Statistical Inference for Food Webs
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1 Statistical Inference for Food Webs Part I: Bayesian Melding Grace Chiu and Josh Gould Department of Statistics & Actuarial Science CMAR-Hobart Science Seminar, March 6,
2 Outline CMAR-Hobart Science Seminar, March 6,
3 Outline Overview: existing vs. statistical approaches for Trophic context Whole-system context: ecological network analysis (ENA) CMAR-Hobart Science Seminar, March 6,
4 Outline Overview: existing vs. statistical approaches for Trophic context Whole-system context: ecological network analysis (ENA) Present ENA techniques CMAR-Hobart Science Seminar, March 6,
5 Outline Overview: existing vs. statistical approaches for Trophic context Whole-system context: ecological network analysis (ENA) Present ENA techniques ENA statistical inference statistical perspective of mass balance Bayesian melding CMAR-Hobart Science Seminar, March 6,
6 Outline Overview: existing vs. statistical approaches for Trophic context Whole-system context: ecological network analysis (ENA) Present ENA techniques ENA statistical inference statistical perspective of mass balance Bayesian melding Example: Chesapeake Bay Mesohaline Network CMAR-Hobart Science Seminar, March 6,
7 Outline Overview: existing vs. statistical approaches for Trophic context Whole-system context: ecological network analysis (ENA) Present ENA techniques ENA statistical inference statistical perspective of mass balance Bayesian melding Example: Chesapeake Bay Mesohaline Network Summary and Conclusion CMAR-Hobart Science Seminar, March 6,
8 Aspects of a Food Web CMAR-Hobart Science Seminar, March 6,
9 Aspects of a Food Web CMAR-Hobart Science Seminar, March 6,
10 Aspects of a Food Web... that was only for trophic relations... CMAR-Hobart Science Seminar, March 6,
11 Aspects of a Food Web: (1) Trophic Food web, trophically CMAR-Hobart Science Seminar, March 6,
12 Aspects of a Food Web: (1) Trophic Food web, trophically structure of interdependence among species predator-prey links = feeding patterns links can be weighted: e.g. predation frequency CMAR-Hobart Science Seminar, March 6,
13 Aspects of a Food Web: (1) Trophic Food web, trophically structure of interdependence among species predator-prey links = feeding patterns links can be weighted: e.g. predation frequency Existing work for trophic analyses: examine interactions between providers (prey) and benefactors (predators) (semi-)quantitative techniques for systematic extraction of the meaning of these interactions CMAR-Hobart Science Seminar, March 6,
14 Aspects of a Food Web: (1) Trophic Food web, trophically structure of interdependence among species predator-prey links = feeding patterns links can be weighted: e.g. predation frequency Existing work for trophic analyses: examine interactions between providers (prey) and benefactors (predators) (semi-)quantitative techniques for systematic extraction of the meaning of these interactions e.g. trophic hierarchy / compartments CMAR-Hobart Science Seminar, March 6,
15 Aspects of a Food Web CMAR-Hobart Science Seminar, March 6,
16 Aspects of a Food Web and now, for whole-system relations... CMAR-Hobart Science Seminar, March 6,
17 Aspects of a Food Web CMAR-Hobart Science Seminar, March 6,
18 Aspects of a Food Web: (2) Whole System Ecological Network (whole-system food web) CMAR-Hobart Science Seminar, March 6,
19 Aspects of a Food Web: (2) Whole System Ecological Network (whole-system food web) trophic compartments and susbstance / energy throughput this interdependence is subject to system balance notion of balance based on thermodynamics CMAR-Hobart Science Seminar, March 6,
