - everybody has an opinionuseful in risk analysisDisadvantage: difficult (sometimes impossible?) todetermine accuracy.Paul Brooks, Jill Hardinuniversity-logo
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityPersonal/Subjective“What do you think are the odds?”What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?Relies on expert information (definition of “expert” is fluid).Advantage:always applicable - everybody has an opinionuseful in risk analysisDisadvantage: difficult (sometimes impossible?) todetermine accuracy.Paul Brooks, Jill Hardinuniversity-logo
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityPersonal/Subjective“What do you think are the odds?”What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?Relies on expert information (definition of “expert” is fluid).Advantage:always applicable - everybody has an opinionuseful in risk analysisDisadvantage: difficult (sometimes impossible?) todetermine accuracy.Paul Brooks, Jill Hardinuniversity-logo
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityPersonal/Subjective“What do you think are the odds?”What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?Relies on expert information (definition of “expert” is fluid).Advantage:always applicable - everybody has an opinionuseful in risk analysisDisadvantage: difficult (sometimes impossible?) todetermine accuracy.Paul Brooks, Jill Hardinuniversity-logo
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityPersonal/Subjective“What do you think are the odds?”What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?Relies on expert information (definition of “expert” is fluid).Advantage:always applicable - everybody has an opinionuseful in risk analysisDisadvantage: difficult (sometimes impossible?) todetermine accuracy.Paul Brooks, Jill Hardinuniversity-logo
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesInterpretations of ProbabilityPersonal/Subjective“What do you think are the odds?”What’s the chance of Florida repeating as NCAA basketballchampion?What’s the probability that a nuclear bomb will be deployedin your lifetime?Relies on expert information (definition of “expert” is fluid).Advantage:always applicable - everybody has an opinionuseful in risk analysisDisadvantage: difficult (sometimes impossible?) todetermine accuracy.Paul Brooks, Jill Hardinuniversity-logo
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesOutline1234Operations Research ModelsAxioms of ProbabilityDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesMarkov ChainsMarkov PropertyBlood Types IISimulationThe Nature of Simulation ModelingAn Example of a Discrete-Event SimulationPaul Brooks, Jill Hardinuniversity-logo
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules1. Probability is always between 0 and 1.probability of an event E is written P(E)2. If event E cannot occur then P(E) 0.E “Jill will grow to be 6 feet tall”. P(E) 0.3. If an event is certain, then P(E) 1.E “Class will end before midnight.” P(E) 1.4. The sum of the probabilities of all possible outcomes is 1.university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules1. Probability is always between 0 and 1.probability of an event E is written P(E)2. If event E cannot occur then P(E) 0.E “Jill will grow to be 6 feet tall”. P(E) 0.3. If an event is certain, then P(E) 1.E “Class will end before midnight.” P(E) 1.4. The sum of the probabilities of all possible outcomes is 1.university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules1. Probability is always between 0 and 1.probability of an event E is written P(E)2. If event E cannot occur then P(E) 0.E “Jill will grow to be 6 feet tall”. P(E) 0.3. If an event is certain, then P(E) 1.E “Class will end before midnight.” P(E) 1.4. The sum of the probabilities of all possible outcomes is 1.university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules1. Probability is always between 0 and 1.probability of an event E is written P(E)2. If event E cannot occur then P(E) 0.E “Jill will grow to be 6 feet tall”. P(E) 0.3. If an event is certain, then P(E) 1.E “Class will end before midnight.” P(E) 1.4. The sum of the probabilities of all possible outcomes is 1.university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules1. Probability is always between 0 and 1.probability of an event E is written P(E)2. If event E cannot occur then P(E) 0.E “Jill will grow to be 6 feet tall”. P(E) 0.3. If an event is certain, then P(E) 1.E “Class will end before midnight.” P(E) 1.4. The sum of the probabilities of all possible outcomes is 1.university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules1. Probability is always between 0 and 1.probability of an event E is written P(E)2. If event E cannot occur then P(E) 0.E “Jill will grow to be 6 feet tall”. P(E) 0.3. If an event is certain, then P(E) 1.E “Class will end before midnight.” P(E) 1.4. The sum of the probabilities of all possible outcomes is 1.university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules1. Probability is always between 0 and 1.probability of an event E is written P(E)2. If event E cannot occur then P(E) 0.E “Jill will grow to be 6 feet tall”. P(E) 0.3. If an event is certain, then P(E) 1.E “Class will end before midnight.” P(E) 1.4. The sum of the probabilities of all possible outcomes is 1.university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules1. Probability is always between 0 and 1.probability of an event E is written P(E)2. If event E cannot occur then P(E) 0.E “Jill will grow to be 6 feet tall”. P(E) 0.3. If