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Elicitation of Experts’ Knowledge for Functional Linear Regression
- Paul-Marie Grollemund, Christophe Abraham, Meïli Baragatti, - Pierre Pudlo
To cite this version:
- Paul-Marie Grollemund, Christophe Abraham, Meïli Baragatti, - Pierre Pudlo. Elicitation of Ex- perts’ Knowledge for Functional Linear Regression. 2018 ISBA World Meeting, Jun 2018, Edinburgh, United Kingdom. 1 p., 2018, �10.13140/RG.2.2.14618.54728�. �hal-02788256�
Elicitation of Experts’ Knowledge for Functional Linear Regression
Grollemund, P.-M., Abraham, C., Pudlo, P. and Baragatti, M.
paul-marie.grollemund@umontpellier.fr
Abstract
We present an approach to elicit experts’ knowledge about the Bliss model [1] which is a parcimonious Bayesian Functional Linear Regression model. We derive an informative prior from elicited information and we define weights to tune prior information contribution on the estimators.
Sparse Step function
βθ(t) =
K
X
k=1
bk1[ak,bk](t) (1)
0.0 0.2 0.4 0.6 0.8 1.0
−101234
Bliss
Bayesian Functional Linear Regression with Sparse Step functions aggregates "weak learners" in a Bayesian way.
Data D: yi, xi(·) where xi(·) ∈ L2([0, 1]) Model: see [2, 5]
yi|xi(·), θ ∼ N
µ +
Z 1 0
xi(t)βθ(t) dt, σ2
(2) where βθ(·) is given in (1).
Noninformative Prior π0(·) and MCMC: [1]
Bayesian Estimator with L2-loss:
βˆ(t) = Z
π0(θ|D)βθ(t) dθ (3) where π0(·|D) is the posterior and βθ as in (1).
R Package
Available at
github.com/pmgrollemund/bliss/
Elicitation
Aim:
Collect experts’ knowledge [4] about the coefficient function Various experts provide: De = yie, xei (·), cei
for i = 1, . . . , ne - pseudo data: yie and xei (·)
- certainty: cei for each pseudo observation
Informative prior
Informative prior is defined as a fractional posterior [3] for which:
- the initial prior is the vaguely noninformative prior π0(·) and - the pseudo data model is (2)
πE(θ) = π(θ|D1, . . . , DE; w) ∝ π0(θ)
E
Y
e=1
p(De|θ; w) where p(De|θ; w) =
ne
Y
i=1
p yie|xei (·), θwie Interpretation : a sequential learning approach
- from initial prior, learn from pseudo data to derive an informative prior - from informative prior, learn from observed data to derive a posterior
Tuning Weights
Naive approach: wie = cei
- Overconfidence Bias, see [4]
Deriving weights from important properties:
- Experts’ interactions (re,f ∈ [0, 1] is the dependence between expert e and expert f) - pseudo data weight ≤ observed data weight
wie = cei × (interaction) × (lower ps. data weight) = cei × 1 1 + P
f6=e re,f × n
ne × E
Illustrations
Prior
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
Posterior
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
0.0 0.2 0.4 0.6 0.8 1.0
−2024
two independent experts low certainty (cei ≈ 0)
many dependent experts (naive approach)
many dependent experts a (tuned weights)
The marginal posterior distributions of βθ(t) (for each t) are represented using heat maps. Red (resp. white) colour is used to represent high (resp. low) posterior densities.
Prior expectation for each expert Prior expectation Posterior expectation Posterior expectation without prior information
References
[1] Grollemund, P.-M., Abraham, C., Baragatti, M. and Pudlo, P. (2018) Bayesian Functional Linear Regression with Sparse Step functions, Bayesian Analysis.
[2] Morris (2015) Functional regression, Annual Review of Statistics and Its Application.
[3] O’Hagan (1995) Fractional Bayes factors for model comparison, JRSSB.
[4] O’Hagan, A., Buck, C., Daneshkhah, A., Eiser, J., Garthwaite, P., Jenkinson, D., . . . and Rakow, T. (2006) Uncertain judgements: eliciting experts’ probabilities, John Wiley & Sons.
[5] Reiss, P., Goldsmith, J., Shang, H.L. and Ogden, T. R. (2015) Methods for scalar-on-function regression, International Statistical Review.
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