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**Bayesian methods for calibrating and comparing process-based vegetation models**

Marcel van Oijen (CEH-Edinburgh)

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**Contents Process-based modelling of forests and uncertainties**

Bayes’ Theorem (BT) Bayesian Calibration (BC) of process-based models Bayesian Model Comparison (BMC) BC & BMC in NitroEurope Examples of BC & BMC in other sciences BC & BMC as tools to develop theory References, Summary, Discussion

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**1. Introduction: Process-based modelling of forests and uncertainties**

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**1.1 Forest growth in Europe**

Project RECOGNITION (FAIRCT ): 15 partner countries across Europe Previous observations RECOGNITION Forests across Europe have started to grow faster in the 20th century: Causes? Future trend? 22 sites Empirical methods + process-based modelling Modelling groups in UK, Sweden and Finland (2), coordinated by CEH-Edinburgh

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**1.2 Forest growth in Europe**

NPP before growth rate increase (1920) CONCLUSION 20th century Growth accelerated by N-deposition. Environmental change : Effects on NPP HOG PFZ HEL KAR PUS RAJ PFF SOL BRI LOP TRI GA2 GA1 ALT AAL SKO BLA JAD PUN KAN KEM KOL % Change in NPP -10 -5 5 10 15 20 25 CO 2 Climate N-deposition CUMULATIVE EFFECTS Latitude EFM CONCLUSION 21st century: Growth likely to be accelerated by climate change and increasing [CO2]. N-deposition CO2 Temperature

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**1.3 Reality check ! How reliable is the European forest study:**

Sufficient data for model parameterization? Sufficient data for model input? Would another model have given different results? In every study using systems analysis and simulation: Model parameters, inputs and structure are uncertain How to deal with uncertainties optimally?

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**1.4 Forest models and uncertainty**

[Levy et al, 2004]

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**1.4 Forest models and uncertainty**

NdepUE (kg C kg-1 N) bgc century hybrid [Levy et al, 2004]

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1.5 Model-data fusion Uncertainties are everywhere: Models (environmental inputs, parameters, structure), Data Uncertainties can be expressed as probability distributions (pdf’s) We need methods that: Quantify all uncertainties Show how to reduce them Efficiently transfer information: data models model application Calculating with uncertainties (pdf’s) = Probability Theory

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2. Bayes’ Theorem

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**2.1 Dealing with uncertainty: Medical diagnostics**

A flu epidemic occurs: one percent of people is ill P(dis) = 0.01 Diagnostic test, 99% reliable P(pos|hlth) = 0.01 P(pos|dis) = 0.99 P(dis|pos) = P(pos|dis) P(dis) / P(pos) Bayes’ Theorem Test result is positive (bad news!) What is P(diseased|test positive)? 0.50 0.98 0.99

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**2.1 Dealing with uncertainty: Medical diagnostics**

A flu epidemic occurs: one percent of people is ill P(dis) = 0.01 Diagnostic test, 99% reliable P(pos|hlth) = 0.01 P(pos|dis) = 0.99 Bayes’ Theorem Test result is positive (bad news!) What is P(diseased|test positive)? 0.50 0.98 0.99 P(dis|pos) = P(pos|dis) P(dis) / P(pos) = P(pos|dis) P(dis) P(pos|dis) P(dis) + P(pos|hlth) P(hlth)

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**2.1 Dealing with uncertainty: Medical diagnostics**

A flu epidemic occurs: one percent of people is ill P(dis) = 0.01 Diagnostic test, 99% reliable P(pos|hlth) = 0.01 P(pos|dis) = 0.99 Bayes’ Theorem Test result is positive (bad news!) What is P(diseased|test positive)? 0.50 0.98 0.99 P(dis|pos) = P(pos|dis) P(dis) / P(pos) = P(pos|dis) P(dis) P(pos|dis) P(dis) + P(pos|hlth) P(hlth) = =

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**2.2 Bayesian updating of probabilities**

