COMPUTATIONAL PROBABILITY
CONTENTS APPLICATIONS AND PRACTICES OF MODELLING, RISK AND UNCERTAINTY PROTECTION AGAINST NATURAL RISK THE POPULAR ‘INITIATOR/FREQUENCY APPROACH’ RECENT DEVELOPMENTS TOWARDS AN ‘EXTENDED FREQUENCY APPROACH’ ENGINEERING DESIGN, SAFETY AND STRUCTURAL RELIABILITY ANALYSIS (SRA) THE DOMAIN OF STRUCTURAL RELIABILITY DETERMINISTIC SAFETY MARGINS AND PARTIAL SAFETY FACTORS PROBABILISTIC STRUCTURAL RELIABILITY ANALYSIS LINKS AND DIFFERENCES WITH NATURAL RISK STUDIES INDUSTRIAL SAFETY, SYSTEM RELIABILITY AND PROBABILISTIC RISK ASSESSMENT (PRA) THE CONTEXT OF SYSTEMS ANALYSIS LINKS AND DIFFERENCES WITH STRUCTURAL RELIABILITY ANALYSIS THE CASE OF ELABORATE PRA (MULTI-STATE, DYNAMIC) INTEGRATED PROBABILISTIC RISK ASSESSMENT (IPRA) MODELLING UNDER UNCERTAINTY IN METROLOGY, ENVIRONMENTAL/SANITARY ASSESSMENT AND NUMERICAL ANALYSIS UNCERTAINTY AND SENSITIVITY ANALYSIS (UASA) SPECI?CITIES IN METROLOGY/INDUSTRIAL QUALITY CONTROL SPECI?CITIES IN ENVIRONMENTAL/HEALTH IMPACT ASSESSMENT NUMERICAL CODE QUALI?CATION (NCQ), CALIBRATION AND DATA ASSIMILATION FORECAST AND TIME-BASED MODELLING IN WEATHER, OPERATIONS RESEARCH, ECONOMICS OR ?NANCE CONCLUSION: THE SCOPE FOR GENERIC MODELLING UNDER RISK AND UNCERTAINTY SIMILAR AND DISSIMILAR FEATURES IN MODELLING, RISK AND UNCERTAINTY STUDIES LIMITATIONS AND CHALLENGES MOTIVATING A UNI?ED FRAMEWORK REFERENCES A GENERIC MODELLING FRAMEWORK THE SYSTEM UNDER UNCERTAINTY DECISIONAL QUANTITIES AND GOALS OF MODELLING UNDER RISK AND UNCERTAINTY THE KEY CONCEPT OF RISK MEASURE OR QUANTITY OF INTEREST SALIENT GOALS OF RISK/UNCERTAINTY STUDIES AND DECISION-MAKING MODELLING UNDER UNCERTAINTY: BUILDING SEPARATE SYSTEM AND UNCERTAINTY MODELS THE NEED TO GO BEYOND DIRECT STATISTICS BASIC SYSTEM MODELS BUILDING A DIRECT UNCERTAINTY MODEL ON VARIABLE INPUTS DEVELOPING THE UNDERLYING EPISTEMIC/ALEATORY STRUCTURE SUMMARY MODELLING UNDER UNCERTAINTY – THE GENERAL CASE PHENOMENOLOGICAL MODELS UNDER UNCERTAINTY AND RESIDUAL MODEL ERROR THE MODEL BUILDING PROCESS COMBINING SYSTEM AND UNCERTAINTY MODELS INTO AN INTEGRATED STATISTICAL ESTIMATION PROBLEM THE COMBINATION OF SYSTEM AND UNCERTAINTY MODELS: A KEY INFORMATION CHOICE THE PREDICTIVE MODEL COMBINING SYSTEM AND UNCERTAINTY COMPONENTS COMBINING PROBABILISTIC AND DETERMINISTIC SETTINGS PRELIMINARY COMMENTS ABOUT THE INTERPRETATIONS OF PROBABILISTIC UNCERTAINTY MODELS MIXED DETERMINISTIC-PROBABILISTIC CONTEXTS COMPUTING AN APPROPRIATE RISK MEASURE OR QUANTITY OF INTEREST AND ASSOCIATED SENSITIVITY INDICES STANDARD RISK MEASURES OR Q I (SINGLE-PROBABILISTIC) A FUNDAMENTAL CASE: THE CONDITIONAL EXPECTED UTILITY RELATIONSHIP BETWEEN RISK MEASURES, UNCERTAINTY MODEL AND ACTIONS DOUBLE PROBABILISTIC RISK MEASURES THE DELICATE ISSUE OF PROPAGATION/NUMERICAL UNCERTAINTY