## 8 Justification of [PERSON]'s Razor
Similarly to the previous paragraph we can approximately count the number of such universes:  
\[Q_{L}^{n} := \{q:\mbox{Length}(q)\leq L\mbox{ and UTM}(q)=u_{1:t}^{true}u_{t+1:n}*\}\; \subset Q_{L}\] \[q_{min}^{n} := \arg\min_{q}\{\mbox{Length}(q):q\in Q_{L}^{n}\}\quad\mbox{and} \quad l_{n}:=\mbox{Length}(q_{min}^{n})\] \[|Q_{L}^{n}| \approx 2^{L-l_{n}}\]  
The probability of being in a universe with future \(u_{t+1:n}\) given \(u_{1:t}^{true}\) is determined by their relative number  
\[P(u_{t+1:n}|u_{1:t}^{true})\;=\;\frac{|Q_{L}^{n}|}{|Q_{L}|}\;\approx\;2^{-(l_ {n}-l)} \tag{6}\]  
which is (asymptotically) independent of \(L\). **Ockham's razor. ** Relation (6) implies that the most likely continuation \(\hat{u}_{t+1:n}:=\arg\max_{u_{t+1:n}}P(u_{t+1:n}|u_{1:t}^{true})\) is (approximately) the one that minimizes \(l_{n}\). By definition, \(q_{min}\) is the shortest program in \(Q_{L}\!=\!\bigcup_{u_{t+1:n}}Q_{L}^{n}\). Therefore  
\[P(\hat{u}_{t+1:n}|u_{1:t}^{true})\approx P(u_{t+1:n}^{q_{min}}|u_{1:t}^{true})\]  
The accuracy of \(\approx\) is clarified later. In words  
_We are most likely in a universe that is (equivalent to) the simplest universe consistent with our past observations._  
This shows that [PERSON]'s razor selects the theory that most likely leads to correct predictions, and hence proves (under the stated assumptions) that [PERSON]'s razor is correct. **Discussion. ** It is important to note that the universal self-sampling assumption has not by itself any bias towards simple models \(q\). Indeed, most \(q\) in \(Q_{L}\) have length close to \(L\), and since we sample uniformly from \(Q_{L}\) this actually represents a huge bias towards large models for \(L\!\rightarrow\!\infty\). The result is also largely independent of the uniform sampling assumption. For instance, sampling a length \(l\!\in\!I\!N\) w.r.t. any reasonable (_i.e._, slower than exponentially decreasing) distribution and then \(q\) of length \(l\) uniformly leads to the same conclusion. How reasonable is the UToE? We have already discussed that it is nearly but not quite as good as any other correct ToE. The philosophical, albeit not practical advantage of UToE is that it is a safer bet, since we can never be sure about the future correctness of a more specific ToE. An a priori argument in favor of UToE is as follows: What is the best candidate for a ToE before, _i.e._, in absence of any observations? If somebody (but how and who?) would tell us that the universe is computable but nothing else, universal self-sampling seems like a reasonable a priori UToE. **Comparison to anthropic self-sampling. ** Our universal self-sampling assumption is related to anthropic self-sampling [18] but crucially different. The anthropic self-sampling assumption states that a priori you are equally likely any of the (human) observers in our universe. First, we sample from any universe and any location (living or dead) in the multiverse and not only among human (or reasonably intelligent) observers. Second, we have no problem of what counts as a reasonable (human) observer. Third, our principle is completely formal. Nevertheless the principles are related since (see inclusion of \(s\)) _given_\(o_{1:t}^{true}\) we also sample from the set of reasonable observers, since \(o_{1:t}^{true}\) includes snapshots of other (human) observers. **No Free Lunch (NFL) myth. ** [PERSON] [23] considers algorithms for finding the minimum of a function, and compares their average performance. The simplest performance measure is the number of function evaluations needed to find the global minimum. The average is taken uniformly over the set of all functions from and to some fixed finite domain. Since sampling uniformly leads with (very) high probability to a totally random function (white noise), it is clear that on average no optimization algorithm can perform better than exhaustive search, and no reasonable algorithm (that is one that probes every function argument at most once) performs worse. That is, all reasonable optimization algorithms are equally bad on average.