Opprinnelsen av Markovs kjeder

Voiceover: When observing the natural world, many of us notice a somewhat beautiful dichotomy. No two things are ever exactly alike, but they all seem to follow some underlying form. Plato believed that the true forms of the universe were hidden from us. Through observation of the natural world, we could merely acquire approximate knowledge of them. They were hidden blueprints. The pure forms were only accessible through abstract reasoning of philosophy and mathematics. For example, the circle he describes as that which has the distance from its circumference to its center everywhere equal. Yet we will never find a material manifestation of a perfect circle or a perfectly straight line. Though interestingly, Plato speculated that after an uncountable number of years, the universe will reach an ideal state, returning to its perfect form. This Platonic focus on abstract pure forms remained popular for centuries. It wasn't until the 16th century when people tried to embrace the messy variation in the real world and apply mathematics to tease out underlying patterns. Bernoulli refined the idea of expectation. He was focused on a method of accurately estimating the unknown probability of some event based on the number of times the event occurs in independent trials. He uses a simple example. Suppose that without your knowledge, 3,000 light pebbles and 2,000 dark pebbles are hidden in an urn, and that to determine the ratio of white versus black by experiment, you draw one pebble after another, with replacement, and note how many times a white pebble is drawn versus black. He went on to prove that the expected value of white versus black observations will converge on the actual ratio as the number of trials increases, known as the weak law of large numbers. He concluded by saying, "If observations "of all events be continued for the entire infinity, "it will be noticed that everything in the world "is governed by precise ratios "and a constant law of change." This idea was quickly extended as it was noticed that not only did things converge on an expected average, but the probability of variation away from averages also follow a familiar, underlying shape, or distribution. A great example of this is Francis Galton's bean machine. Imagine each collision as a single independent event, such as a coin flip. After 10 collisions or events, the bean falls into a bucket representing the ratio of left versus right deflection, or heads versus tails. This overall curvature, known as the binomial distribution, appears to be an ideal form as it kept appearing everywhere any time you looked at the variation of a large number of random trials. It seems the average fate of these events is somehow predetermined, known today as the central limit theorem. This was a dangerous philosophical idea to some. Pavel Nekrasov, originally a theologian by training, later took up mathematics and was a strong proponent of the religious doctrine of free will. He didn't like the idea of us having this predetermined statistical fate. He made a famous claim that independence is a necessary condition for the law of large numbers, since independence just describes these toy examples using beans or dice, where the outcome of previous events doesn't change the probability of the current or future events. However, as we all can relate, most things in the physical world are clearly dependent on prior outcomes, such as the chance of fire or sun or even our life expectancy. When the probability of some event depends, or is conditional, on previous events, we say they are dependent events, or dependent variables. This claim angered another Russian mathematician, Andrey Markov, who maintained a very public animosity towards Nekrasov. He goes on to say in a letter that "this circumstance "prompts me to explain in a series of articles "that the law of large numbers can apply "to dependent variables," using a construction which he brags Nekrasov cannot even dream about. Markov extends Bernoulli's results to dependent variables using an ingenious construction. Imagine a coin flip which isn't independent, but dependent on the previous outcome, so it has short-term memory of one event. This can be visualized using a hypothetical machine which contains two cups, which we call states. In one state we have a 50-50 mix of light versus dark beads, while in the other state we have more dark versus light. One cup we can call state zero. It represents a dark having previously occurred, and the other state, we can call one, it represents a light bead having previously occurred. To run our machine, we simply start in a random state and make a selection. Then we move to either state zero or one, depending on that event. Based on the outcome of that selection, we output either a zero if it's dark, or a one if it's light. With this two-state machine, we can identify four possible transitions. If we are in state zero and a black occurs, we loop back to the same state and select again. If a light bead is selected, we jump over to state one, which can also loop back on itself, or jump back to state zero if a dark is chosen. The probability of a light versus dark selection is clearly not independent here, since it depends on the previous outcome. But Markov proved that as long as every state in the machine is reachable, when you run these machines in a sequence, they reach equilibrium. That is, no matter where you start, once you begin the sequence, the number of times you visit each state converges to some specific ratio, or a probability. This simple example disproved Nekrasov's claim that only independent events could converge on predictable distributions. But the concept of modeling sequences of random events using states and transitions between states became known as a Markov chain. One of the first and most famous applications of Markov chains was published by Claude Shannon.