an event is certain, then P(E) 1.E “Class will end before midnight.” P(E) 1.4. The sum of the probabilities of all possible outcomes is 1.university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules5. For two events A and B:P(A or B) P(A) P(B) P(A and B)P(A and B) P(A) P(B) P(A or B)university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules6. For two events A and BP(B occurs given that A occurs) P(B A) P(A and B)/P(A) P(A and B) P(B A)P(A)If P(B A) P(B) and P(A B) P(A) then A and B are saidto be independent.That is, knowing that one event will occur doesn’t give us anyinformation about the other.If A and B are independent, then P(A and B) P(A)P(B).university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules6. For two events A and BP(B occurs given that A occurs) P(B A) P(A and B)/P(A) P(A and B) P(B A)P(A)If P(B A) P(B) and P(A B) P(A) then A and B are saidto be independent.That is, knowing that one event will occur doesn’t give us anyinformation about the other.If A and B are independent, then P(A and B) P(A)P(B).university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules6. For two events A and BP(B occurs given that A occurs) P(B A) P(A and B)/P(A) P(A and B) P(B A)P(A)If P(B A) P(B) and P(A B) P(A) then A and B are saidto be independent.That is, knowing that one event will occur doesn’t give us anyinformation about the other.If A and B are independent, then P(A and B) P(A)P(B).university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability Rules6. For two events A and BP(B occurs given that A occurs) P(B A) P(A and B)/P(A) P(A and B) P(B A)P(A)If P(B A) P(B) and P(A B) P(A) then A and B are saidto be independent.That is, knowing that one event will occur doesn’t give us anyinformation about the other.If A and B are independent, then P(A and B) P(A)P(B).university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability RulesDefinitionMutually exclusive events are non-overlapping— that is, theycannot happen at the same time.A randomly chosen person is maleB randomly chosen person if femaleThese events are mutually exclusive.A randomly chosen person has blue eyesB randomly chosen person has brown hairThese events are not mutually exclusive.university-logoPaul Brooks, Jill Hardin
Operations Research ModelsAxioms of ProbabilityMarkov ChainsSimulationDefinitionInterpretations of ProbabilityProbability RulesExample: Blood TypesRandom VariablesProbability RulesBayes’ RuleSuppose A1 , A2 , . . . , Ak are mutually exclusive events sothatP(Ai ) 0, i 1, . . . , kP(A1 ) P(A2 ) · · · P(Ak ) 1 (i.e., they are exhaustive).Let B be another event with P(B) 0. ThenP(Ai B) P(Ai and B)P(B Ai )P(Ai ) kP(B)PP(B Ai )P(Ai )i 1university-logoPaul Brooks, J
Operations Research Models Axioms of Probability Markov Chains Simulation Probabilistic Operations Research Models Paul Brooks Jill Hardin Department of Statistical Sciences and Operations Research Virginia Commonwealth University BNFO 691 D
non-Bayesian approach, called Additive Regularization of Topic Models. ARTM is free of redundant probabilistic assumptions and provides a simple inference for many combined and multi-objective topic models. Keywords: Probabilistic topic modeling · Regularization of ill-posed inverse problems · Stochastic matrix factorization · Probabilistic .
deterministic polynomial-time algorithms. However, as argued next, we can gain a lot if we are willing to take a somewhat non-traditional step and allow probabilistic veriflcation procedures. In this primer, we shall survey three types of probabilistic proof systems, called interactive proofs, zero-knowledge proofs, and probabilistic checkable .
A Model for Uncertainties Data is probabilistic Queries formulated in a standard language Answers are annotated with probabilities This talk: Probabilistic Databases 9. 10 Probabilistic databases: Long History Cavallo&Pitarelli:1987 Barbara,Garcia-Molina, Porter:1992 Lakshmanan,Leone,Ross&Subrahmanian:1997
5 Probabilistic Relational Models Lise Getoor, Nir Friedman, Daphne Koller, Avi Pfeffer and Ben Taskar Probabilisticrelationalmodels (PRMs) are a rich representationlanguagefor struc-tured statistical models. They combine a frame-based logical representation with probabilistic semantic
What is this tutorial NOT about? Classification methods Kernel methodsKernel methods Discriminative models – Linear Discriminant Analysis (LDA) – Canonical Correlation Analysis (CCA) Probabilistic latent variable models – Probabilistic PCA – Probabilistic latent semantic indexin
Machine Learning: A Probabilistic Perspective Kevin Murphy, MIT Press, 2012 Probabilistic Graphical Models Daphne Koller & Nir Friedman, MIT Press, 2009 Supplemental Texts Pattern Recognition & Machine Learning, C.M. Bishop, Springer, 2007. Especially Chapter 8 Artificial Intellige
study on the probabilistic analysis of structural instability in underground openings. er efore, in this research a tunnel probabilistic structural stability analysis (TuPSA) was developed using rst order reliability method. For this purpose, at r st, the reliability problem and rst order reliability method (FORM) are explained, and
The blue light protocol is subject to CTR Policy exemplar standard 11 as follows “CTRs and any related recording or disclosure of personal information will be with the express consent of the individual (or when appropriate someone with parental responsibility for them), or if they lack capacity, assessed to be in their best interests applying the Mental Capacity Act 2005 and its Code of .
