Eliezer S. Yudkowsky pointed out in a 2003 "agi" post titled "Breaking Solomonoff induction... well, not really" [1] that Solomonoff Induction is flawed because it fails to incorporate anthropic reasoning. But apparently he thought this doesn't "really" matter because in the long run Solomonoff Induction will converge with the correct reasoning. Here I give two counterexamples to show that this convergence does not necessarily occur.

The first example is a thought experiment where an induction/prediction machine is first given the following background information: Before predicting each new input symbol, it will be copied 9 times. Each copy will then receive the input "1", while the original will receive "0". The 9 copies that received "1" will be put aside, while the original will be copied 9 more times before predicting the next symbol, and so on. To a human upload, or a machine capable of anthropic reasoning, this problem is simple: no matter how many "0"s it sees, it should always predict "1" with probability 0.9, and "0" with probability 0.1. But with Solomonoff Induction, as the number of "0"s it receives goes to infinity, the probability it predicts for "1" being the next input must converge to 0. In the second example, an intelligence wakes up with no previous memory and finds itself in an environment that apparently consists of a set of random integers and some of their factorizations. It finds that whenever it outputs a factorization for a previously unfactored number, it is rewarded. To a human upload, or a machine capable of anthropic reasoning, it would be immediately obvious that this cannot be the true environment, since such an environment is incapable of supporting an intelligence such as itself. Instead, a more likely explanation is that it is being used by another intelligence as a codebreaker. But Solomonoff Induction is incapable of

`reaching such a conclusion no matter how much time we give it, since it`

`takes fewer bits to algorithmically describe just a set of random numbers`

`and their factorizations, than such a set embedded within a universe capable`

`of supporting intelligent life. (Note that I'm assuming that these numbers`

`are truly random, for example generated using quantum coin flips.)`

A different way to "break" Solomonoff Induction takes advantage of the fact that it restricts Bayesian reasoning to computable models. I wrote about this in "is induction unformalizable?" [2] on the "everything" mailing list. Abram Demski also made similar points in recent posts on this mailing list. [1] http://www.mail-archive.com/agi@v2.listbox.com/msg00864.html

`[2]`

`http://groups.google.com/group/everything-list/browse_frm/thread/c7442c13ff1396ec/804e134c70d4a203`

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