20 Aspects of a Food Web: (2) Whole System Ecological Network (whole-system food web) trophic compartments and susbstance / energy throughput this interdependence is subject to system balance notion of balance based on thermodynamics Existing Work: Ecological Network Analysis (ENA) deterministic biophysical theory in a balance model quantity of exchange of substance / energy among compartments extract characteristics of these quantities = describe interactions among compartments CMAR-Hobart Science Seminar, March 6,
21 Disclaimer Images of food webs were generated by a Google search. CMAR-Hobart Science Seminar, March 6,
22 Trophic analysis and ENA Main Goal: to understand / predict (e.g. over time) within-web interactions based on the quantities associated with the edges (arrows) between pairs of species / compartments CMAR-Hobart Science Seminar, March 6,
23 Trophic analysis and ENA Main Goal: to understand / predict (e.g. over time) within-web interactions based on the quantities associated with the edges (arrows) between pairs of species / compartments ISSUES: randomness of quantities ignored no formal statistical inference of interaction patterns or predictions for ENA, randomness = observed quantities don t balance some quantities are unobservable from field computer algorithms generate missing quantities to minimize imbalance (e.g. EcoSim / EcoPath) CMAR-Hobart Science Seminar, March 6,
24 Trophic analysis and ENA Main Goal: to understand / predict (e.g. over time) within-web interactions based on the quantities associated with the edges (arrows) between pairs of species / compartments ISSUES: randomness of quantities ignored no formal statistical inference of interaction patterns or predictions for ENA, randomness = observed quantities don t balance some quantities are unobservable from field computer algorithms generate missing quantities to minimize imbalance (e.g. EcoSim / EcoPath) = further complicating any inference attempts! CMAR-Hobart Science Seminar, March 6,
25 A statistician s ideas... CMAR-Hobart Science Seminar, March 6,
26 A statistician s ideas... Alternative Trophic Analysis: take a completely quantitative regression approach: accounts for randomness of data! accounts for substance exchange! and other variables proper inference possible from regression modelling includes prediction inference for pairwise links and compartments AND over time simple scatterplots to identify and interpret compartments CMAR-Hobart Science Seminar, March 6,
27 A statistician s ideas... Alternative Trophic Analysis: take a completely quantitative regression approach: accounts for randomness of data! accounts for substance exchange! and other variables proper inference possible from regression modelling includes prediction inference for pairwise links and compartments AND over time simple scatterplots to identify and interpret compartments that ll be Statistical Inference Part II CMAR-Hobart Science Seminar, March 6,
28 Statistical Inference Part I CMAR-Hobart Science Seminar, March 6,
29 Statistical Inference Part I ENA Statistical Inference: can overcome empirical imbalance by incorporating randomness through Bayesian Melding can fill in missing quantities by prediction inference within Bayesian framework can be extended to temporal model without explicit calibration (as opposed to, e.g. morphing multiple static analyses into a single dynamics model) CMAR-Hobart Science Seminar, March 6,
30 Statistical Inference Part I ENA Statistical Inference: can overcome empirical imbalance by incorporating randomness through Bayesian Melding can fill in missing quantities by prediction inference within Bayesian framework can be extended to temporal model without explicit calibration (as opposed to, e.g. morphing multiple static analysis into a single dynamics model) CMAR-Hobart Science Seminar, March 6,
