## 7 Extensions
(In the special case of \(Q(u^{\prime})\!=\!1\) for \(u^{\prime}\!=\!u\) and 0 else, \(Q\) describes the deterministic universe \(u\).) Similarly, the observational process may be probabilistic. Let \(S(o|u)\) be the probability of observing \(o\) in universe \(u\). Together, \((Q,\!S)\) is a probabilistic CToE that predicts observation \(o\) with probability \(P(o)\!=\!\sum_{u}\!S(o|u)Q(u)\). A computable probabilistic CToE is one for which there exist programs (of lengths Length\((Q)\) and Length\((S)\)) that compute the functions \(Q(\cdot)\) and \(S(\cdot|\cdot)\). Consider now the true observation \(o^{true}_{1:t}\). The larger \(P(o^{true}_{1:t})\) the \"better\" is \((Q,\!S)\). In the degenerate deterministic case, \(P(o^{true}_{1:t})\!=\!1\) is maximal for a correct CToE, and 0 for a wrong one. In every other case, \((Q,\!S)\) is only a partial theory that needs completion, since it does not compute \(o^{true}_{1:t}\). Given \(P\), it is possible to code \(o^{true}_{1:t}\) in \(|\!\log_{2}\!P(o^{true}_{1:t})|\) bits (arithmetic or Shannon-Fano code). Assuming that \(o^{true}_{1:t}\) is indeed sampled from \(P\), one can show that with high probability this is the shortest possible code. So there exists an effective description of \(o^{true}_{1:t}\) of length  
\[\text{Length}(Q)+\text{Length}(S)+|\!\log_{2}\!P(o^{true}_{1:t})| \tag{5}\]  
This expression should be used (minimized) when comparing probabilistic CToEs. The principle is reminiscent of classical two-part Minimum Encoding Length principles like MML and MDL [4, 5]. Note that the noise corresponds to the errors and the log term to the error table of the previous paragraphs. **Probabilistic examples. ** Assume \(S(o|o)=1\,\forall o\) and consider the observation sequence \(o_{1:t}^{true}=u_{1:t}^{true}=11001001000011111101010100\). If we assume this is a sequence of fair coin flips, then \(Q(o_{1:t})=P(o_{1:t})=2^{-t}\) are very simple functions, but \(|\log_{2}P(o_{1:t})|=t\) is large. If we assume that \(o_{1:t}^{true}\) is the binary expansion of \(\pi\) (which it is), then the corresponding deterministic \(Q\) is somewhat more complex, but \(|\log_{2}P(o_{1:t}^{true})|=0\). So for sufficiently large \(t\), the deterministic model of \(\pi\) is selected, since it leads to a shorter code (5) than the fair-coin-flip model. Quantum theory is (argued by physicists to be) truly random. Hence all modern ToE candidates (P+G, S, C, M) are probabilistic. This yields huge additive constants \(|\log_{2}P(o_{1:t}^{true})|\) to the otherwise quite elegant theories \(Q\). [PERSON] [9, 15] argues that all apparent physical randomness is actually only pseudo random, _i.e._, generated by a small program. If this is true and we could find the random number generator, we could instantly predict all apparent quantum-mechanical random events. This would be a true improvement of existing theories, and indeed the corresponding CToE would be significantly shorter. In [20, Sec.8.6.2] I give an argument why believing in true random noise may be an unscientific position. **Theories with parameters. ** Many theories in physics depend on real-valued parameters. Since observations have finite accuracy, it is sufficient to specify these parameters to some finite accuracy. Hence the theories including their finite-precision parameters can be coded in finite length. There are general results and techniques [4, 5] that allow a comfortable handling of all this. For instance, for smooth parametric models, a parameter accuracy of \(O(1/\sqrt{n})\) is needed, which requires \(\frac{1}{2}\log_{2}n+O(1)\) bits per parameter. The explicable \(O(1)\) term depends on the smoothness of the model and prevents 'cheating' (e.g. zipping two parameters into one). **Infinite and continuous universes. ** So far we have assumed that each time-slice through our universe can be described in finitely many bits and time is discrete. Assume our universe were the infinite continuous 3+1 dimensional Minkowski space \(I\!\!R^{4}\) occupied by (tiny) balls (\"particles\").