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Suppose that we have four possible events $e_0, e_1, e_2 $ and $e_3$. These event are exclusive $e_i \cap e_j = \emptyset$ and exhaustive $\cup e_i = \Omega$. Now we ask two experts on their opinion as to the probabilities of the individual $e_i$. We get the results in the following table:

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There are numerous ways to combine their opinion to get a resulting probability distribution. There are two papers in particular that provide a very good overview of the entire topic:

Genest, C., & Zidek, J. V. (1986). Combining Probability Distributions: A Critique and an Annotated Bibliography. Statistical Science, 1(1), 114–135. https://doi.org/10.1214/ss/1177013825

Dietrich, F., & List, C. (2017). Probabilistic Opinion Pooling. (A. Hájek & C. Hitchcock, Eds.), Social Choice and Welfare (Vol. 1). Oxford University Press. https://doi.org/10.1093/oxfordhb/9780199607617.013.37

Some of the more prominent methods are:

  1. linear: take a weighted arithmetic mean of probabilities for each event

  2. geometric: take a weighted geometric mean of probabilities for each event and then normalize

  3. multiplicative: take the product of probabilities for each event and then normalize

The results of these methods are also included in the table. Each method can be justified based on firm mathematical results.

Now here is the catch. Suppose that we are no longer interested in the individual probabilities of $e_2$ and $e_3$, but we would just like to know what is the probability of $e_2 \cup e_3$. There are two ways in which we can proceed:

  1. We can sum the resulting combined probabilities for $e_1$ and $e_2$.
  2. We can sum the experts' opinions for $e_1$ and $e_2$ and then combine the probabilities.

As you see in the table, these two approaches do not yield the same results. For example, for the geometric combinator method 1 yields 0.71 and method 2 yields 0.712.

If the two methods would be guaranteed to produce the same result, we would call this the marginalization property. Lindley in his paper ''Reconciliation of discrete probability distributions'' (sorry, I cannot find the exact citation) tries to argue that the entire concept of marginalization property is flawed, since in the first case, we are given more information by the experts. He also gives a numerical explanation for his supra-Bayesian model, but I find it very hard to follow. Furthermore, I am not very convinced by his explanation of receiving "more information". His numeric results hold, but I have the feeling that this is due to the fact that his model is much more complicated and must account for the individual correlations between his experts.

Are there any intuitive arguments that would reconcile the marginalization property with the simple geometric and multiplicative operators?

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The concept of the marginalization property as something desirable is flawed because the two ways of summing the probabilities provide answers to different questions when the pooling method is nonlinear. Like the different pooling methods, both pre-pooling and post-pooling summation can be epistemically justified based on the context and whether we want consistency in the individual or pooled estimates.

Let's consider polling expert opinions on the outcome of the U.S. presidential election as an example. First, let $e_0,e_1$ be the events that a Democrat wins with the popular vote or with the electoral college only, respectively, and let $e_2,e_3$ be the events that a different party wins with or without the popular vote. Suppose the experts were first asked to estimate $P(e_0,e_1)$ and $P(e_2,e_3)$ and then asked for the more specific probabilities. Since the experts considered the probability of the joint event $e_2 \cup e_3$ as an intermediate step in estimating the probability of the separate events, pre-pooling summation would make sense since it treats the intermediate estimates the same as the experts' other estimates.

Now, let $e_0,e_1,e_2,e_3$ be the events that Croatia, Belgium, England, or France wins the World Cup. Suppose we will be using our model to make several bets on possible outcomes, including betting that any one of two or three given countries wins. If we want our final estimates of expected return on joint bets to be consistent, then we would use post-pooling summation.

So a lack of a marginalization property is not a disadvantage of the multiplicative and geometric methods. It just means there is another parameter (pre- vs. post-pooling summation) to consider in the model. You could argue the linear method then is better because it is simpler and consistent in both cases, but this is because it is not capturing the nuances that would have justified using the geometric or multiplicative methods in the first place.

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