31 Conventional ENA Simple e.g. from Ulanowicz (Comput. Biol. & Chem., 2004): Fix a transfer medium, e.g. nitrogen or heat. Let i, j = compartment label T ij = rate of transfer of medium from i to j X i = rate of exogenous input of medium to i E i = rate of external transfer of medium from i R i = rate of dissipation of medium from i CMAR-Hobart Science Seminar, March 6,
32 Conventional ENA Simple e.g. from Ulanowicz (Comput. Biol. & Chem., 2004): Fix a transfer medium, e.g. nitrogen or heat. Let i, j = compartment label T ij = rate of transfer of medium from i to j X i = rate of exogenous input of medium to i E i = rate of external transfer of medium from i R i = rate of dissipation of medium from i Balance Equation: total inflow rate = total outflow rate = X i + j T ji = j T ij + E i + R i CMAR-Hobart Science Seminar, March 6,
33 Conventional ENA Simple e.g. from Ulanowicz (Comput. Biol. & Chem., 2004): Fix a transfer medium, e.g. nitrogen or heat. Let i, j = compartment label T ij = rate of transfer of medium from i to j X i = rate of exogenous input of medium to i E i = rate of external transfer of medium from i R i = rate of dissipation of medium from i Balance Equation: total inflow rate = total outflow rate = X i + j T ji = j T ij + E i + R i Ideally, do this over all n compartments and K media. CMAR-Hobart Science Seminar, March 6,
34 Conventional ENA: Ideally System of n K equations: X (1) 1 + T (1) +1 = T (1) 1+ + E (1) 1 + R (1) 1. X (1) n + T (1) +n = T (1) n+ + E (1) n + R (1) n X (2) 1 + T (2) +1 = T (2) 1+ + E (1) 1 + R (2) 1. X (2) n + T (2) +n = T (2) n+ + E (1) n + R (2) n. X (K ) 1 + T (K ) +1 = T (K ) 1+ + E (K ) 1 + R (K ) 1. X (K ) n + T (K ) +n = T (K ) n+ + E (K ) n + R (K ) n CMAR-Hobart Science Seminar, March 6,
35 Conventional ENA X (1) 1 + T (1) +1 = T (1) 1+ + E (1) 1 + R (1) 1. X (1) n + T (1) +n = T (1) n+ + E (1) n + R (1) n CMAR-Hobart Science Seminar, March 6,
36 Conventional ENA X (1) 1 + T (1) +1 = T (1) 1+ + E (1) 1 + R (1) 1. X (1) n + T (1) +n = T (1) n+ + E (1) n + R (1) n Randomness = field data never satisfy equality CMAR-Hobart Science Seminar, March 6,
37 Conventional ENA X (1) 1 + T (1) +1 = T (1) 1+ + E (1) 1 + R (1) 1. X (1) n + T (1) +n = T (1) n+ + E (1) n + R (1) n Randomness = field data never satisfy equality Worse yet, only some quantities are observable from field... CMAR-Hobart Science Seminar, March 6,
38 Conventional ENA Example for n=4, K =1 Observe: X i, E i, R i for all i; T ij for all (i, j) except i=3 Unknown: T 31, T 32, T 34 = T 3+ X 1 + T 21 + T 31 + T 41 = T 12 + T 13 + T 14 + E 1 + R 1 X 2 + T 12 + T 32 + T 42 = T 21 + T 23 + T 24 + E 2 + R 2 X 3 + T 13 + T 23 + T 43 = T 31 + T 32 + T 34 + E 3 + R 3 X 4 + T 14 + T 24 + T 34 = T 41 + T 42 + T 43 + E 4 + R 4 CMAR-Hobart Science Seminar, March 6,
39 Conventional ENA Example for n=4, K =1 Observe: X i, E i, R i for all i; T ij for all (i, j) except i=3 Unknown: T 31, T 32, T 34 = T 3+ X 1 + T 21 + T 31 + T 41 = T 12 + T 13 + T 14 + E 1 + R 1 X 2 + T 12 + T 32 + T 42 = T 21 + T 23 + T 24 + E 2 + R 2 X 3 + T 13 + T 23 + T 43 = T 31 + T 32 + T 34 + E 3 + R 3 X 4 + T 14 + T 24 + T 34 = T 41 + T 42 + T 43 + E 4 + R 4 no balance = no theoretical solution for T 3+ CMAR-Hobart Science Seminar, March 6,
40 Conventional ENA Remedy: deduce T 3j s from observable auxiliary quantities CMAR-Hobart Science Seminar, March 6,
41 Conventional ENA Remedy: deduce T 3j s from observable auxiliary quantities e.g. theoretical relationship among T ij compartment production biomass. f (P ij, B ij,...) = T ij CMAR-Hobart Science Seminar, March 6,
42 Conventional ENA Remedy: deduce T 3j s from observable auxiliary quantities e.g. theoretical relationship among T ij compartment production biomass. f (P ij, B ij,...) = T ij even if a deduced T ij is free of uncertainty... CMAR-Hobart Science Seminar, March 6,
