At long last, I am finally returning to the blog. I want to address my earlier point about Hume and review Uwe Saint-Mont's recent paper Induction: A Logical Analysis to provide some clarification on the application of probability and statistics.
If it seems like I am belaboring the point it's because I am. When learning probability the inductive approach comes up early and often. Having a sense of the guide posts early on will hopefully lead to better understanding of when and where different methods are applicable.
I mentioned in a previous post that Hume attacked induction with force. He disputed "that inductive conclusions, i.e., the very method of generalizing, can be justified at all." Since his work induction has been looked at with skepticism even to this day. Popper and Miller went so far as to provide a proof of the impossibility of inductive probability.
As Pillai would point out, we use induction in probability and statistics because in certain - but broad-ranging cases - it works. Supporting Pillai, Jaynes comments (p. 699) on Popper and Millner, their proof is “written for scientists... like trying to prove the impossibility of heavier-than-air flight to an assembly of professional airline pilots.” Understanding where it is induction is applicable is key.
I'd like to go into Uwe Saint-mont's paper in more detail later, but here I'll just summarize. Saint-mont seeks to deal with Hume's objections to induction in a constructive way. The key concepts are boundedness and information.
For bounded problems, induction is a reasonable approach to logic. You can use a deck of cards or a game of dice as examples of bounded problems. The rules of the game are well understood at the outset and outcomes are bounded.
With regards to information Saint-mont considers a basic model where you have tiers going from more general to less general. This introduces the concept of layers of information and the distance between those layers. If the distance is bounded, well-structured inductive leaps between less general and more general can be made in "small" steps. If the distance is unbounded, "a leap of faith from [less general] to [more general] never seems to be well-grounded." This implies that the less general tier is a subset of the more general tier. In such instances (the realm of "Mediocristan" as Taleb would write), the law of large numbers holds and statistics provides a valid response to Hume's arguments.
As Saint-Mont writes "without any assumptions, boundary conditions or restrictions - beyond any bounds - there are no grounds for induction." Thus, when using inductive logic it is important that your boundaries be well defined in order for the "logic funnel" to be applicable.
I can easily imagine constructs where you can create consistent models for induction (games of chance, certain closed financial systems, isolated computer algorithms). The question remains, however, how applicable can these constructs be to real life? Understanding that is key to useful application of probability and statistics since in unbounded systems uncertainty will dominate and induction can rapidly lead you astray.