IMPORTANCE RANKING AND SENSITIVITY ANALYSIS SUMMARY: MAIN STEPS OF THE STUDIES AND LATER ISSUES EXERCISES A GENERIC TUTORIAL EXAMPLE: NATURAL RISK IN AN INDUSTRIAL INSTALLATION PHENOMENOLOGY AND MOTIVATION OF THE EXAMPLE THE HYDRO COMPONENT THE SYSTEM’S RELIABILITY COMPONENT THE ECONOMIC COMPONENT UNCERTAIN INPUTS, DATA AND EXPERTISE AVAILABLE A SHORT INTRODUCTION TO GRADUAL ILLUSTRATIVE MODELLING STEPS STEP ONE: NATURAL RISK STANDARD STATISTICS STEP TWO: MIXING STATISTICS AND A QRA MODEL STEP THREE: UNCERTAINTY TREATMENT OF A PHYSICAL/ENGINEERING MODEL (SRA) STEP FOUR: MIXING SRA AND QRA STEP ?VE: LEVEL-2 UNCERTAINTY STUDY ON MIXED SRA-QRA MODEL STEP SIX: CALIBRATION OF THE HYDRO COMPONENT AND UPDATING OF RISK MEASURE STEP SEVEN: ECONOMIC ASSESSMENT AND OPTIMISATION UNDER RISK AND/OR UNCERTAINTY SUMMARY OF THE EXAMPLE UNDERSTANDING NATURES OF UNCERTAINTY, RISK MARGINS AND TIME BASES FOR PROBABILISTIC DECISION-MAKING NATURES OF UNCERTAINTY: THEORETICAL DEBATES AND PRACTICAL IMPLEMENTATION DE?NING UNCERTAINTY – AMBIGUITY ABOUT THE REFERENCE RISK VS UNCERTAINTY – AN IMPRACTICAL DISTINCTION THE ALEATORY/EPISTEMIC DISTINCTION AND THE ISSUE OF REDUCIBILITY VARIABILITY OR UNCERTAINTY – THE NEED FOR CAREFUL SYSTEM SPECI?CATION OTHER DISTINCTIONS UNDERSTANDING THE IMPACT ON MARGINS OF DETERMINISTIC VS PROBABILISTIC FORMULATIONS UNDERSTANDING PROBABILISTIC AVERAGING, DEPENDENCE ISSUES AND DETERMINISTIC MAXIMISATION AND IN THE LINEAR CASE UNDERSTANDING SAFETY FACTORS AND QUANTILES IN THE MONOTONOUS CASE PROBABILITY LIMITATIONS, PARADOXES OF THE MAXIMAL ENTROPY PRINCIPLE DETERMINISTIC SETTINGS AND INTERVAL COMPUTATION – USES AND LIMITATIONS CONCLUSIVE COMMENTS ON THE USE OF PROBABILISTIC AND DETERMINISTIC RISK MEASURES HANDLING TIME-CUMULATED RISK MEASURES THROUGH FREQUENCIES AND PROBABILITIES THE UNDERLYING TIME BASIS OF THE STATE OF THE SYSTEM UNDERSTANDING FREQUENCY VS PROBABILITY FUNDAMENTAL RISK MEASURES DE?NED OVER A PERIOD OF INTEREST HANDLING A TIME PROCESS AND ASSOCIATED SIMPLI?CATIONS MODELLING RARE EVENTS THROUGH EXTREME VALUE THEORY CHOOSING AN ADEQUATE RISK MEASURE – DECISION-THEORY ASPECTS THE SALIENT GOAL INVOLVED THEORETICAL DEBATE AND INTERPRETATIONS ABOUT THE RISK MEASURE WHEN SELECTING BETWEEN RISKY ALTERNATIVES (OR CONTROLLING COMPLIANCE WITH A RISK TARGET) THE CHOICE OF ?NANCIAL RISK MEASURES THE CHALLENGES ASSOCIATED WITH USING DOUBLE-PROBABILISTIC OR CONDITIONAL PROBABILISTIC RISK MEASURES SUMMARY RECOMMENDATIONS DIRECT STATISTICAL ESTIMATION TECHNIQUES THE GENERAL ISSUE INTRODUCING ESTIMATION TECHNIQUES ON INDEPENDENT SAMPLES ESTIMATION BASICS GOODNESS-OF-?T AND MODEL SELECTION TECHNIQUES A NON-PARAMETRIC METHOD: KERNEL MODELLING ESTIMATING PHYSICAL VARIABLES IN THE ?OOD EXAMPLE DISCRETE EVENTS AND TIME-BASED STATISTICAL