43 Conventional ENA Remedy: deduce T 3j s from observable auxiliary quantities e.g. theoretical relationship among T ij compartment production biomass. f (P ij, B ij,...) = T ij even if a deduced T ij is free of uncertainty... System of equations fails regardless... what then? CMAR-Hobart Science Seminar, March 6,
44 Conventional ENA Computer software to the rescue! CMAR-Hobart Science Seminar, March 6,
45 Conventional ENA Computer software to the rescue! tinker with quantities subject to certain criteria until equality (almost) holds criteria built into software can be mysterious to user restrict tinkering to deduced quantities only, if possible CMAR-Hobart Science Seminar, March 6,
46 Conventional ENA Computer software to the rescue! Lingo tinker with quantities subject to certain criteria until equality (almost) holds criteria built into software can be mysterious to user restrict tinkering to deduced quantities only, if possible (Program) Input: observed and deduced quantities (Program) Output: balanced quantities CMAR-Hobart Science Seminar, March 6,
47 Conventional ENA In light of im balance theory-based deduction coerced balance... CMAR-Hobart Science Seminar, March 6,
48 Conventional ENA In light of im(possible)balance theory-based deduction coerced balance... CMAR-Hobart Science Seminar, March 6,
49 Conventional ENA In light of im(possible)balance theory-based deduction coerced balance... Million $ question: How confident are we in the numbers?? CMAR-Hobart Science Seminar, March 6,
50 Conventional ENA: confidence Existing attempts: Sensitivity analyses perturb program input monitor behavior of program output CMAR-Hobart Science Seminar, March 6,
51 Conventional ENA: confidence Existing attempts: Sensitivity analyses perturb program input monitor behavior of program output but... How to make inference for underlying network structure? CMAR-Hobart Science Seminar, March 6,
52 Perspectives of Mass Balance CMAR-Hobart Science Seminar, March 6,
53 Perspectives of Mass Balance Physics: in = out CMAR-Hobart Science Seminar, March 6,
54 Perspectives of Mass Balance Physics: in = out Statistics: E( in ) = E( out ) CMAR-Hobart Science Seminar, March 6,
55 Perspectives of Mass Balance Physics: in = out Statistics: E( in ) = E( out ) Simplest example Let W i = X i + T +i w/ mean µ W U i = T i+ + E i w/ mean µ U R i w/ mean µ R CMAR-Hobart Science Seminar, March 6,
56 Perspectives of Mass Balance Physics: in = out Statistics: E( in ) = E( out ) Simplest example Let W i = X i + T +i w/ mean µ W U i = T i+ + E i w/ mean µ U R i w/ mean µ R = balance model: µ W = µ U + µ R CMAR-Hobart Science Seminar, March 6,
57 Perspectives of Mass Balance Physics: in = out Statistics: E( in ) = E( out ) Simplest example Let W i = X i + T +i w/ mean µ W U i = T i+ + E i w/ mean µ U R i w/ mean µ R = balance model: µ W = µ U + µ R single balance equation of unobservable quantities estimation and confidence statements via statistical inference CMAR-Hobart Science Seminar, March 6,
58 Bayesian Melding for ENA Inference CMAR-Hobart Science Seminar, March 6,
59 Bayesian Melding for ENA Inference Elements of deterministic modeling deterministic model M( ) model input θ model output φ CMAR-Hobart Science Seminar, March 6,
60 Bayesian Melding for ENA Inference Elements of deterministic modeling deterministic model M( ) model input θ model output φ M : θ φ or φ := M(θ) CMAR-Hobart Science Seminar, March 6,
61 Bayesian Melding for ENA Inference Elements of deterministic modeling deterministic model M( ) model input θ = E(observables) model output φ = E(unobservables) M : θ φ or φ := M(θ) Rationale for choice of input/output: CMAR-Hobart Science Seminar, March 6,