Bayes’ Theorem: Prior probability → Posterior prob. Medical diagnostics: P(disease) → P(disease|test result) Model parameterization: P(params) → P(params|data) Model selection: P(models) → P(model|data) SPAM-killer: P(SPAM) → P(SPAM| header) Weather forecasting: … Climate change prediction: … Oil field discovery: … GHG-emission estimation: … Jurisprudence: … …

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**2.2 Bayesian updating of probabilities**

Bayes’ Theorem: Prior probability → Posterior prob. Model parameterization: P(params) → P(params|data) Model selection: P(models) → P(model|data) Application of Bayes’ Theorem to process-based models (not analytically solvable): Markov Chain Monte-Carlo (Metropolis algorithm)

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2.3 What and why? We want to use data and models to explain and predict ecosystem behaviour Data as well as model inputs, parameters and outputs are uncertain No prediction is complete without quantifying the uncertainty. No explanation is complete without analysing the uncertainty Uncertainties can be expressed as probability density functions (pdf’s) Probability theory tells us how to work with pdf’s: Bayes Theorem (BT) tells us how a pdf changes when new information arrives BT: Prior pdf Posterior pdf BT: Posterior = Prior x Likelihood / Evidence BT: P(θ|D) = P(θ) P(D|θ) / P(D) BT: P(θ|D) P(θ) P(D|θ)

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**3. Bayesian Calibration (BC) of process-based models**

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**3.1 Process-based forest models**

Environmental scenarios Height NPP Initial values Soil C Parameters Model

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**3.2 Process-based forest model BASFOR**

40+ parameters 12+ output variables BASFOR

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**3.3 BASFOR: outputs Carbon in trees Volume (standing + thinned)**

Carbon in soil

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**3.4 BASFOR: parameter uncertainty**

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**3.5 BASFOR: prior output uncertainty**

Carbon in trees (standing + thinned) Volume (standing) Carbon in soil

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**3.6 Data Dodd Wood (R. Matthews, Forest Research)**

Carbon in trees (standing + thinned) Volume (standing) Carbon in soil

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**3.7 Using data in Bayesian calibration of BASFOR**

Prior pdf Data Bayesian calibration Posterior pdf

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**3.8 Bayesian calibration: posterior uncertainty**

Carbon in trees (standing + thinned) Volume (standing) Carbon in soil

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**P(|D) = P() P(D| ) / P(D) P() P(D|f())**

3.9 How does BC work again? f = the model, e.g. BASFOR P(|D) = P() P(D| ) / P(D) P() P(D|f()) “Posterior distribution of parameters” “Prior distribution of parameters” “Likelihood” of data, given mismatch with model output

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**Bayesian calibration in action!**

Bayes’ Theorem: P( |D) P() P(D|(f()) Parameter prob. distr. Output Data

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**3.10 Calculating the posterior using MCMC**

MCMC trace plots P(|D) P() P(D|f()) Start anywhere in parameter-space: p1..39(i=0) Randomly choose p(i+1) = p(i) + δ IF: [ P(p(i+1)) P(D|f(p(i+1))) ] / [ P(p(i)) P(D|f(p(i))) ] > Random[0,1] THEN: accept p(i+1) ELSE: reject p(i+1) i=i+1 4. IF i < 104 GOTO 2 Bayesian Calibration of the forest model BASFOR cannot be done analytically. Therefore we use a Markov Chain Monte Carlo (MCMC) method. The MCMC does not give us a formula for the posterior parameter pdf, but it generates a representative sample from the posterior. The simplest MCMC-method is used: the Metropolis et al (1953) algorithm. Metropolis et al (1953) Sample of parameter vectors from the posterior distribution P(|D) for the parameters

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3.11 MCMC in action BC3D.AVI

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**3.12 Using data in Bayesian calibration of BASFOR**

Prior pdf Data Bayesian calibration Posterior pdf

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**3.13 Parameter correlations**

39 parameters 39 parameters

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**3.14 Continued calibration when new data become available**

Prior pdf Posterior pdf Bayesian calibration Prior pdf New data

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**3.14 Continued calibration when new data become available**

Prior pdf Posterior pdf Prior pdf New data Bayesian calibration

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**3.15 Bayesian projects at CEH-Edinburgh**

