I had an idea for Wei Dai's "What is Probability, Anyway?," but after actually typing up I became rather unsure that I was actually saying anything new. Is this something that hasn't been brought up before, or did I just write up a "durr"? (If it's not, I'll probably expand it into a full Discussion post later.)
The fundamental idea is, imagining a multiverse of parallel universes, define all identical conscious entities as a single cross-universal entity, and define probability of an observation E as (number of successors to the entity which observed E) / (total number of successors to the entity). Observations constrain the entity to particular universes, as do decisions, but in different ways; so that we occasionally find ourselves on either side of an observation, but never see ourselves move counter to a decision (except in the sense that what we decide as a brain is not always what we consciously decide.)
Fair warning: I attempted to formalize the concept, but as a undergrad non-math major, the result may look less than impressive to trained eyes. My apologies if this is the case.
The idea is as follows:
Define a conscious observer as some algorithm P(0). P(0) computes on available data and returns a new observer P(1) to act on new available data. Note that it is possible to generate a set of all possible outputs P(n); on human timescales and under the limitation of a human lifetime, it is plausible that such a set would match with the intuitive concept of a "character" who undergoes development.
Assume many-worlds. There are now a very large number of identical algorithms P(n) scattered across the many worlds. Since P(n)=P(n), no local experiment can distinguish between algorithms; therefore scratch the concept of them being separate entirely, and consider them all to be a single conscious entity P.
P does not know which universe it is in (by definition) to start. It can change this by making an observation: it updates itself on sensory data. Regardless of which result is recorded, P(n+1) has lesser measure than P(n): P(n+1) occupies precisely half of the universes P does. P(n+1) has learned more about the universe it is in, so ist space of possible universes has diminished.
An example: consider the example of observing a fair coin - for example, observing the spin of an electron. All of P(n) runs the same algorithm: read the single bit corresponding to the spin, add bit to memory with a suitable wrapper: "Result of experiment: 0/1". This is the new P(n+1), which regardless of result is a new entity. Let us designate successors to P(n) which observed a positive spin Q+, and those which observed a negative spin Q-. Since Q+ and Q- are not equal - they differ in one bit - they are not the same entity, even though they are both successors to (and are part of the same "character" as) P(n). Thus each of Q+/- observe only one version of the experiment.
As a lead-in to decision-making: consider what would happen if P(n) had precommitted to producing Q+, and never produced a Q-. Then the universe "Character P observes a negative spin" is inconsistent, and does not exist (barring, say, a random cosmic ray changing the algorithm.) Such a mind would never observe a spin-down event. This is distinct from quantum immortality/suicide - whereas a quantum suicide leaves behind a "world without you," precommitting in this way means that a given world is inconsistent and never existed in the first place. Barring improbability, no successor of P(n) observes a spin-down event.
In this sense, we can define a decision as a "false observation." P(n) decides to cause event E by choosing to only output successor functions in which event E is observed. (Note that this wording is excessively confusing; a brain which outputs a "move arm" signal is highly unlikely to be in a state where the arm does not move, and so can be said to have "decided" to move the arm.) A decision, then, as expected, also narrows the field of possible universes - but, at least hypothetically, in a purposeful manner.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.