62 Bayesian Melding for ENA Inference Elements of deterministic modeling deterministic model M( ) model input θ = E(observables) model output φ = E(unobservables) M : θ φ or φ := M(θ) Rationale for choice of input/output: to make statistical inference, need assumptions about statistical behavior for quantities experience with observed quantities can be basis of such assumptions then statistical behavior of E(unobservables) is defined by M through those of E(observables) CMAR-Hobart Science Seminar, March 6,
63 Bayesian Melding for ENA Inference Among W i, U i, R i, suppose R i is unobservable CMAR-Hobart Science Seminar, March 6,
64 Bayesian Melding for ENA Inference Among W i, U i, R i, suppose R i is unobservable = rewrite balance model as or µ R = µ W µ U φ = M(θ) CMAR-Hobart Science Seminar, March 6,
65 Bayesian Melding for ENA Inference Among W i, U i, R i, suppose R i is unobservable = rewrite balance model as or µ R = µ W µ U φ = M(θ) where φ = µ R model output θ = (µ W, µ U ) model input M(θ) = (1, 1)θ the model M : θ φ CMAR-Hobart Science Seminar, March 6,
66 Bayesian Melding for ENA Inference Bayesian Inference CMAR-Hobart Science Seminar, March 6,
67 Bayesian Melding for ENA Inference Bayesian Inference specify prior and likelihood for φ, θ obtain posterior for φ, θ = posterior inference CMAR-Hobart Science Seminar, March 6,
68 Bayesian Melding for ENA Inference Bayesian Inference specify prior and likelihood for φ, θ obtain posterior for φ, θ = posterior inference Bayesian Melding (due to Poole & Raftery, 2000) CMAR-Hobart Science Seminar, March 6,
69 Bayesian Melding for ENA Inference Bayesian Inference specify prior and likelihood for φ, θ obtain posterior for φ, θ = posterior inference Bayesian Melding (due to Poole & Raftery, 2000) two priors for φ: specified prior, h(φ) induced prior, h (φ), from cranking prior for θ through M CMAR-Hobart Science Seminar, March 6,
70 Bayesian Melding for ENA Inference θ (1) M(θ (1) ) = φ (1).. θ (m) M(θ (m) ) = φ (m) g(θ) input prior h (φ) induced output prior CMAR-Hobart Science Seminar, March 6,
71 Bayesian Melding for ENA Bayesian Inference specify prior and likelihood for φ, θ obtain posterior for φ, θ = posterior inference predictions for missing T 3i s via posterior predictive Bayesian Melding (due to Poole & Raftery, 2000) two priors for φ: specified prior, h(φ) induced prior, h (φ), from cranking prior for θ through M CMAR-Hobart Science Seminar, March 6,
72 Bayesian Melding for ENA Bayesian Inference specify prior and likelihood for φ, θ obtain posterior for φ, θ = posterior inference predictions for missing T 3i s via posterior predictive Bayesian Melding (due to Poole & Raftery, 2000) two priors for φ: specified prior, h(φ) induced prior, h (φ), from cranking prior for θ through M combine both priors = melded prior for φ, q(φ) CMAR-Hobart Science Seminar, March 6,
73 Bayesian Melding for ENA Inference Melded prior for φ is for some γ (0,1). q(φ) h (φ) γ h(φ) 1 γ CMAR-Hobart Science Seminar, March 6,
74 Bayesian Melding for ENA Inference Melded prior for φ is for some γ (0,1). q(φ) h (φ) γ h(φ) 1 γ What s γ? arbitrary in principle can be specified to reflect expert opinions on relative reliability between h and h (or M) CMAR-Hobart Science Seminar, March 6,
75 Bayesian Melding for ENA Inference Bayesian Inference specify prior and likelihood for φ, θ obtain posterior for φ, θ = posterior inference predictions for missing T 3i s via posterior predictive Bayesian Melding (due to Poole & Raftery, 2000) two priors for φ: specified prior, f (φ) induced prior, f (φ), from cranking prior for θ through M combine both priors = melded prior for φ, q(φ) CMAR-Hobart Science Seminar, March 6,
