Probability Aggregation Methods in Geoscience

Abstract

The need for combining different sources of information in a probabilistic framework is a frequent task in earth sciences. This is a need that can be seen when modeling a reservoir using direct geological observations, geophysics, remote sensing, training images, and more. The probability of occurrence of a certain lithofacies at a certain location for example can easily be computed conditionally on the values observed at each source of information. The problem of aggregating these different conditional probability distributions into a single conditional distribution arises as an approximation to the inaccessible genuine conditional probability given all information. This paper makes a formal review of most aggregation methods proposed so far in the literature with a particular focus on their mathematical properties. Exact relationships relating the different methods is emphasized. The case of events with more than two possible outcomes, never explicitly studied in the literature, is treated in detail. It is shown that in this case, equivalence between different aggregation formulas is lost. The concepts of calibration, sharpness, and reliability, well known in the weather forecasting community for assessing the goodness-of-fit of the aggregation formulas, and a maximum likelihood estimation of the aggregation parameters are introduced. We then prove that parameters of calibrated log-linear pooling formulas are a solution of the maximum likelihood estimation equations. These results are illustrated on simulations from two common stochastic models for earth science: the truncated Gaussian model and the Boolean. It is found that the log-linear pooling provides the best prediction while the linear pooling provides the worst.

Appendices

Appendix A: Maximum Entropy

Let us define Q(A,D ₀,D ₁,…,D _n ) the joint probability distribution maximizing its entropy $H(Q) = -\sum_{A \in{\mathcal{A}}} Q(D_{0},D_{1},\dots,D_{n})(A) \ln Q(D_{0},D_{1},\dots,D_{n}) (A)$ subject to the following constraints.

• 1. Q(A,D ₀)=Q(A∣D ₀)Q(D ₀)∝P ₀(A), for all $A \in {\mathcal{A}}$.

• 2. Q(A,D ₀,D _i )=Q(A∣D _i )Q(D _i )Q(D ₀)∝P _i (A), for all $A \in{\mathcal{A}}$ and all i=1,…,n.

We will first show that

$$Q(A,D_0,D_1,\dots,D_n) \propto P_0(A)^{1-n} \prod_{i=1}^{n}P_i(A), $$

from which the conditional probability

is immediately derived. For ease of notation, we will use ∑ _A as a short notation for $\sum_{A \in{\mathcal{A}}}$.

Proof

The adequate approach is to use the Lagrange multiplier technique on the objective function

where μ _A and λ _A,i are Lagrange multipliers. For finding the solution Q optimizing the constrained problem, we set all partial derivatives to 0. This leads to the system of equations

(54)

(55)

(56)

From Eqs. (54) and (55), we get

$$Q(A,D_0) = e^{-1}\prod_A e^{\mu_A} \propto P_0(A). $$

Similarly, from Eqs. (54) and (56), we get

$$Q(A,D_0,D_i) = Q(A,D_0) \prod_A e^{\lambda_{A,i}} \propto P_i(A),\quad \hbox{for}\ i=1,\dots,n, $$

from which we find

$$\prod_A e^{\lambda_{A,i}} \propto P_i(A)/P_0(A),\quad \hbox{for}\ i=1,\dots,n. $$

Plugging this in Eq. (54) yields

$$Q(A,D_0,D_1,\dots,D_n) \propto P_0(A) \prod_{i=1}^n \frac{P_i(A)}{P_0(A)}. $$

Hence,

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Appendix B: Conditional Probabilities for the Trinary Event Example

1. Let us first compute the conditional probability

where $G^{2}_{2}(t,t;\rho)$ is the bivariate cpf of a (0,1) bi-Gaussian random vector with correlation ρ. For symmetry reasons, one has P(I(s′)=2∣I(s)=1)=P(I(s′)=3∣I(s)=1), from which it follows immediately

2. We consider now

3. The picture is slightly more complicated for P(I(s′)=2∣I(s)=2)

There is no closed-form expression for the double integral which must be evaluated numerically. Then P(I(s′)=3∣I(s)=2) is computed as the complement to 1.

4. The conditional probabilities of I(s′) given that I(s)=3 are then obtained by symmetry.