MODELS (FREQUENCIES, RELIABILITY MODELS, TIME SERIES) ENCODING PHENOMENOLOGICAL KNOWLEDGE AND PHYSICAL CONSTRAINTS INSIDE THE CHOICE OF INPUT DISTRIBUTIONS MODELLING DEPENDENCE LINEAR CORRELATIONS RANK CORRELATIONS COPULA MODEL MULTI-DIMENSIONAL NON-PARAMETRIC MODELLING PHYSICAL DEPENDENCE MODELLING AND CONCLUDING COMMENTS CONTROLLING EPISTEMIC UNCERTAINTY THROUGH CLASSICAL OR BAYESIAN ESTIMATORS EPISTEMIC UNCERTAINTY IN THE CLASSICAL APPROACH CLASSICAL APPROACH FOR GAUSSIAN UNCERTAINTY MODELS (SMALL SAMPLES) ASYMPTOTIC COVARIANCE FOR LARGE SAMPLES BOOTSTRAP AND RESAMPLING TECHNIQUES BAYESIAN-PHYSICAL SETTINGS (SMALL SAMPLES WITH EXPERT JUDGEMENT) UNDERSTANDING RARE PROBABILITIES AND EXTREME VALUE STATISTICAL MODELLING THE ISSUE OF EXTRAPOLATING BEYOND DATA – ADVANTAGES AND LIMITATIONS OF THE EXTREME VALUE THEORY THE SIGNI?CANCE OF EXTREMELY LOW PROBABILITIES COMBINED MODEL ESTIMATION THROUGH INVERSE TECHNIQUES INTRODUCING INVERSE TECHNIQUES HANDLING CALIBRATION DATA MOTIVATIONS FOR INVERSE MODELLING AND ASSOCIATED LITERATURE KEY DISTINCTIONS BETWEEN THE ALGORITHMS: THE REPRESENTATION OF TIME AND UNCERTAINTY ONE-DIMENSIONAL INTRODUCTION OF THE GRADUAL INVERSE ALGORITHMS DIRECT LEAST SQUARE CALIBRATION WITH TWO ALTERNATIVE INTERPRETATIONS BAYESIAN UPDATING, IDENTI?CATION AND CALIBRATION AN ALTERNATIVE IDENTI?CATION MODEL WITH INTRINSIC UNCERTAINTY COMPARISON OF THE ALGORITHMS ILLUSTRATIONS IN THE ?OOD EXAMPLE THE GENERAL STRUCTURE OF INVERSE ALGORITHMS: RESIDUALS, IDENTI?ABILITY, ESTIMATORS, SENSITIVITY AND EPISTEMIC UNCERTAINTY THE GENERAL ESTIMATION PROBLEM RELATIONSHIP BETWEEN OBSERVATIONAL DATA AND PREDICTIVE OUTPUTS FOR DECISION-MAKING COMMON FEATURES TO THE DISTRIBUTIONS AND ESTIMATION PROBLEMS ASSOCIATED TO THE GENERAL STRUCTURE HANDLING RESIDUALS AND THE ISSUE OF MODEL UNCERTAINTY ADDITIONAL COMMENTS ON THE MODEL-BUILDING PROCESS IDENTI?ABILITY IMPORTANCE FACTORS AND ESTIMATION ACCURACY SPECI?CITIES FOR PARAMETER IDENTI?CATION, CALIBRATION OR DATA ASSIMILATION ALGORITHMS THE BLUE ALGORITHM FOR LINEAR GAUSSIAN PARAMETER IDENTI?CATION AN EXTENSION WITH UNKNOWN VARIANCE: MULTIDIMENSIONAL MODEL CALIBRATION GENERALISATIONS TO NON-LINEAR CALIBRATION BAYESIAN MULTIDIMENSIONAL MODEL UPDATING DYNAMIC DATA ASSIMILATION INTRINSIC VARIABILITY IDENTI?CATION A GENERAL FORMULATION LINEARISED GAUSSIAN CASE NON-LINEAR GAUSSIAN EXTENSIONS MOMENT METHODS RECENT ALGORITHMS AND RESEARCH ?ELDS CONCLUSION: THE MODELLING PROCESS AND OPEN STATISTICAL AND COMPUTING CHALLENGES COMPUTATIONAL METHODS FOR RISK AND UNCERTAINTY PROPAGATION CLASSIFYING THE RISK MEASURE COMPUTATIONAL ISSUES RISK MEASURES IN RELATION TO CONDITIONAL AND COMBINED UNCERTAINTY DISTRIBUTIONS EXPECTATION-BASED SINGLE PROBABILISTIC RISK MEASURES SIMPLI?ED INTEGRATION OF SUB-PARTS WITH