Parameterization and uncertainty quantification of 3-PG model of forest growth & C-stock (Genevieve Patenaude, Ronnie Milne, M. v.Oijen) [CO2] Uncertainty in earth system resilience (Clare Britton & David Cameron) Time Selection of forest models Data Assimilation forest EC data (David Cameron, Mat Williams, M.v.Oijen) Risk of frost damage in grassland Uncertainty in UK C-sequestration (Marcel van Oijen, Jonathan Rougier, Ron Smith, Tommy Brown, Amanda Thomson)

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**3.16 BASFOR: forest C-sequestration 2005-2076**

Change in annual mean Temperature Change in potential C-seq. Uncertainty in change of potential C-seq. UKCIP Uncertainty due to model parameters only, NOT uncertainty in inputs / upscaling

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**3.17 Integrating RS-data (Patenaude et al.)**

Model 3-PG BC RS-data: Hyper-spectral, LiDAR, SAR

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**3.18 What kind of measurements would have reduced uncertainty the most ?**

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**3.19 Prior predictive uncertainty & height-data**

Prior pred. uncertainty Height Biomass Height data Skogaby

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**3.20 Prior & posterior uncertainty: use of height data**

Prior pred. uncertainty Height Biomass Posterior uncertainty (using height data) Height data Skogaby

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**3.20 Prior & posterior uncertainty: use of height data**

Prior pred. uncertainty Height Biomass Posterior uncertainty (using height data) Height data (hypothet.)

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**3.20 Prior & posterior uncertainty: use of height data**

Prior pred. uncertainty Height Biomass Posterior uncertainty (using height data) Posterior uncertainty (using precision height data)

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**3.21 Summary for BC procedure**

Prior P() Model f Data D ± σ “Error function” e.g. N(0, σ) MCMC Samples of (104 – 105) Samples of f() P(D|f()) Posterior P(|D) PCC Calibrated parameters, with covariances Sensitivity analysis of model parameters Uncertainty of model output

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**3.22 Summary for BC vs tuning**

Model tuning Define parameter ranges (permitted values) Select parameter values that give model output closest (r2, RMSE, …) to data Do the model study with the tuned parameters (i.e. no model output uncertainty) Bayesian calibration Define parameter pdf’s Define data pdf’s (probable measurement errors) Use Bayes’ Theorem to calculate posterior parameter pdf Do all future model runs with samples from the parameter pdf (i.e. quantify uncertainty of model results) BC can use data to reduce parameter uncertainty for any process-based model

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**4. Bayesian Model Comparison (BMC)**

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**4.1 RECOGNITION revisited: model uncertainty**

EFM 25 20 15 10 5 Latitude

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**4.1 RECOGNITION revisited: model uncertainty**

EFM 25 20 Q 20 15 10 15 5 10 5 -5 Latitude -10 -5 40 HOG PFZ HEL KAR PUS RAJ PFF SOL BRI LOP TRI GA2 GA1 ALT AAL SKO BLA JAD PUN KAN KEM KOL EFIMOD FinnFor 30 20 20 10 10 Latitude HOG PFZ HEL KAR PUS RAJ PFF SOL BRI LOP TRI GA2 GA1 ALT AAL SKO BLA JAD PUN KAN KEM KOL Latitude -10 HOG PFZ HEL KAR PUS RAJ PFF SOL BRI LOP TRI GA2 GA1 ALT AAL SKO BLA JAD PUN KAN KEM KOL

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**4.2 Bayesian comparison of two models**

Bayes Theorem for model probab.: P(M|D) = P(M) P(D|M) / P(D) The “Integrated likelihood” P(D|Mi) can be approximated from the MCMC sample of outputs for model Mi (*) P(M1) = P(M2) = ½ P(M2|D) / P(M1|D) = P(D|M2) / P(D|M1) The “Bayes Factor” P(D|M2) / P(D|M1) quantifies how the data D change the odds of M2 over M1 (*) harmonic mean of likelihoods in MCMC-sample (Kass & Raftery, 1995)

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4.3 BMC: Tuomi et al. 2007

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**4.4 Bayes Factor for two big forest models**

