Bayes' Theorem and Stochastic Processes

(Short paper I wrote for a CS23: Discrete Maths course at Cabrillo College. — March 2014)

How does the concept of a stochastic process relate to say, conditional probabilities and Bayes' Theorem (which is often referred to as inverted conditional probabilities). Are they the same? Is one a subset of the other? What is a stochastic process? What does it mean for a stochastic process to be discrete or continuous? What are some current and traditional areas of research where stochastic processes are used?

As humans, we find it important to analyse, classify, and understand the world around us. Some systems, such as spring contraptions (or other physics related systems), are considered deterministic systems, because they can be explained with a strict set of rules (usually explained with differential equations). However there are many other systems that are intrinsically random (or we simply don’t understand them to the full extent and therefore they “seem” random) and cannot be explained with a discrete set of equations, instead we need to rely on probability. We call these “random” systems, stochastic systems.

We refer to stochastic process as the collection of random data/variables which represent change in a stochastic system. Think of this as the set of variables obtained when you research the population of majestic lions in the African Savannah. These process can be considered Discrete or Continuous:

  • Discrete process are those in which its variables are discrete numbers (e.g. If you toss a coin you can get a finite number of possibilities:: TAILS, HEADS, or if you're lucky EDGE.) Instead there are
  • Continuous process in which the variables are actually representing ranges of data (e.g. A penguin weight can be anywhere from 8-10 pounds. Notice the possibilities inside this range are effectively uncountably infinite)

Since this systems are random, the only way we can understand and model them are through probabilities. Now let's say we are trying to model a typical Blackjack (aka. 21) game and we want to know our probabilities of winning with a perfect 21 points. Obviously, there are multiple ways of winning (e.g: 9+9+3 ; King+8 ; Ace[1]+Ace[11]+9 ). This is a discrete random system that we will only be able to model probabilistically. Furthermore, our models will have to adapt and change depending on our initial conditions, since our chance of winning are not the same if our first card is a 3 rather than a 13. This is where Probability Theory comes into play., and more specifically, conditional probability. Conditional probability is used to measure the probability of something happening if some other correlated event happens first. We denote this as P(A|B) “Probability of A given B”. A simple example would be: In a class where 20 assignments make up 100% of the grade, what is the probability that a student will pass the class if he got an A on his first assignment?

But now, what would happen if I inverted the previous question, “In a class where 20 assignments make up 100% of the grade, what is the probability that if student passes, he got an A on his first assignment”. This is where Bayes theorem comes in handy. It is a common mistake to think that P(A|B) has to equal P(B|A). The only way P(A|B)=P(B|A) is if P(A)=P(B), otherwise you have to know that P(A|B) =P(A|B)\cdot \frac{P(A)}{P(B)}I  wouldn’t say Bayes Theorem is a subset of Conditional Probability, they share more of a converse relationship (if we were to treat them as statements CP could be something similar to: p→ q while BT could be: q→ p).

As you can imagine, these ideas of stochastic process and conditional probability are used for almost anything you can think of: Programming Computer AI, Understanding electron behavior around the nucleus, predicting population growth, trying to beat the odds at a casino, weather prediction patterns, videogames, and disease patterns and pandemic spread predictions, among others. Pretty much anything that involves a random set of variables is usually modeled with this ideas and theorems.

To give an interesting and useful example, there is a model known as Stochastic Oscillators that are used to predict overall momentum of a stock. In other words, it helps predict whether the stock trend is going up or down in a set period of time. It is an interesting model, since you can have stock falling in the short run that actually have strong rising momentum. Stock fluctuations are typical in the stock market, since random events, such as congressional decisions, global or even regional events, might have impacts on the selling price of a stock, however the overall momentum of the stock will tell you if these bumps will have an overtly negative effect, or if they are just transitional for your selected time range (e.g: if you select a yearly time frame with a strong stock with rising momentum, a congressional decision might make the stock fall for perhaps a few months, but at the end of the year the stock will have risen further than its original price.)

One particularly flabbergasting example of the use of Stochastic models in Physics, is the a new theory for Quantum Mechanics generally referred to as  Stochastic interpretation of Quantum Mechanics. This theory models physics with the assumption that space-time structure is actually undergoing metric and topological fluctuations (layman’s terms: It changes and distorts, so maybe one second is actually a tad longer than another second or a meter is actually smaller in some places than other). When analyzing space-time from a large scale view, these tiny fluctuations average out and we get a convenient system that can be described with discrete models as the ones described in Classical Physics (aka. Force = Mass * Gravity). However, when we try to analyse our universe at a small scale, we can’t even out these fluctuations and eventually our classical views breakdown. At this point we start talking about Quantum Mechanics and we need to rely on Stochastic models and Probability Theory to describe its behavior. (e.g. Schrodinger's Equation, which gives us a probability wave of where we might find small particles). What this basically hypothesizes is that the universe’s fluctuations create Quantum Mechanics and not the other way around (which is how it is usually considered).

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