Thinking About Probability #3

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.

Onwards!

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.

Thinking About Probability #1

This is my first attempt at adding some of the Notability maths into the blog – we’ll see how it goes.

As I mentioned in my previous post I am starting to chronicle my re-learnings in probability and statistics starting with the textbook “Probability, Random Variables and Stochastic Processes” by Papoulis and Pillai. The writing is pretty engaging for a math book and I’m hope to make it at least a chapter before getting distracted. Joe Norman incorporated parts of the book in his Applied Complexity Science course I took recently and I grabbed it off Amazon where I was pleasantly surprised to find a positive review from Taleb.

On to the learning. It’s late here so I’ll start from the beginning and maybe get through some thoughts on a few ideas/equations.

We start with an intro to probability and I think one of the key takeaways is that probability is dealing specifically with mass phenomena. How many events? Well, it depends. More on this later I suppose.

A physical interpretation of probability is provided by the following equation (Notability works great btw):

Where P means probability, A stands for an event, n(A) is the number of times the event occurred and n is the number of experiments run.

In a classic example of flipping a coin, if you flip the coin 10 times and you get heads 4 times out of the 10, then your probability P(A) of getting head (funny undergrad story on this I might tell later) is 4/10 or 0.4.

Easy enough… but is it?

This equation is an approximation of the probability and has a couple key assumptions baked in (assumptions to me are like kernels or rules of an automaton, perhaps more on this later). The first assumption is that the relative frequency of the occurrence of A is close to P(A) provided that n is sufficiently large. Obvious question: how large is large enough? The second assumption is that provided you have meet the first demand then only with a “high degree of certainty” is the equation valid.

Any use of this equation for prediction in the real word takes us down the path of induction. You’ll never be able to run an experiment an infinite amount of times and so to estimate the probability of an event occurring on the n+1 experiment it is going to have to be based on a priori knowledge. Say you flip a coin 100 times and see heads half of the time, then you would induce that the next 100 flips would yield heads half of the time.

David Hume had a problem with inductive logic. Gauche had the following summary of his argument:

“(i)Any verdict on the legitimacy of induction must result from deductive or inductive arguments, because those are the only kinds of reasoning.

(ii)A verdict on induction cannot be reached deductively. No inference from the observed to the unobserved is deductive, specifically because nothing in deductive logic can ensure that the course of nature will not change.

(iii)A verdict cannot be reached inductively. Any appeal to the past successes of inductive logic, such as that bread has continued to be nutritious and that the sun has continued to rise day after day, is but worthless circular reasoning when applied to induction’s future fortunes.

Therefore, because deduction and induction are the only options, and because neither can reach a verdict on induction, the conclusion follows that there is no rational justification for induction.”

Whoops. Now what?

It’s fucking late now and I’m just one page into Papoulis. I’m having fun though so I’ll keep doing this (daily?) and hopefully have time to go back and properly reference things.

Next post, I want to pick this up with a way-out of the inductive reasoning trap which I found in a recent paper by Uwe Saint-Mont. Then maybe I’ll move on to deductive (math-fun) reasoning.