Partial answer: I input Solomonoff (2009), "Algorithmic Probability: Theory and Applications" into ChatGPT and Claude, ansking the following question:

The attached article describes Solomonoff induction. Does it answer the question if there's always an UTM that induces a Solomonoff prior that approximates arbitrarily well a given positive distribution?

ChatGPT annoyingly waffles back what's in the article, as it often did when I tried it on such things in the past. Claude instead is concise and precise. Complete answers follow.

Claude:

No, the article does not directly address the question of whether there always exists a universal Turing machine (UTM) that induces a Solomonoff prior that approximates arbitrarily well a given positive distribution.

The article focuses on describing Solomonoff induction and algorithmic probability more generally. It discusses properties like completeness, incomputability, diversity, and subjectivity of the Solomonoff prior, as well as applications to sequence prediction and grammar induction.

But it does not analyze the specific question of whether for any positive distribution there exists a choice of UTM that makes the corresponding Solomonoff prior arbitrarily close to that distribution. The article does state that the choice of reference machine affects the Solomonoff prior, and that by modifying the computer language it is possible to "express any conceivable a priori probability distributions." However, it does not formally prove that any positive distribution can be approximated arbitrarily well.

So in summary, while the article provides good background on Solomonoff induction, it does not directly address or prove the specific claim in the question. The universality of approximating arbitrary positive distributions with a Solomonoff prior by choosing the right UTM is not explicitly analyzed.

ChatGPT:

The article "Algorithmic Probability—Theory and Applications" by Ray J. Solomonoff discusses the concept of Algorithmic Probability (ALP) and its properties, including its completeness and incomputability. ALP is described as a method that is theoretically capable of discovering any regularity in a dataset using a relatively small sample. This feature, referred to as the completeness of ALP, implies that it can approximate the true probabilities of symbols in data generated by any unknown probabilistic source P, with a very small error rate. The error in probability approaches zero more rapidly than 1/n, indicating rapid convergence to correct probabilities. This convergence holds for any P describable by a computer program, including many functions that are formally incomputable​​.

However, ALP is also characterized by its incomputability, meaning it is impossible to find the truly best models with any certainty due to the infinity of models to be tested, some of which take an unacceptably long time to evaluate. Despite this, ALP's incomputability does not inhibit its practical use for prediction. In practice, approximations to ALP can be used, though it is impossible to determine how close these approximations are to the true ALP. The practical approach involves using approximations of ALP and methods to estimate their effectiveness in future problems (Out of Sample Error), methods to search for good models, and quick and simple methods to compare models. Despite its theoretical incomputability, ALP's approximations can approach completeness, offering little error in prediction if sufficient time is spent searching for good models. This aspect makes ALP a desirable feature in practical prediction systems despite its incomputable nature​​.

In summary, the article does address the question of whether there's always an Universal Turing Machine (UTM) that induces a Solomonoff prior approximating arbitrarily well a given positive distribution. It explains that while ALP in its true form is incomputable, in practice, approximations of ALP can be employed effectively. These approximations can approach the completeness of ALP, thereby allowing for accurate predictions and the discovery of regularities in data, albeit without the ability to ascertain how close these approximations are to the true ALP. This implies that while a UTM can be used to induce a Solomonoff prior, the exactness of this approximation in practice is limited by the incomputability of ALP and relies on the effectiveness of the approximation methods employed.