76 Bayesian Melding for ENA Inference Bayesian Inference specify prior and likelihood for φ, θ obtain posterior for φ, θ = posterior inference predictions for missing T 3i s via posterior predictive Bayesian Melding (due to Poole & Raftery, 2000) two priors for φ: specified prior, f (φ) induced prior, f (φ), from cranking prior for θ through M combine both priors = melded prior for φ, q(φ) crank melded prior q(φ) through M 1 = melded prior for θ, p(θ) CMAR-Hobart Science Seminar, March 6,
77 Bayesian Melding for ENA Inference Bayesian Inference specify prior and likelihood for φ, θ obtain posterior for φ, θ = posterior inference predictions for missing T 3i s via posterior predictive Bayesian Melding (due to Poole & Raftery, 2000) two priors for φ: specified prior, f (φ) induced prior, f (φ), from cranking prior for θ through M combine both priors = melded prior for φ, q(φ) crank melded prior q(φ) through M 1 = melded prior for θ, p(θ) there are tricks to overcome non-invertibility of M CMAR-Hobart Science Seminar, March 6,
78 Bayesian Melding for ENA Inference Bayesian Inference specify prior and likelihood for φ, θ obtain posterior for φ, θ = posterior inference predictions for missing T 3i s via posterior predictive Bayesian Melding (due to Poole & Raftery, 2000) two priors for φ: specified prior, f (φ) induced prior, f (φ), from cranking prior for θ through M combine both priors = melded prior for φ, q(φ) crank melded prior q(φ) through M 1 posterior inference based on p and M there are tricks to overcome non-invertibility of M = melded prior for θ, p(θ) CMAR-Hobart Science Seminar, March 6,
79 Bayesian Melding for ENA Inference posterior for θ: π θ (θ) p(θ) L θ (θ) L φ ( M(θ) ) CMAR-Hobart Science Seminar, March 6,
80 Bayesian Melding for ENA Inference posterior for θ: π θ (θ) p(θ) L θ (θ) L φ ( M(θ) ) posterior for φ: M : π θ (θ) π φ (φ) CMAR-Hobart Science Seminar, March 6,
81 Bayesian Melding for ENA Inference posterior for θ: π θ (θ) p(θ) L θ (θ) L φ ( M(θ) ) posterior for φ: M : π θ (θ) π φ (φ) Hallelujah! CMAR-Hobart Science Seminar, March 6,
82 Application: Chesapeake Bay Summer Network based on Baird & Ulanowicz (1989), Ecological Monographs 59, and ulan/ntwk/datall.zip CMAR-Hobart Science Seminar, March 6,
83 Application: Chesapeake Bay Summer Network CMAR-Hobart Science Seminar, March 6,
84 Application: Chesapeake Bay Summer Network CMAR-Hobart Science Seminar, March 6,
85 Application: Chesapeake Bay Summer Network transfer medium: carbon (g/m 2 /summer) i=1,..., 36 compartments CMAR-Hobart Science Seminar, March 6,
86 Application: Chesapeake Bay Summer Network transfer medium: carbon (g/m 2 /summer) i=1,..., 36 compartments Elements of Bayesian inference Likelihood: W i, U i, R i θ, φ independent exponentials specified prior: (θ, φ) trivariate log-normal CMAR-Hobart Science Seminar, March 6,
87 Application: Chesapeake Bay Summer Network transfer medium: carbon (g/m 2 /summer) i=1,..., 36 compartments Elements of Bayesian inference Likelihood: W i, U i, R i θ, φ independent exponentials specified prior: (θ, φ) trivariate log-normal = W i, U i, R i are marginally dependent (as would be necessary due to theoretical balance) CMAR-Hobart Science Seminar, March 6,
88 Application: Chesapeake Bay Summer Network NOTE: Instead of painstaking extraction of (unbalanced) data from the flow diagram, we adopted the online data (already balanced) would ideally use former. CMAR-Hobart Science Seminar, March 6,
89 Application: Chesapeake Bay Summer Network Data: W U R Specified prior ( ) and Melded Posterior ( ): θ θ φ CMAR-Hobart Science Seminar, March 6,
90 Application: Chesapeake Bay Summer Network Estimates and 95% Confidence intervals: θ 1 = µ W θ 2 = µ U φ = µ R Melding Posterior Mean HPD interval (72.69, ) (47.99, 80.84) (17.57, 33.23) Classical CLT interval (33.24, ) (20.70, ) (4.24, 43.80) CMAR-Hobart Science Seminar, March 6,
