# Conditional events that are not in the event algebra?

The Wikip. page on conditional event algebra states that:

David Lewis showed that in orthodox probability theory, only certain trivial Boolean algebras with very few elements contain, for any given A and B, an event X which satisfies P(X) = P(B|A). Later extended by others, this result stands as a major obstacle to any talk about logical objects that can be the bearers of conditional probabilities.

I haven't read the paper by Lewis. But I'm aware that in Bayesian analysis, the conditional likelihood P(B|A) is not a probability distribution (B here represents data or evidence, while A represents model parameters, see, eg Guyonnet & Ferson "Bayesian methods in risk assessment" p.11) There must be a connection here right?

So I'm primarily interested in Bayesian data analysis but would appreciate an explanation in terms of basic probability theory to relate events and probability measures: What are some simple examples of conditional events that are not in the event algebra?

Do such example involve measure-zero events (which are always problematic)?

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in orthodox probability theory, only certain trivial Boolean algebras with very few elements contain, for any given A and B, an event X which satisfies P(X) = P(B|A).

Really? On $(\Omega,\mathcal F,\mathbb P)=([0,1],\mathcal B([0,1]),\mathrm{Leb})$, for every $A$ and $B$, there exists an event $X$ such that $\mathbb P(X)=\mathbb P(B\mid A)$ (as soon as the RHS exists): pick $X=[0,x]$ where $x=\mathbb P(B\mid A)$.

...in Bayesian analysis, the conditional likelihood P(B|A) is not a probability distribution...

Is that so? When it exists, the mapping $\mathbb P(\ \mid A)$ is very much a probability measure.

What are some simple examples of conditional events that are not in the event algebra?

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Not sure Lewis pretended to show what is written on the WP page. –  Did Mar 3 '13 at 18:59
These are probably better suited to comments on the question than as an answer. –  robjohn Mar 3 '13 at 19:36
@Did, I'm not sure about the first issue - I was also surprised by the existence of "conditional event algebra" - though that it is developed by a philosopher rather than a mathematician invites skepticism. However, my Q I added a link to a paper that discusses the Bayesian conditional. –  alancalvitti Mar 3 '13 at 19:50

Did is correct that P(_|A) can be treated as a probability function. It's what you get by starting with a probability function P and then revising it by conditionalizing on A (= learning that A is definitely true). So that's not an issue. The Lewis result is also not about zero-measure events, although they do come up eventually if you really dig into this topic.

I just updated the Wiki article (written by me a number of years ago) to read:

...only certain trivial Boolean algebras with very few elements contain, for any given A and B, an event X such that P(X) = P(B|A) is true for any probability function P.

In other words, in general there's no single event X, fixed by the choice of A and B, such that P(X) tracks P(B|A) as we vary the probability function. That's the Lewis result, also referenced in the Wiki article on Causal Decision Theory.

I think Lewis's result can be visualized as follows. Imagine a square pool with a flat bottom. We can make waves as big as we like, so the water surface is by no means flat. But the total volume is of course constant, and is equal to 1 (that is, 1 pool's worth).

We can draw Venn diagrams on the flat bottom, to represent events. The volume of water above circle A corresponds to the probability of event A. Because the water can slosh around, a fixed circle can have varying probabilities.

The conditional probability P(B given A) is a ratio: the volume above the intersection of A and B, divided by the volume above A. Lewis is saying that there's no circle (or other fancy shape) that you can draw, anywhere on the bottom, that will reliably have a volume above it numerically equal to that ratio. Draw any circle you like, you will always have the ability to move water around so that the volume above that circle is different from the ratio of the A-and-B volume to the A-volume.

The nonstandard domains used for conditional event algebras would then, by analogy, begin with an idea such as "Imagine that every circle A there's a matching pool, suspended in the air above it..." Just to convey the flavor.

What are conditional events? If you're drawing balls from an urn, a conditional event would be something like "If the third ball is red, the fourth ball is blue." When we treat the if-then as a material conditional (that is, "If A then B" being equivalent to "Either B or not-A"), then the event in question occurs if the third ball is something other than red. Whereas proponents of conditional event algebras would say the event cannot be judged to have happened or failed to happen until the condition "third ball red" is met.

Lewis didn't (that I'm aware) have anything to do with developing conditional event algebras, except to produce this finding which prompted others to explore ways of getting around it.

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