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  • 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.

  • And 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 an abstract pure forms

  • remained popular for centuries.

  • And 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.

  • And 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.

  • And 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.

  • And after 10 collisions, or events,

  • the bean falls into a bucket, representing

  • the ratio of left versus right deflection.

  • Or heads versus tails.

  • And this overall curvature, known

  • as a 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.

  • But this was a dangerous philosophical idea to some.

  • And 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.

  • And 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.

  • And when the probability of some event

  • depends, or is conditional on previous events,

  • we say they are dependent events, or dependent variables.

  • So this claim angered another Russian mathematician,

  • Andre 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 could not even dream about.

  • So 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 a short term memory of one event.

  • And 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.

  • Now 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.

  • Now to run our machine, we simply start in a random state,

  • and make a selection.

  • And then we move to either state zero or one,

  • depending on that event.

  • And based on the outcome of that selection,

  • we output either a zero, if it's dark, or a one, if it's light.

  • And 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 probability.

  • Now 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.

  • And one of the first, and most famous applications

  • of Markov chains, was published by Claude Shannon.

When observing the natural world, many of us

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B2 US markov state probability dependent dark event

Markov chains (Language of Coins: 11/16)

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    Lypan posted on 2015/10/18
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