91 Application: Chesapeake Bay Summer Network Estimates and 95% Confidence intervals: θ 1 = µ W θ 2 = µ U φ = µ R Melding Posterior Mean HPD interval (72.69, ) (47.99, 80.84) (17.57, 33.23) Classical CLT interval (33.24, ) (20.70, ) (4.24, 43.80) Classical intervals are MUCH WIDER! CMAR-Hobart Science Seminar, March 6,
92 Application: Chesapeake Bay Summer Network Dependence among θ 1, θ 2, φ: θ φ φ θ θ θ 2 CMAR-Hobart Science Seminar, March 6,
93 Application: Chesapeake Bay Summer Network Dependence among θ 1, θ 2, φ: θ φ φ θ 1 High dependence is presumed among all 3 due to theoretical balance. Inference indicates dependence is only high between µ W and µ U = new insight! Note: inference on dependence structure NOT possible with classical inference (which treats µ s as constants). CMAR-Hobart Science Seminar, March 6, θ 1 θ 2
94 Summary Physics Statistics (1) each quantity (variable) no random variable needs must satisfy within- to satisfy balance compartment balance = no compartment needs = collapse compart- to satisfy balance ments to allow balance random variables have = wrong biology expectations that satisfy within-system balance = the more compartments (i.e. data) the better! CMAR-Hobart Science Seminar, March 6,
95 Summary Physics Statistics (1) each quantity (variable) no random variable needs must satisfy within- to satisfy balance compartment balance = no compartment needs = collapse compart- to satisfy balance ments to allow balance random variables have = wrong biology expectations that satisfy within-system balance = the more compartments (i.e. data) the better! can be a need as long as replicated observations are available per compartment CMAR-Hobart Science Seminar, March 6,
96 Summary Physics Statistics (2) unobservable variables statistical prediction are deduced from (inferential) possible auxiliary theory in certain scenarios e.g. variable observed for some compartments e.g. deduce unobserved through formal regression (both in progress) (3) no formal inference / possible for confidence statements (a) any quantity in for any quantity system-balance equation (b) unobservable variable in some cases (see (2)) (4) multiple media on perceivably straightsame system hard forward to impossible (in progress) CMAR-Hobart Science Seminar, March 6,
97 Conclusion CMAR-Hobart Science Seminar, March 6,
98 Conclusion So why statistical inference for food webs? CMAR-Hobart Science Seminar, March 6,
99 Conclusion So why statistical inference for food webs? statistical models are more honest: properly acknowledge uncertainty statistical inference-based techniques are tractable proper prediction inference is possible (straightforward within Bayesian framework) CMAR-Hobart Science Seminar, March 6,
100 Conclusion So why statistical inference for food webs? statistical models are more honest: properly acknowledge uncertainty statistical inference-based techniques are tractable proper prediction inference is possible (straightforward within Bayesian framework) ENA with Bayesian melding additionally overcomes empirical imbalance under theoretical balance soundly integrates statistical practice with physical theory often preferred by scientists over purely empirically driven approaches CMAR-Hobart Science Seminar, March 6,
101 Thank you! This presentation is available from: gchiu/talks/csiro-hobart09.pdf Articles: Chiu & Gould (submitted). Chiu & Gould (2008), U of Waterloo Working Paper # navigation/techreports/08workingpapers/08-07.pdf CMAR-Hobart Science Seminar, March 6,
FULL TITLE Statistical Inference for Food Webs with Emphasis on Ecological Networks via Bayesian Melding. SHORT TITLE For Peer Review
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An Application of Bayesian Melding to Ecological Networks. Joshua Michael Gould. A research paper presented to the. University of Waterloo
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