Concept Architecture
Concept
Theoretically, a Logarithmic Pool is a mathematical aggregation method used to combine multiple probability distributions into a single consensus distribution by calculating a weighted geometric mean of the individual probability densities. It is founded on Bayesian probability theory and opinion pooling and preserves proportional agreement among expert beliefs. In health economics, logarithmic pooling is used in expert elicitation to combine probability distributions from multiple experts when parameter uncertainty cannot be estimated directly from empirical evidence.
Mathematically, the logarithmic pool is constructed by multiplying each expert's probability distribution after raising it to a specified weight and then normalising the resulting product so that it integrates to one. The weights represent the relative influence assigned to each expert and may be equal or based on calibration performance or other predefined criteria.
In practice, logarithmic pooling is applied after expert elicitation by assigning weights to individual expert distributions, calculating the weighted geometric mean and normalising the combined distribution. The resulting consensus distribution is then used as an input for probabilistic sensitivity analysis, Bayesian evidence synthesis and health economic decision models when empirical evidence is limited or unavailable.
Purpose
Used to combine multiple expert probability distributions into a single consensus distribution while preserving proportional agreement and supporting probabilistic modelling under uncertainty.
Mathematical Formulae
Primary Formula
p(x) = c ? ?? p?(x)??
where:
p(x) = pooled probability density
p?(x) = probability density from expert i
w? = weight assigned to expert i
c = normalising constant
Supporting Formulae
Weight constraint:
?? w? = 1
Normalising constant:
c = 1 / ? ?? p?(x)?? dx
Related Mathematical Methods
- Linear Pool
- Expert Elicitation
- Bayesian Analysis
- Bayesian Evidence Synthesis
- Probability Distribution
- Performance Weights
- Probabilistic Sensitivity Analysis
Example
Three clinical experts provide probability distributions for the long-term effectiveness of a new treatment. Equal weights of one-third are assigned to each expert. The logarithmic pool combines the three distributions by calculating their weighted geometric mean and normalising the result to produce a single consensus probability distribution. This pooled distribution is subsequently used as an input for probabilistic sensitivity analysis in a cost-effectiveness model.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| LN | =LN(B2) | Calculate the natural logarithm of an expert probability density. |
| EXP | =EXP(SUMPRODUCT(Weights,LN_Values)) | Calculate the weighted geometric mean of expert probability densities. |
| SUMPRODUCT | =SUMPRODUCT(B2:D2,$F$2:$F$4) | Calculate weighted logarithmic contributions from multiple experts. |
| SUM | =SUM(ResultRange) | Calculate the normalising constant for the pooled distribution. |
VBA (Optional)
VBA can automate logarithmic pooling, normalise pooled probability distributions and generate consensus distributions for probabilistic sensitivity analysis.
Sources
- Genest C, Zidek JV. Combining probability distributions: A critique and an annotated bibliography. Statistical Science. 1986;1(1):114?135.
- Clemen RT, Winkler RL. Combining probability distributions from experts in risk analysis. Risk Analysis. 1999;19(2):187?203.
- O'Hagan A, Buck CE, Daneshkhah A, et al. Uncertain Judgements: Eliciting Experts' Probabilities. Wiley.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- ISPOR Good Practices for Expert Elicitation.
Related Concepts (2)
Library
Publications
1
Value of Information Analytical Methods: Report 2 of the ISPOR Value of Information Analysis Emerging Good Practices Task Force — Rothery, Strong, Koffijberg, Basu, Ghabri, Knies, Murray, Sanders Schmidler, Steuten & Fenwick, Vol. 23, No. 3 ed., 2020 (Value in Health)
The methods companion to the ISPOR VOI series, giving detailed algorithms and software guidance for computing EVPI, EVPPI, EVSI and the expected net benefit of sampling, with recommendations for selecting methods by decision-problem features.
Journal ArticleView source →
Frequently Asked Questions (6)
What is the logarithmic pool?
A method combining multiple experts' probability distributions by taking a weighted geometric average rather than a simple arithmetic average.
Source: Cooke 1991
What effect does the logarithmic pool have on areas of expert agreement?
The logarithmic pool combines experts' distributions by a weighted geometric average, which multiplies the distributions together rather than adding them. This concentrates the pooled distribution where the experts agree and suppresses regions any expert judged very unlikely, producing a tighter, usually single-peaked result. The effect is a sharper consensus than the linear pool gives, at the risk of understating genuine disagreement by discounting values that even one expert rejected. It rewards agreement. O'Hagan and colleagues (2006) describe this method.
Source: O'Hagan et al. 2006
How does the logarithmic pool combine distributions?
The logarithmic pool combines distributions by multiplying each expert's distribution raised to the power of its weight and normalising the product, which is a weighted geometric average. Because it multiplies the densities, an outcome given low probability by any expert receives low probability in the pool, so the result is concentrated where the experts' distributions overlap. The pooled distribution is typically unimodal and tighter than the linear pool, representing a compromise that reflects the common ground among the experts rather than preserving their separate peaks.
Source: O'Hagan et al. 2006
What are the properties of the logarithmic pool?
The logarithmic pool typically yields a unimodal, more concentrated distribution than the linear pool, since multiplying densities emphasises where experts agree and suppresses where any expert assigns low probability. It is externally Bayesian, meaning that pooling the experts and then updating with data gives the same result as updating each expert first and then pooling, a coherence property the linear pool lacks. Its tighter result represents a compromise. These properties make the logarithmic pool attractive where a concentrated consensus and Bayesian coherence are wanted.
Source: Cooke 1991
How does the logarithmic pool differ from the linear pool?
The logarithmic pool takes a weighted geometric average, giving a typically unimodal, concentrated distribution that suppresses outcomes any expert deems unlikely and is externally Bayesian, whereas the linear pool takes a weighted arithmetic average, giving a mixture that can be multimodal, retains individual peaks, and spreads to reflect disagreement but is not externally Bayesian. So the logarithmic pool reconciles views into a tighter compromise, while the linear pool represents disagreement as a wider, possibly multimodal spread. The choice depends on how disagreement should be treated.
Source: Cooke 1991
What are the limitations of the logarithmic pool?
The logarithmic pool can be overconfident when experts genuinely disagree, since multiplying densities produces a narrow distribution that may understate the true uncertainty by suppressing the tails where views differ. It gives near-zero probability to any outcome an expert rules out, which can be undesirable, and its result depends on the weights. It is less intuitive than a simple average. These limitations mean the logarithmic pool is applied with care where experts differ substantially, weighing its concentrated, coherent result against the risk of understating disagreement.
Source: O'Hagan et al. 2006
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 31 Oct 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/logarithmic-pool
- Term code
- HE-EM-VI-019
Stable URI · Machine-readable · Resolvable · CC BY 4.0