A closed model is one that is interpreted as representing all truths about that which is modeled. An open model is instead interpreted as making a specific set of assertions, and leaving the rest undecided. Formally, we might say that a closed model is interpreted to include all of the truths, so that any other statements are false. This is also known as the closed-world assumption.
A typical example of an open model is a set of statements in predicate logic. This could be changed to a closed model simply by applying the closed-world assumption. A possibly more typical example of a closed-world model is a computer program that outputs the data so far (and predicts specific future output), as in Solomonoff induction. These two types of model are very different! One important difference is that we can simply *add* to an open model if we need to account for new data, while we must always *modify* a closed model if we want to account for more information. The key difference I want to ask about here is: a length-based bayesian prior seems to apply well to closed models, but not so well to open models. First, such priors are generally supposed to apply to entire joint states; in other words, probability theory itself (and in particular bayesian learning) is built with an assumption of an underlying space of closed models, not open ones. Second, an open model always has room for additional stuff somewhere else in the universe, unobserved by the agent. This suggests that, made probabilistic, open models would generally predict universes with infinite description length. Whatever information was known, there would be an infinite number of chances for other unknown things to be out there; so it seems as if the probability of *something* more being there would converge to 1. (This is not, however, mathematically necessary.) If so, then taking that other thing into account, the same argument would still suggest something *else* was out there, and so on; in other words, a probabilistic open-model-learner would seem to predict a universe with an infinite description length. This does not make it easy to apply the description length principle. I am not arguing that open models are a necessity for AI, but I am curious if anyone has ideas of how to handle this. I know that Pei Wang suggests abandoning standard probability in order to learn open models, for example. --Abram Demski ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=111637683-c8fa51 Powered by Listbox: http://www.listbox.com