DISCRETE INPUTS NON-EXPECTATION BASED SINGLE PROBABILISTIC RISK MEASURES OTHER RISK MEASURES (DOUBLE PROBABILISTIC, MIXED DETERMINISTIC-PROBABILISTIC) THE GENERIC MONTE-CARLO SIMULATION METHOD AND ASSOCIATED ERROR CONTROL UNDERTAKING MONTE-CARLO SIMULATION ON A COMPUTER DUAL INTERPRETATION AND PROBABILISTIC PROPERTIES OF MONTE-CARLO SIMULATION CONTROL OF PROPAGATION UNCERTAINTY: ASYMPTOTIC RESULTS CONTROL OF PROPAGATION UNCERTAINTY: ROBUST RESULTS FOR QUANTILES (WILKS FORMULA) SAMPLING DOUBLE-PROBABILISTIC RISK MEASURES SAMPLING MIXED DETERMINISTIC-PROBABILISTIC MEASURES CLASSICAL ALTERNATIVES TO DIRECT MONTE-CARLO SAMPLING OVERVIEW OF THE COMPUTATION ALTERNATIVES TO MCS TAYLOR APPROXIMATION (LINEAR OR POLYNOMIAL SYSTEM MODELS) NUMERICAL INTEGRATION ACCELERATED SAMPLING (OR VARIANCE REDUCTION) RELIABILITY METHODS (FORM-SORM AND DERIVED METHODS) POLYNOMIAL CHAOS AND STOCHASTIC DEVELOPMENTS RESPONSE SURFACE OR META-MODELS MONOTONY, REGULARITY AND ROBUST RISK MEASURE COMPUTATION SIMPLE EXAMPLES OF MONOTONOUS BEHAVIOURS DIRECT CONSEQUENCES OF MONOTONY FOR COMPUTING THE RISK MEASURE ROBUST COMPUTATION OF EXCEEDANCE PROBABILITY IN THE MONOTONOUS CASE USE OF OTHER FORMS OF SYSTEM MODEL REGULARITY SENSITIVITY ANALYSIS AND IMPORTANCE RANKING ELEMENTARY INDICES AND IMPORTANCE MEASURES AND THEIR EQUIVALENCE IN LINEAR SYSTEM MODELS SOBOL SENSITIVITY INDICES SPECI?CITIES OF BOOLEAN INPUT/OUTPUT EVENTS – IMPORTANCE MEASURES IN RISK ASSESSMENT CONCLUDING REMARKS AND FURTHER RESEARCH NUMERICAL CHALLENGES, DISTRIBUTED COMPUTING AND USE OF DIRECT OR ADJOINT DIFFERENTIATION OF CODES OPTIMISING UNDER UNCERTAINTY: ECONOMICS AND COMPUTATIONAL CHALLENGES GETTING THE COSTS INSIDE RISK MODELLING – FROM ENGINEERING ECONOMICS TO ?NANCIAL MODELLING MOVING TO COSTS AS OUTPUT VARIABLES OF INTEREST – ELEMENTARY ENGINEERING ECONOMICS COSTS OF UNCERTAINTY AND THE VALUE OF INFORMATION THE EXPECTED UTILITY APPROACH FOR RISK AVERSION NON-LINEAR TRANSFORMATIONS ROBUST DESIGN AND ALTERNATIVES MIXING COST EXPECTATION AND VARIANCE INSIDE THE OPTIMISATION PROCEDURE THE ROLE OF TIME – CASH ?OWS AND ASSOCIATED RISK MEASURES COSTS OVER A TIME PERIOD – THE CASH ?OW MODEL THE ISSUE OF DISCOUNTING VALUING TIME ?EXIBILITY OF DECISION-MAKING AND STOCHASTIC OPTIMISATION COMPUTATIONAL CHALLENGES ASSOCIATED TO OPTIMISATION STATIC OPTIMISATION (UTILITY-BASED) STOCHASTIC DYNAMIC PROGRAMMING COMPUTATION AND ROBUSTNESS CHALLENGES THE PROMISE OF HIGH PERFORMANCE COMPUTING THE COMPUTATIONAL LOAD OF RISK AND UNCERTAINTY MODELLING THE POTENTIAL OF HIGH-PERFORMANCE COMPUTING CONCLUSION: PERSPECTIVES OF MODELLING IN THE CONTEXT OF RISK AND UNCERTAINTY AND FURTHER RESEARCH OPEN SCIENTI?C CHALLENGES CHALLENGES INVOLVED BY THE DISSEMINATION OF ADVANCED MODELLING IN THE CONTEXT OF RISK AND UNCERTAINTY