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 #2

I left off my previous post with Hume’s problem of induction and a way forward coming from Uwe Saint-Mont.

For this post, I was originally planning to do a deep dive on the structure of logic and epistemic uncertainty to help frame future discussions. Fortunately Sean McLure has already done a better job than I could ever do in a recent series on his podcast, NonTrivial.

I highly recommend the listen as a primer for logical analysis to help understand where logic can be useful and where risk management becomes a better proxy, particularly in complex situations.

He spends a lot of time talking about inductive logic and Popperian falsification, which I think will become foundational as this probability series progresses.

Here is the link to the first part of the Facts and Logic Series on NonTrivial: https://overcast.fm/+ctunQ1iqg

Here is a link to Sean’s comprehensive list of logical fallacies: https://www.notes2tree.com/published_tree/?publish_tree=h7aGQYBbDd

Next post I’ll get back to probability starting with the Uwe Saint-Mont paper.

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.

Returning to the Blog

I have had this nagging sensation that I need to write again. Had everything set up to go on my iPad for some late night writing sessions and then my keyboard died. I wasn’t sure what the issue was but finally just decided to replace it.

Lot’s of things going on since my last post. Top of mind is $GME and the recent hedge fund blow ups. I like GME a lot and think they have some long term potential to turn around into a major gaming media company. Also, if we turn the corner on coronavirus and can go back to small public gatherings I can see some huge potential for board gaming as a turnaround strategy for the retail gaming. Nowhere near justifying the $300+ price point, but a good direction for the company nonetheless, I think.

I have also been spending a lot of time circling around complex systems. I think I was originally introduced to the topic through a Coursera class from Scott Page years ago. Some of the recent work from Yaneer Bar-Yam, Nassim Taleb, Joe Norman and Ole Peters has kindled the old fire and I have been reading everything I can get my hands on. I am finding my main limitation so far has been my lack of a deep working knowledge of probability and statistics so I am starting from scratch with a goal of eventually understanding Nassim’s recent writings in his Technical Incerto.

I recently learned that Notability now has an automated handwriting to equation tool so I’ll be posting some of the learning as I go along. It’s mostly for my sake to help express my understanding but feel free to follow along.

I’ll start with Papoulis and see where it goes from there. I have a tendency to bounce around books a bit as I find textbook statistics incredibly boring. I figure by the end of it I’ll have a self-made understanding of probability and stats and hopefully some interesting insights along the way.

Lastly, just before I pop off, I have been digging into Christopher Alexander’s Nature of Order recently. I am currently on Book 1 and its pretty damn interesting already. I’m sensing a deep dive into David Bohm’s physics work on implicate order in the near future.

Lastly, lastly, I had the pleasure of running across a recent blog post from Jerry Neumann on LinkedIn. The first piece I read was called Strategy Under Uncertainty and it is a wonderful combination of Christiansen, Taleb and his personal experiences in Angel and VC startup investing. His blog is called the Reaction Wheel (after some kind of satellite part) and I slogged through all 270 posts over the Christmas holidays. It was time well spent.

Han Zero

To be accompanied with the music video “Han Zero (lofi)” https://youtu.be/xm2hvD-NF7Q

“Han!” she yelled.

“What?” He replied in a tired tone.

“We’re moving.” She said, pointing out the window.

Han sat up rubbing his eyes. He got out of the bed and moved over to where Agatha was standing. Peering out the window he could see the telltale signs of motion as the station they had been docked at slowly became smaller and smaller.

Sure enough, they were on a trajectory, but when had he put in the coordinates? He drifted off into his mind trying to piece together events in the last few hours. As he re-entered that mental space between sleep and wakefulness he remembered that he was startled earlier by that easily forgotten clattering of the Bernoulli Drive.

Snapping to attention he reached down to grab his pants and got up to walk over to the Bernoulli Drive. It was a surprisingly simple machine that liked to play games – always leaving the pilot guessing as to whether it would work or not. They had gotten enough thrust from the previous burn that they were still moving, but without a proper kick their orbit around the station would decay and they would collide back on the Rebel station.

