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A NON-BINARY VIEW OF ERGODICITY
From a mathematical point of view, a model describing an activity is either ergodic or isn't.
In real life, instead, it makes sense to describe ergodicity as a non-binary property.
Here is why.
Thread, 1/9
From a mathematical point of view, a model describing an activity is either ergodic or isn't.
In real life, instead, it makes sense to describe ergodicity as a non-binary property.
Here is why.
Thread, 1/9
2/ Take two non-ergodic activities: crossing the street and skydiving.
For both, the outcome of performing them an infinite number of times is death.
However, it is intuitive to most that crossing the street is on a different risk category.
How to express this mathematically?
For both, the outcome of performing them an infinite number of times is death.
However, it is intuitive to most that crossing the street is on a different risk category.
How to express this mathematically?
3/ People don't care about infinite time frames. They have limited life expectancy.
They do not care about the question, "does crossing the street have a chance to kill me?"
They care about, "does crossing the street have a meaningful chance to kill me within my lifetime?"
They do not care about the question, "does crossing the street have a chance to kill me?"
They care about, "does crossing the street have a meaningful chance to kill me within my lifetime?"
4/ Hence, for most people, the common question to determine whether an activity is ergodic ("does the time average correspond to the ensemble average?") has only meaning over time frames comparable to their life expectancy.
(or, their family's / company's life expectancy)
(or, their family's / company's life expectancy)
5/ On infinite time frames any activity has either a 0% or 100% chance of hitting an absorbing barrier. It makes sense to think of ergodicity as a binary property
On medium-term time frames, most activities have non-0 non-100 chance. It makes to think of ergodicity as non-binary
On medium-term time frames, most activities have non-0 non-100 chance. It makes to think of ergodicity as non-binary
6/ Why does it matter?
It matters because if ergodicity is non-binary, then we can use expressions such as "increase the ergodicity of an activity."
These expressions are useful for they focus us on practical solutions that can make us better off.
It matters because if ergodicity is non-binary, then we can use expressions such as "increase the ergodicity of an activity."
These expressions are useful for they focus us on practical solutions that can make us better off.
7/ As an example of the danger of binary thinking, take face masks.
Do they work 100%? No.
That means that we shouldn't use them if we use binary thinking (dangerous). Instead, using non-binary thinking, we can adopt practical solutions such as "better with than without".
Do they work 100%? No.
That means that we shouldn't use them if we use binary thinking (dangerous). Instead, using non-binary thinking, we can adopt practical solutions such as "better with than without".
8/ Hence the importance of non-binary considerations. This thread gave a justification to use ergodicity as a non-binary property in practical scenarios.
9/ I discuss this topic in larger details in my book, "Ergodicity: Definition, Examples, And Implications, As Simple As Possible" http://gum.co/ergodicity
