Name: 馬爾科夫鏈(硬幣的語言：11/16) (Markov chains (Language of Coins: 11/16))
Uploaded: 2021-01-14T05:46:59.000Z
Duration: 7 min 15 s
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副詞型分詞構句

When observing the natural world, many of us

but they all seem to follow some underlying form.

And Plato believed that the true forms of the universe

Through observation of the natural world,

we could merely acquire approximate knowledge of them.

through abstract reasoning of philosophy and mathematics.

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.

that after an uncountable number of years,

This platonic focus on an abstract pure forms

when people tried to embrace the messy variation

in the real world, and apply mathematics to tease out

Bernoulli refined the idea of expectation.

He was focused on a method of accurately estimating

based on the number of times the event occurs

3,000 light pebbles and 2,000 dark pebbles

And that, to determine the ratio of white versus black

after another, with replacement, and note how many times

He went on to prove that the expected value of white

versus black observations will converge on the actual ratio,

And he concluded by saying, if observations of all events

it will be noticed that everything in the world

is governed by precise ratios and a constant law of change.

converge on an expected average, but the probability

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,

the bean falls into a bucket, representing

the ratio of left versus right deflection.

to be an ideal form, as it kept appearing everywhere,

It seems the average fate of these events

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

He didn't like the idea of us having this predetermined

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

However, as we all can relate, most things

Such as the chance of fire, or sun, or even

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

He goes on to say in a letter that, this circumstance prompts

So Markov extends Bernoulli's results to dependent variables,

Imagine a coin flip, which isn't independent, but dependent

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馬爾科夫鏈(硬幣的語言：11/16) (Markov chains (Language of Coins: 11/16))

state

random

claim

light