He smacked the drive with his hand and it sputtered and wheezed a breathy mechanical note. They waited. There was nothing, no lights not even a hum or a hiss. It had been doing this more often recently. 

“Shit.” He said. 

The Bernoulli Drive’s function was always discontinuous, bursting between fantastic possibilities of one moment and eery silence in another.  He much preferred the predictability of the Ensemble drives. While they were impossible to understand – complex beyond interpretability – at least they produced something! Han knew that was a cop-out though. Ensemble drives were prone to overfitting and you could easily run off the Graph if you didn’t have a team of technicians constantly tuning the device and retroactively explaining deviations.

Ensemble drive aside he still would have liked to retool the Bernoulli Drive at the last docking. The unpredictability was really starting to get on his nerves! Unfortunately his handler for this trip, an unpleasant fellow by the name of Darth was eager to get his cargo to the Axis. They would have to put up with the finicky drive and hope the input parameters didn’t drift any further than they already had.

He slapped the drive again, this time with a noise so loud Agatha’s attenuators whizzed to life. 

“Easy!” Agatha demanded, putting her hands on her ears with a languishing scoff as she adjusted her settings.

The Bernoulli Drive was silent. He tried to think of nothing. Let the drive give him clues. Dealing with pure randomness was like trying to peer into an additional dimension beyond your senses. He caught the Moment and renormalized. With a pinch he started the drive and it sputtered into life in discrete bursts. Chop! Chop chop! Then, with an instantaneous jolt and a loud Wham! they were actively accelerating. He felt satisfied with himself even though he intimately knew that controlling randomness was a rouse.

“I fucking hate that thing,” Han said, exasperated “it’s never predictable when you need it to be.” He leaned back into his pilot seat giving a cursory glance at the Graph monitor. The trajectory was set, there was no changing that now. All they could do was wait.

He adjusted the seat setting to lift his feet up and relaxed. With the last output from the Bernoulli Drive they had all the momentum they needed to make it to their destination.

“Han?” Agatha said after several minutes.

He snapped back to attention and looked over at her. As he acknowledged her with a dumb smile, he gazed past her trying to remember where he had found her again. Was it on the Edge during one of his cargo missions? Or was it the inner worlds of the Axis on the last delivery? Trips down the Graph always left the short-term memory a little hazy. It didn’t matter any more. 

There was one thing that was clear: Agatha was an AI babe he couldn’t live without.

Han had first met Agatha in the Machine Learning Academy where she had been studying irreducible systems, one of the last subjects that AI’s had not completely overturned since the Awakening. He had completed his pilot apprenticeship a few years before and had been back at the Academy’s cantina for a reunion. Agatha had spotted him in one of the smoky corners of the establishment. Han wasn’t your usual smuggler shooting holes cargo containers like the Talebian fixers busting holes through shaky logic. No, Han understood the real world and had learned his trade through experiences that only a proper apprenticeship could provide. It was the deviation from her training that attracted Agatha to Han. He just wasn’t like other things she had seen before. It was galactic romance at first sight.

Han recalled giving up a gig to spend an extra month with her on the Academy world before finally getting back on the graph with a cargo bounty he couldn’t refuse. That was the last time he saw her until he picked her up at the Foundry on one the Edge worlds – that’s where he had found her!

“What happens when we get there?” She asked.

His thought interrupted,, Han reflected on that question. How many times had he been to the Axis since he had gotten his ship? Nine? Ten? Every trip was the same. Grab some illicit cargo in one of the Edge worlds and travel down the Graph to make a delivery to one of the stations or dock worlds near the Axis. Nobody had ever actually been to the Axis before, that was just a saying. People living near the Axis were comfortable living at the limit, never daring to crossover to the Indeterminate. Once the cargo was delivered he would head towards the Axis getting flung out to the Edge again by the Torus-like geometry of space there.

“I guess we do it again.” Han said. He cracked open a beer and sat down reflecting on the unknown lying in wait for him at the Axis…