卡尔纳普式的归纳逻辑的局限与所罗门诺夫先验的优势

The Limits of Carnapian Inductive Logic and the Advantages of Solomonoff Prior

20世纪50年代,卡尔纳普发展了归纳逻辑来表示证据相对于假设的"确证度"。随后,利斯塔、古德-图灵等人提出了各种平滑方法,这些平滑方法可以看作广义的卡尔纳普式的归纳逻辑。这些方法虽然都可以从某个层次的"无差别原则"导出,但这并不能构成其理论基础。"无差别原则"无论作用在这里的哪一层都不合适,根据机器学习领域的无免费午餐定理,都不具有通用性,只有作用在可能世界的产生方式这一层次上导出的所罗门诺夫先验才具有通用性,能够逼近任何可计算的模式。而且,不仅如此,在同时满足奥卡姆剃刀原则和最大熵原则的意义上,所罗门诺夫先验具有最优性。

In 1950 s,Carnap develops inductive logic to express the degree of confirmation of some hypothesis relative to some evidence. After that,Ristad,Good develops serveral smoothing methods. Most of these smoothing methods can be taken as sort of general Carnapian inductive logic. Although they can be deduced from applying the so-called"indifference principle"to different levels,they are still lack of solid theoretical foundations. It seems that none of these"indifference principles"work,except the"indifference principle"on the level of the generation of possible worlds,which leads us to Solomonoff’s prior. According to the no-free-lunch theorem,the Carnapian logics are as good and as poor as any random algorithm,while Solomonoff’s induction model could achieve"universality". Besides,Solomonoff’s prior is optimal with respect to the constraint of Occam’s razor and maximum entropy principle.