The Advanced Math Behind Caramel Squares and Buttered Bread Hitting the Floor

Some people just don't appreciate the math that underpins our classical universe. Like today, for instance.  When I dropped this cooling rack of buckwheat caramel squares, exactly half of them landed sticky-side down on the floor.  It was the same experiment to test if a piece of buttered bread or buttered toast would end butter-side up or down. It was an amazing probability distribution of 0.5 in spite of the multi-variate inputs, force of gravity, varying weights, force of swearing, etc.  The randomness of it all conformed to theoretical statistical probability in this ad hoc Monte Carlo method.  She-that-continously-disapproves-of-me was not impressed with this experimental result and the exactitude of the value of observations.  I guess some people just don't have a mind for maths.  (In case you counted the squares and came up with an odd number, I ate the one that suffered the most topological deviation when it transferred its potential and kinetic energy to the surface of the floor.  Deceleration has no impact on flavor on impact.  Inquiring minds want to know these things.)

How Not To Convince Warren Buffett - Bayesian Approach To Revenue Forecasting For Startups

While waiting for Honda Xcelerator in Silicon Valley to evaluate my latest disruptive auto tech pitch, I got a little weary of documenting the API and creating more entry points, so I was thinking about revenue streams and startups. I received the Warren Buffet biography for Christmas, and by coincidence, I came across a passage in the book where a startup was pitched to Warren. It gave me pause to think.

Warren had bought the Wall Street firm Salomon Brothers, and it was a problem-child investment. The company was caught up in treasury bond scandal, and Warren had to beg and plead with the government and regulators not to shut them down, and destroy his investment. As a mea culpa, heads had to roll, and one of the heads was John "JM" Meriwether. JM had reported the transgression of one his employees that caused the evolving scandal, and JM's superiors sat on the information without immediately reporting it to the regulators. After it was all said and done, JM was a victim as well because of his position, although he had no culpability in hiding the fraud. He left Salomon Brothers and started a hedge fund called Long Term Capital. He approached Warren Buffett to invest in it. It was Meriwethers' approach that got my attention.

Warren was still on good terms with JM after the DCBM (contractors and consultants know this term -- it is "Don't Come Back on Monday"). Although JM got the DCBM, he was still welcome at Warren's table. If you are in Warren's inner circle, you get invited to a steak dinner at Gorat's in Omaha -ha-ha Nebraska. JM had a history of arbitrage and trading at Salomon and he compiled the numerical results of his successes and failures while heading the arb team. If you know anything about statistics, now you should be able to at least start feeling the heat in terms of the Bayesian Approach.

Over the course of ingesting the finer bovine parts, JM pulled out a schedule to show Buffett different probabilities (another Bayesian bell rings) of results and how much money his hedge fund, Long Term, could make, based on those probabilities. Also in the schedule was the probabilities of various strategies involving small or large trades with different parameters of leveraged capital. To someone like me, the approach was brilliant. It was totally Bayesian and it provided some evidence of pro forma revenues other than wishful thinking and shots in the dark at a dart board.

Every venture capitalist knows that over 99.999% of the business plans that they receive, show pro forma revenues of over a million dollars after two years. It is almost a de rigueur feature of a business plan and pitch deck. And we all know almost all of them never hit that benchmark. Taking a Bayesian Approach to revenue forecasting could be a breath of fresh air to business plans, pitch decks and venture capitalism in general, even though it didn't work on Warren Buffett.

So what is the Bayesian Approach? Bayes’ theorem is named after Rev. Thomas Bayes (1701–1761), who first provided an equation that allows new evidence to update beliefs (Wikipedia). The formula in mathematical terms is given as:

P(A|B) = P(B|A) x P(A) / P(B)

Describing it in words goes like this: A and B are related events and the probability of B happening is not 0. The probability of A happening, given that B has happened = the probability that B will happen given A, times the probability of B, all divided the the probability of B.