Calculation of P(D|BASFOR) MCMC 5000 steps Data Rajec: Emil Klimo Calculation of P(D|BASFOR+) MCMC 5000 steps

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**4.5 Bayes Factor for two big forest models**

Calculation of P(D|BASFOR) P(D|M1) = 7.2e-016 P(D|M2) = 5.8e-15 MCMC 5000 steps Bayes Factor = 7.8, so BASFOR+ supported by the data Data Rajec: Emil Klimo Calculation of P(D|BASFOR+) MCMC 5000 steps

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**4.6 Summary of BMC procedure**

Data D Prior P(1) Updated parameters MCMC Samples of 1 (104 – 105) Posterior P(1|D) Model 1 MCMC Prior P(2) Model 2 Samples of 2 (104 – 105) Posterior P(2|D) Updated parameters P(D|M1) P(D|M2) Bayes factor Updated model odds

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5. BC & BMC in NitroEurope

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**5.1 NitroEurope & uncertainty**

What is the effect of reactive nitrogen supply on the direction and magnitude of net greenhouse gas budgets for Europe? This CEH co-ordinated IP builds on CEH’s involvement in other previous and current European GHG projects such as GREENGRASS, CarboMont and CarboEurope IP

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**5.2 NitroEurope & Uncertainty**

NitroEurope (NEU): non-CO2 GHG Europe experiments at plot-scale, observations at regional scale models at plot- and regional scale protocols for good-modelling practice and for uncertainty quantification and analysis (collab. with CEU in JUTF) Modellers NEU (2006)

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**5.3 Uncertainty assessment NEU models**

BC BC Plot scale forest model added in 2007: DAYCENT BC Yes

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**5.4 NEU – Forest model comparison 2007-8**

4 models (DNDC, BASFOR, COUP, DayCENT) Models frozen Calibration of models using data Höglwald (D) {Mainly N2O & NO-emission rates} Comparison of models using data AU & DK Bayesian Model Comparison (BMC) Bayesian Calibration (BC)

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Bayesian Calibration (BC) and Bayesian Model Comparison (BMC) of process-based models in NitroEurope: Theory, implementation and guidelines

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**Methods for doing BC: MCMC and Accept-Reject **

Bayesian Calibration (BC) and Bayesian Model Comparison (BMC) of process-based models in NitroEurope: Theory, implementation and guidelines Theory of BC and BMC Methods for doing BC: MCMC and Accept-Reject 3.1 Standard Metropolis algorithm 3.2 Metropolis with a modified proposal generating mechanism (“Reflection method”) 3.3 Accept-Reject algorithm FAQ – Bayesian Calibration References Appendix 1: MCMC code in MATLAB: the Metropolis algorithm Appendix 2: MCMC code in MATLAB: Metropolis-with-Reflection Appendix 3: ACCEPT-REJECT code in MATLAB Appendix 4: MCMC code in R: the Metropolis algorithm

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**5.6 BASFOR changes for NEU Soil temperature calculated**

Mineralisation of litter and SOM = f(Tsoil): Gaussian curve (Tuomi et al. 2007): f = exp[ (T-10) (2Tm-T-10) / 2σ2 ] Nemission split up into N2O and NO: Hole-In-the-Pipe (HIP) approach (Davidson & Verchot, 2000): fN2O = 1 / ( 1 + exp[-r(WFPS-WFPS50)] ) fN2O (-) Water-Filled Pore Space (WFPS) (-)

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**5.12 BC results: Prior & Posterior**

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**5.13 BC results: simulation uncertainty & data**

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**5.15 Data have information content, which is additive**

= +

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**BASFOR with T-sensitivity**

5.16 BMC BASFOR BASFOR with T-sensitivity log P(D) = Data log P(D) = BF = 1131.0 log P(D) = Data log P(D) = BF = 0.33

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**6. Examples of BC & BMC in other sciences**

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**6.1 Bayes in other disguises**

Linear regression using least squares BC, except that uncertainty is ignored Model: straight line Prior: uniform Likelihood: Gaussian (iid) = Note: Realising that LS-regression is a special case of BC opens up possibilities to improve on it, e.g. by having more information in the prior or likelihood (Sivia 2005) All Maximum Likelihood estimation methods can be seen as limited forms of BC where the prior is ignored (uniform) and only the maximum value of the likelihood is identified (ignoring uncertainty) BC, e.g. for spatiotemporal stochastic modelling with spatial correlations included in the prior Hierarchical modelling =

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**6.2 Bayes in other disguises (cont.)**

Inverse modelling (e.g. to estimate emission rates from concentrations) Geostatistics, e.g. Bayesian kriging Data Assimilation (KF, EnKF etc.)

