Often people think that various forms of anthropic reasoning require you to change your beliefs in ways other than conditionalizing on evidence. This is false, at least in the cases I know of. I shall talk about Frank Arntzenius‘ paper Some Problems for Conditionalization and Reflection [gated] because it explains the issue well, though I believe his current views agree with mine.
He presents five thought experiments: Two Roads to Shangri La, The Prisoner, John Collins’s Prisoner, Sleeping Beauty and Duplication. In each of them, it seems the (arguably) correct answer violates van Fraassen’s reflection principle, which basically says that if you expect to believe something in the future without having been e.g. hit over the head between now and then, you should believe it now. For instance the thirder position in Sleeping Beauty seems to violate this principle because before the experiment Beauty believes there is a fifty percent chance of heads, and that when she wakes up she will think there is a thirty three percent chance. Arntzenius argued that these seemingly correct answers really are the correct ones, and claimed that they violate the reflection principle because credences can evolve in two ways other than by conditionalization.
First he said credences can shift, for instance through time. I know that tomorrow I will have a higher credence in it being Monday than I do today, and yet it would not be rational for me to increase my credence in it being Monday now on this basis. They can also ‘spread out’. For instance if you know you are in Fairfax today, and that tomorrow a perfect replica of your brain experiencing Fairfax will be made and placed in a vat in Canberra, tomorrow your credence will go from being concentrated in Fairfax to being spread between there and Canberra. This is despite no damage having been done to your own brain. As Arntzenius pointed out, such an evolution of credence looks like quite the opposite of conditionalization, since conditionalization consists of striking out possibilities that your information excludes – it never opens up new possibilities.
I agree that beliefs should evolve in these two ways. However they are both really conditionalization, just obscured. They make sense as conditionalization when you think of them as carried out by different momentary agents, based on the information they infer from their connections to other momentary agents with certain beliefs (e.g. an immediately past self).
Normal cases can be considered this way quite easily. Knowing that you are the momentary agent that followed a few seconds after an agent who knew a certain set of facts about the objective world, and who is (you assume) completely trustworthy, means you can simply update the same prior with those same facts and come to the same conclusion. That is, you don’t really have to do anything. You can treat a stream of moments as a single agent. This is what we usually do.
However sometimes being connected in a certain way to another agent does not make everything that is true for them true for you. Most obviously, if they are a past self and know it is 12 o clock, your connection via being their one second later self means you should exclude worlds where you are not at time 12:00:01. You have still learned from your known relationship to that agent and conditionalized, but you have not learned that what is true of them is true of you, because it isn’t. This is the first way Arntzenius mentioned that credences seem to evolve through time not by by conditionalization.
The second way occurs when one person-moment is at location X, and another person moment has a certain connection to the person at X, but there is more than one possible connection of that sort. For instance when two later people both remember being an earlier person because the earlier person was replicated in some futuristic fashion. Then while the earlier person moment could condition on their exact location, the later one must condition on being in one of several locations connected that way to the earlier person’s location, so their credence spreads over more possibilities than that of the earlier self. If you call one of these later momentary agents the same person as the earlier one, and say they are conditionalizing, it seems they are doing it wrong. But considered as three different momentary people learning from their connections they are just conditionalizing as usual.
What exactly the later momentary people should believe is a matter of debate, but I think that can be framed entirely as a question of what their state spaces and priors look like.
Momentary humans almost always pass lots of information from one to the next, chronologically along chains of memory through non-duplicated people, knowing their approximate distance from one another. So most of the time they can treat themselves as single units who just have to update on any information coming from outside, as I explained. But conditionalization is not specific to these particular biological constructions; and when it is applied to information gained through other connections between agents, the resulting time series of beliefs within one human will end up looking different to that in a chain with no unusual extra connections.
This view also suggests that having cognitive defects, such as memory loss, should not excuse anyone from having credences, as for instance Arntzenius argued it should in his paper Reflections on Sleeping Beauty: “in the face of forced irrational changes in one’s degrees of belief one might do best simply to jettison them altogether”. There is nothing special about credences derived from beliefs of a past agent you identify with. They are just another source of information. If the connection to other momentary agents is different to usual, for instance through forced memory loss, update on it as usual.
