I realized that although the idea of a deformed Solomonoff semi-measure is correct, the multiplication prescription I suggested is rather ad hoc. The following construction is a much more natural and justifiable way of combining M and S.
Fix t0 a time parameter. Consider a stochastic process S(-t0) that begins at time t = -t0, where t = 0 is the time our agent A “forms”, governed by the Solomonoff semi-measure. Consider another stochastic process M(-t0) that begins from the initial conditions generated by S(-t0) (I’m assuming M only carries information about dynamics and not about initial conditions). Define S’ to be the conditional probability distribution obtained from S by two conditions:
a. S and M coincide on the time interval [-t0, 0]
b. The universe contains A at time t=0
Thus t0 reflects the extent to which we are certain about M: it’s like telling the agent we have been observing behavior M for time period t0.
There is an interesting side effect to this framework, namely that A can exert “acausal” influence on the utility by affecting the initial conditions of the universe (i.e. it selects universes in which A is likely to exist). This might seem like an artifact of the model but I think it might be a legitimate effect: if we believe in one-boxing in Newcomb’s paradox, why shouldn’t we accept such acausal effects?
For models with a concept of space and finite information velocity, like cellular automata, it might make sense to limit the domain of “observed M” in space as well as time, to A’s past “light-cone”
I realized that although the idea of a deformed Solomonoff semi-measure is correct, the multiplication prescription I suggested is rather ad hoc. The following construction is a much more natural and justifiable way of combining M and S.
Fix t0 a time parameter. Consider a stochastic process S(-t0) that begins at time t = -t0, where t = 0 is the time our agent A “forms”, governed by the Solomonoff semi-measure. Consider another stochastic process M(-t0) that begins from the initial conditions generated by S(-t0) (I’m assuming M only carries information about dynamics and not about initial conditions). Define S’ to be the conditional probability distribution obtained from S by two conditions:
a. S and M coincide on the time interval [-t0, 0]
b. The universe contains A at time t=0
Thus t0 reflects the extent to which we are certain about M: it’s like telling the agent we have been observing behavior M for time period t0.
There is an interesting side effect to this framework, namely that A can exert “acausal” influence on the utility by affecting the initial conditions of the universe (i.e. it selects universes in which A is likely to exist). This might seem like an artifact of the model but I think it might be a legitimate effect: if we believe in one-boxing in Newcomb’s paradox, why shouldn’t we accept such acausal effects?
For models with a concept of space and finite information velocity, like cellular automata, it might make sense to limit the domain of “observed M” in space as well as time, to A’s past “light-cone”