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Bayesian Analysis of Bayesian Analysis of Stochastic System Stochastic System
DynamicsDynamics
Rudolf KulhavýRudolf Kulhavý
Why to study stochastic systems?Why to study stochastic systems?
� Dynamic modeling of the overall performance of
– value chains
– value networks
– virtual enterprisesValueadded
Valueadded
Price
Price
CC
AA
BB
Value added
Density
Why to study stochastic systems?Why to study stochastic systems?
� Dynamic modeling of the overall performance of
– value chains
– value networks
– virtual enterprises
� Estimation of probabilitiesof critical events or specificquantiles of randomvariables
Valueadded
Valueadded
Price
Price
CC
AA
BB
Value added
Density
Assets – Liabilities
Probabilitydensityfunction
Dynamic model ofassets & liabilities
Insolvencyprobability
Stochastic dynamic modelStochastic dynamic model
Generalization B
Conditionalprobability
density functions
Generalization A
Sampling period
State andmeasurement
“noise”
Discrete-timevalues
Timeindex
Stochastic dynamic modelStochastic dynamic model
Stochasticdifferential equationrepresentation
Conditional probability
representation
Markov chainMarkov chain
States… …
Controlled Markov chainControlled Markov chain
Exogenous inputs
States… …
Partially observed, controlled Markov Partially observed, controlled Markov
chainchain
Exogenous inputs
Measurements
States… …
Partially observed, controlled Markov Partially observed, controlled Markov
chain, with unknown parameterschain, with unknown parameters
… …
Parameters
Inputs
States
Measurements
Partially observed, controlled Markov Partially observed, controlled Markov
chain, with unknown parameterschain, with unknown parameters
… …
Parameters
Inputs
States
Measurements
Unknown model parameters can be Unknown model parameters can be
treated as extra statestreated as extra states
The augmentation of the state vector
� increases the dimensionality of the problem (and, thereby, uncertainty of the original states);
� adds additional nonlinearities.
On the other hand, it allows for explicit modelingof parameter variations.
Summary of modelSummary of model
Exogenousinputs
Measurements
States andparameters
…
Estimation of states (and parameters)Estimation of states (and parameters)
Time update
Measurement updateObservations
Past data sequences
QuantificationQuantification
of all uncertaintyof all uncertainty
via via probabilityprobability
BayesianBayesianinferenceinference
Functional recursionsFunctional recursions
Time updateTime update
Measurement updateMeasurement update
Likelihood PriorPosterior
Transitionprobabibility
PosteriorNext-stepprior
Product ruleProduct rule
Sum ruleSum rule
ProbabilityTheory
Sequential Monte Carlo approximationSequential Monte Carlo approximation
Time update
Measurement update
Replacing probabilities with samples
Particle filter (step 1)Particle filter (step 1)
Particle filter (step 2)Particle filter (step 2)
Data updateData update
Likelihood
Posterior
stateAlgorithm:
Resample from posterior samples with probabilities proportional to the likelihood values, then draw a sample from the corres-ponding kernel
density
pdf Replace posterior w/ samples
state
Likelihood
pdf Replace samples w/ smoothkernels
state
Likelihood
No need for explicit sampling, except Step 1
Samples come from the preced-ing iteration step
Particle filter (steps 3, 4)Particle filter (steps 3, 4)
Time updateTime update
Algorithm:
Pick up randomly one of the posterior samples, then draw a new sample from the corresponding transition probability density function
Particle filter has Particle filter has
been applied successbeen applied success--
fully in many areasfully in many areas
� Automated target recognitionAnuj Srivastava, Michael Miller , Ulf Grenander
� Bayesian networksDaphne Koller; Kevin Murphy
� Computational anatomyUlf Grenander, Michael Miller
� Mobile roboticsDieter Fox, Wolfram Burgard, Sebastian Thrun
� Neural networksNando de Freitas
� Signal processingPetar Djurić
� Tracking and guidanceDavid Salmond, Neil Gordon
� Visual shape and motionAndrew Blake, Michael Isard, John MacCormick
Illustrative exampleIllustrative example
� Consider a service company whose economic results depend critically on the performance of both the sales and service staff.
� The stocks of Sales Capacity and Service Capacityare measured in multiples of full-time equivalents (FTE) of an average sales or service person.
– This relates the labor capacity to the total performance of a team rather than the number of physical persons.
– Thus, hiring an additional person can increase the stock by more or less than one, depending on the actual person’s productivity.
Model structureModel structure
SimulationSimulation
resultsresults
estimate
EstimationEstimation
resultsresults
estimate
measurement
Model comparisonModel comparison
Posterior probability functionPosterior probability function
Model class index
Predictive density functionPredictive density function
Model class index
Approximate model comparisonApproximate model comparison
Monte Carlo approximationMonte Carlo approximation
Samples from posterior pdfSamples from posterior pdf
ReverendThomas BayesPierre-Simon,
Marquis de Laplace
So, what has So, what has Bayesian InferenceBayesian Inference to do to do
with System Dynamics?with System Dynamics?
Jay Wright Forrester
System Dynamics x Bayesian InferenceSystem Dynamics x Bayesian Inference
� System dynamics provides the modeler with practical methodology for convert-ing prior information into a dynamic model structure (highly informative priors).
Bayesian inference
� gives precise meaning to all modeling concepts;
� yields a coherent frame-work for consistently up-dating the prior state of knowledge with numerical evidence at hand;
� captures and combines all manifestations of uncertainty (stochastic fluctuations, measurement errors, unknown model parameters, unknown model structure).
Theoretical modelsWhite-box models
Phenomenological modelsGrey-box models
Empirical modelsBlack-box models
Problemoriented
Structurefocused
SDfocus
“[T]here is a loose connection between simplicity and plausibility, because the more complicated a set of possible hypotheses, the larger the manifold of conceivable alternatives, and so the smaller must be the prior probability of any particular hypothesis in the set.”
Edwin Jaynes, Probability Theory: The Logic of Science
Thus, among models of comparable predictive power, Bayesian inference assigns higher posterior probability to “simpler” ones.
Occam’s razorOccam’s razor
The amount of prior probability contained in the high likelihood region of parameter space
The maximum likelihood value
ConclusionConclusion
� The progress made in sequential Monte Carlo methods has made Bayesian inference an attractive option for system dynamics modeling, especially for problems where quantification of the state (and parameter) uncertainty is critical.