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**6.3 Regional application of plot-scale models**

Upscaling method Model structure Modelling uncertainty 1. Stratify into homogeneous subregions & Apply Unchanged P(θ) unchanged Upscaling unc. 2. Apply to selected points (plots) & Interpolate Unchanged (but extend w. geostatistical model) P(θ) unchanged (Bayesian kriging only), Interpolation uncertainty 3. Reinterpret the model as a regional one & Apply New BC using regional I-O data 4. Summarise model behav. & Apply exhaustively (deterministic metamodel) E.g. multivariate regression model or simple mechanistic New BC needed of metamodel using plot-data 5. As 4. (stochastic emulator) E.g. Gaussian process emulator Code uncertainty (Kennedy & O’H.) 6. Summarise model behaviour & Embed in regional model Unrelated new model

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**7. References, Summary, Discussion**

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**7.1 Bayesian methods: References**

Bayes’ Theorem MCMC BMC Forest models Crop models Probability theory Complex process-based models, MCMC Bayes, T. (1763) Metropolis, N. (1953) Kass & Raftery (1995) Green, E.J. / MacFarlane, D.W. / Valentine, H.T. , Strawderman, W.E. (1996, 1998, 1999, 2000) Jansen, M. (1997) Jaynes, E.T. (2003) Van Oijen et al. (2005)

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**7.2 Discussion statements / Conclusions**

Uncertainty (= incomplete information) is described by pdf’s Plausible reasoning implies probability theory (PT) (Cox, Jaynes) Main tool from PT for updating pdf’s: Bayes Theorem Parameter estimation = quantifying joint parameter pdf Model evaluation = quantifying pdf in model space requires at least two models

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**7.2 Discussion statements / Conclusions**

Uncertainty (= incomplete information) is described by pdf’s Plausible reasoning implies probability theory (PT) (Cox, Jaynes) Main tool from PT for updating pdf’s: Bayes Theorem Parameter estimation = quantifying joint parameter pdf BC Model evaluation = quantifying pdf in model space requires at least two models BMC Practicalities: When new data arrive: MCMC provides a universal method for calculating posterior pdf’s Quantifying the prior: Not a key issue in env. sci.: (1) many data, (2) prior is posterior from previous calibration MaxEnt can be used (Jaynes) Defining the likelihood: Normal pdf for measurement error usually describes our prior state of knowledge adequately (Jaynes) Bayes Factor shows how new data change the odds of models, and is a by-product from Bayesian calibration (Kass & Raftery) Overall: Uncertainty quantification often shows that our models are not very reliable

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App2.1 How to do BC The problem: You have: (1) a prior pdf P(θ) for your model’s parameters, (2) new data. You also know how to calculate the likelihood P(D|θ). How do you now go about using BT to calculate the posterior P(θ|D)? Methods of using BT to calculate P(θ|D): Analytical. Only works when the prior and likelihood are conjugate (family-related). For example if prior and likelihood are normal pdf’s, then the posterior is normal too. Numerical. Uses sampling. Three main methods: MCMC (e.g. Metropolis, Gibbs) Sample directly from the posterior. Best for high-dimensional problems Accept-Reject Sample from the prior, then reject some using the likelihood. Best for low-dimensional problems Model emulation followed by MCMC or A-R

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**Should we measure the “sensitive parameters”?**

Yes, because the sensitive parameters: are obviously important for prediction ? No, because model parameters: are correlated with each other, which we do not measure cannot really be measured at all So, it may be better to measure output variables, because they: are what we are interested in are better defined, in models and measurements help determine parameter correlations if used in Bayesian calibration Key question: what data are most informative?

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