It doesn't sound like much, but the Bayes formula has staggering implications. It solves practical questions that were unanswerable by any other means: the defenders of Captain Dreyfus used it to demonstrate his innocence in the Dreyfus spying affair; insurance actuaries used it to set rates; Alan Turing used it to decode the German Enigma cipher and arguably save the Allies from losing the Second World War; the U.S. Navy used it to search for a missing H-bomb and to locate Soviet subs; RAND Corporation used it to assess the likelihood of a nuclear accident; and Harvard and Chicago researchers used it to verify the authorship of the Federalist Papers (The Less Wrong Blog). It is also the basis of some machine learning and artificial intelligence.

I think that it is a brilliant strategy for demonstrating revenue possibilities for start-ups. You could take a pool of known customers, a customer conversion rate (which is a probability based on your efforts to date) coupled to a variety of strategies to converting them, coupled to a variety of probabilies of what they will pay, and if you have done your homework, you will come up with a believable, but less spectacular pro forma revenue statement for your startup.

While the approach is brilliant, it didn't work on Warren Buffett. Why? Warren & crew had this to say about it: "We thought that they were very smart people. But we were a little leery of the complexity and leverage of their business. We were very leery of being used as a sales lead. We knew that others would follow if we got in." (Munger - The Snowball). Warren thought that there was a flaw in the original premise of how they were going to use their leverage. He didn't want to be a Judas goat -- a wise old goat that is used for it entire lifetime to daily lead other goats to slaughter.

So while it didn't convince billionaire Buffett, taking a Bayesian approach to revenue forecasting for a startup, just might land you a round of financing.

A Returned-Probability Artificial Neural Network - The Quantum Artificial Neural Network

Artificial Neural Networks associated with Deep Learning, Machine Learning using supervised and unsupervised learning are fairly good at figuring out deterministic things. For example they can find an open door for a robot to enter. They can find patterns in a given matrix or collection, or field.

However, sometimes there is no evident computability function. In other words, suppose that you are looking at an event or action that results from a whole bunch of unknown things, with a random bit of chaos thrown in.  It is impossible to derive a computable function without years of study and knowing the underlying principles. And even then, it still may be impossible to quantify with an equation, regression formula or such.

But Artificial Neural Nets can be trained to identify things without actually knowing anything about the background causes.  If you have a training set with the answers or results of size k (k being a series of cases), then you can always train your Artificial Neural Networks or Multilayer Perceptrons on k-1 sets, and evaluate how well you are doing with the last set. You measure the error rate and back propagate, and off you go to another training epoch if necessary.

This is happening with predicting solar flares and the resultant chaos that it cause with electronics and radio communications when these solar winds hit the earth.  Here is a link to the article, where ANN does the predicting:

http://www.dailymail.co.uk/sciencetech/article-2919263/The-computer-predict-SUN-AI-forecasts-devastating-solar-flares-knock-power-grids-Earth.html

In this case, the ANN's have shown that there is a relationship between vector magnetic fields of the surface of the sun, the solar atmosphere and solar flares.  That's all well and dandy for deterministic events, but what if the determinism was a probability and not a direct causal relationship mapped to its input parameters? What if there were other unknown or unknownable influence factors?

That's were you need an ANN (Artificial Neural Network) to return a probability as the hypothesis value. This is an easy task for a stats package working on database tables, churning out averages, probabilities, degrees of confidence, standard deviations etc, but I am left wondering if it could be done internally in the guts of the artificial neuron.

The artificial neuron is pretty basic. It sums up all of the inputs and biases multiplied by their weights, and feeds the result to an activation function.  It does this many times over in many layers.  What if you could encode the guts of the neuron to spit out the probability of the results of what is being inputted? What if somehow you changed the inner workings of the perceptron or neuron to calculate the probability.  It seems to me that the activation function is somehow ideally suited to adaptation to do this, because it can be constructed to deliver an activation value of between 0 and 1, which matches probability notation.

Our human brains work well with fuzziness in our chaotic world.   We unconsciously map patterns and assign probabilities to them. There is another word for fuzzy values. It is a "quantum" property. The more you know about one property of an object, the less you know about another.  Fuzziness. The great leap forward for Artificial Neural Networks, is to become Quantum and deliver a probability.  Once we can get an Artificial Neural Net machine to determine probability, then we can apply Bayesian mechanics. That's when it can make inferences, and get a computer on the road to thinking from first principles -- by things that it has learned by itself.