## 4 Open Problems in Universal Induction  
The induction problem is a fundamental problem in philosophy [1, 5] and science [142]. [PERSON]'s model is a promising universal solution of the induction problem. In [31], an attempt has been made to collect the most important fundamental philosophical and statistical problems, regarded as open, and to present arguments and proofs that [PERSON]'s theory overcomes them. Despite the force of the arguments, they are likely not yet sufficient to convince the (scientific) world that the induction problem is solved. The discussion needs to be rolled out much further, say, at least one generally accessible article per one allegedly open problem. Indeed, this endeavor might even discover some catch in [PERSON]'s theory. Some problems identified and outlined in [31] worth to investigate in more detail are:  
* **The zero prior problem**. The problem is how to _confirm universal hypotheses_ like \(H\! :=\)\"all balls in some urn (or all ravens) are black\". A natural model is to assume that balls (or ravens) are drawn randomly from an infinite population with fraction \(\theta\) of black balls (or ravens) and to assume _some_ prior _density_ over \(\theta\!\in\! [0;\!1]\) (a uniform density gives the Bayes-Laplace model). Now we draw \(n\) objects and observe that they are all black. The problem is that the posterior proability \(P[H|\mbox{black}_{1}...\mbox{black}_{n}]\!\equiv\!0\), since the prior probability \(P[H]\!=\!P[\theta\!=\!1]\!\equiv\!0\). [PERSON]'s [143] approach does not solve the problem [31]. * **The black raven paradox** by [PERSON] goes as follows [144, Ch.11.4]: Observing \(B\)lack \(R\)avens confirms the hypothesis \(H\) that all ravens are black. In general, \((i)\) hypothesis \(R\!\rightarrow\!B\) is confirmed by \(R\)-instances with property \(B\). Formally substituting \(R\!\leadsto\
eg B\) and \(B\!\leadsto\
eg R\) leads to \((ii)\) hypothesis \(\
eg B\!\rightarrow\
eg R\) is confirmed by \(\
eg B\)-instances with property \(\
eg R\). But \((iii)\) since \(R\!\rightarrow\!B\) and \(\
eg B\!\rightarrow\
eg R\) are logically equivalent, \(R\!\rightarrow\!B\) must also be confirmed by \(\
eg B\)-instance with property \(\
eg R\). Hence by \((i)\), observing Black Ravens confirms Hypothesis \(H\), so by \((iii)\), observing White Socks also confirms that all Ravens are Black, since White Socks are non-Ravens which are non-Black. But this conclusion is absurd. Again, neither [PERSON]'s nor any other approach solves this problem. * **The Grue problem**[145]. Consider the following two hypotheses: \(H1:=\)\"All emeralds are green\", and \(H2:=\)\"All emeralds found till year 2020 are green, thereafter all emeralds are blue\". Both hypotheses are equally well supported by empirical evidence. [PERSON]'s razor seems to favor the more plausible hypothesis \(H1\), but by using new predicates _grue_:=\"green till y2020 and blue thereafter\" and _bleen_:=\"blue till y2020 and green thereafter\", \(H2\) gets simpler than \(H1\). * **Reparametrization invariance**[146]. The question is how to extend the symmetry principle from finite hypothesis classes (all hypotheses are equally likely) to infinite hypothesis classes. For \"compact\" classes, [PERSON]'s prior [147] is a solution, but for non-compact spaces like \(I\!N\) or \(I\!R\), classical statistical principles lead to improper distributions, which are often not acceptable. * **Old-evidence/updating problem** and _ad-hoc hypotheses_[148]. How shall a Bayesian treat the case when some evidence \(E{\hat{=}}x\) (e.g. Mercury's perihelion advance) is known well-before the correct hypothesis/theory/model \(H{\hat{=}}\mu\) ([PERSON]'s general relativity theory) is found? How shall \(H\) be added to the Bayesian machinery a posteriori? What is the prior of \(H\)? Should it be the belief in \(H\) in a hypothetical counterfactual world in which \(E\) is not known? Can old evidence \(E\) confirm \(H\)? After all, \(H\) could simply be constructed/biased/fitted towards \"explaining\" \(E\).