Value of Information Concept Note - Kriging with Known Measurement Error and EVSI for Monitor Siting

Summary

Concept note, no raw source. With a GP prior and a monitor of known error variance, the posterior variance after a reading is available in closed form and does not depend on the value read, so the EVSI of a candidate site is a closed-form function of how much it shrinks the variance of the decision-relevant quantity. Greedy selection is justified when gains show diminishing returns; whether joint value is submodular must be checked, not assumed.

Overview

E89 (Where should the next air monitors go?) uses EPA AirData daily PM2.5 (419 sites in 19 states) and county populations with a “marginal Matern GP with known measurement error, exact county conditionals, closed-form EVSI checked by brute force, lazy greedy site selection, simulated joint value against summed gains, a search that lets county means move, a paired-instrument refit”. Krause et al.’s paper supplies the design theory (Krause 2008 - Greedy Algorithm and the 1-1 over e Guarantee); this note links it to EVSI.

Main Content

Kriging with known measurement error

Latent field $f\sim GP(0,K)$ observed as $y_i=f(x_i)+{ϵ}_i$, ${ϵ}_i\sim N(0,{\sigma }_i^2)$ with known ${\sigma }_i^2$ (a monitor’s error variance). The posterior of $f(x^{\ast })$ given sites $A$ is $N({\mu }_{\ast |A},{\sigma }_{\ast |A}^2)$ with ${\sigma }_{\ast |A}^2=K(x^{\ast },x^{\ast })-k_{\ast A}(K_{AA}+{\Sigma }_A{)}^{-1}{k}_{A\ast }$, ${\Sigma }_A=\mathrm{diag}({\sigma }_i^2)$ (Krause 2008 - GP Model and Posterior Variance, with noise).

EVSI of one new reading for a threshold decision

Let a decision depend on whether a county mean ${\mu }_c$ exceeds a threshold ${\tau }_0$ (for example, issue an advisory), with net benefit $\propto ({\mu }_c-{\tau }_0)$ for acting. Before a reading, ${\mu }_c\sim N(m,s_0^2)$ (the GP marginal). A reading with known error ${\sigma }^2$ at a site correlated with ${\mu }_c$ yields a posterior with variance $s_1^2<s_0^2$ and a preposterior posterior mean $\sim N(m,s_0^2-s_1^2)$. By the closed form of Value of Information Concept Note - EVPI, EVPPI and EVSI,

$${\text{EVSI}}_c=w_c[\tau \phi ((m-{\tau }_0)/\tau )+(m-{\tau }_0)\Phi ((m-{\tau }_0)/\tau )-\max (m-{\tau }_0,0)],{\tau }^2=s_0^2-s_1^2,$$

with $w_c$ the stakes (for example population). EVSI is largest where the prior mean sits near the threshold and the site shrinks variance most.

Greedy siting. Maintain gains ${\delta }_y=\text{EVSI}(A\cup y)-\text{EVSI}(A)$ and use lazy evaluation (Krause 2008 - Lazy Evaluation and Local Kernels); lazy greedy is exact for any function with diminishing returns. MI is submodular (Krause Lemma 3), but EVSI and expected variance reduction are not guaranteed to be: Krause’s counterexample for information gain (Krause 2008 - Related Work and Why Other Criteria Lack Guarantees) shows complementary sites (a second site worth more after the first) are possible. E89’s “simulated joint value against summed gains” is the right empirical check for whether summing gains over-counts.

Examples

Verified numbers (reusing the normal closed form): a county with prior mean at the threshold ($m-{\tau }_0=0$) and $s_0=1$; a reading shrinking variance to $s_1^2=0.5$ gives ${\tau }^2=0.5$, $\text{EVSI}=\tau \phi (0)=0.707\times 0.3989=0.282$ (per unit stakes). A reading that only shrinks to $s_1^2=0.9$ gives $\tau =0.316$ and EVSI $0.126$: halving the value of a weak monitor.

import numpy as np
from scipy import stats
def evsi_site(m, tau0, s0sq, s1sq, w=1.0):
    d, tau = m - tau0, np.sqrt(s0sq - s1sq)
    return w*(tau*stats.norm.pdf(d/tau) + d*stats.norm.cdf(d/tau) - max(d, 0))
print(evsi_site(0, 0, 1, 0.5), evsi_site(0, 0, 1, 0.9))   # 0.2821 0.1262

PyMC-style sketch (untested): pm.gp.Marginal with sigma= set to the known per-site error SD; obtain $s_1^2$ for candidate sites from gp.predict(Xnew, point=..., diag=True) after conditioning on hypothetical sites.

Connections

• Marketing: choosing which geos to instrument (holdout or measurement) when the decision turns on whether a geo’s lift clears a hurdle; stakes $w_c$ are spend or population.
• Related designs: Expected Information Gain, Bayesian Optimisation, Acquisition Functions.

See Also

• Krause 2008 - Near-Optimal Sensor Placements in Gaussian Processes - Overview
• Value of Information Concept Note - When an Exact Gaussian Process Earns Its Price

Sources

• Krause, Singh and Guestrin (2008); Raiffa and Schlaifer (1961); Cressie (1991) on kriging; E89 method list at https://redam94.github.io/pymc-challenges/examples.html. Closed forms derived here.