Bayesian prior: Difference between revisions

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Then:
Then:
{{L3}}You have already made several category errors, should not be talking about art, and if you are, no-one should be listening; but <li>
{{L3}}You have already made several category errors, should not be talking about art, and if you are, no-one should be listening; but <li>
If nonetheless you still are, and they still are, and you are trying to estimate the statistical likelihood of a specific Elizabethan playwright being the best in history, then your knowledge that there were vastly fewer playwrights active in the Elizabethan period than have existed in all of history until now — which is a Bayesian “prior distribution” — might help you conclude that the odds of that Elizabethan playwright really being the best are vanishingly low.</ol>
If nonetheless you still are, and they still are, and you are trying to estimate the statistical likelihood of a specific Elizabethan playwright being the best in history, then your knowledge that there were vastly fewer playwrights active in the Elizabethan period than have existed in all of history until now — which is a Bayesian “prior distribution” — might help you conclude that the odds of that Elizabethan playwright really being the best are vanishingly low.</ol>  


At the same time, everyone else will conclude that you have no idea about literature and a shaky grasp even of Bayesian statistics.
At the same time, everyone else will conclude that you have no idea about literature and a shaky grasp even of Bayesian statistics.
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{{Drop|B|ayesian statistics have}}, in our dystopian techno-determinist age, a lot to answer for.
{{Drop|B|ayesian statistics have}}, in our dystopian techno-determinist age, a lot to answer for.


They take us from a surprising proof of how the odds work in a game of chance — that will help you choose wisely between goats and cars — but once it departs the tightly-controlled conditions of that statistical experiment, it is easily misapplied and may get badly lost in evaluating Shakespeare, our debt to distant future generations, and the onrushing [[apocalypse]], courtesy of which, it seems to say, there won’t ''be'' any distant future generations to worry about anyway.
In their place they can unravel surprising odds in a game of chance that human brains intuitively misapprehend this will help should you be asked to choose wisely between goats and cars — but outside the tight swim lanes of statistical experiment, they can be easily misapplied and may get badly lost in weighing up the risks of the market, the merits of Shakespeare, our debt to distant future generations, and the prospect of onrushing [[apocalypse]], courtesy of which, some theorists tell us, there won’t ''be'' many future generations to worry about anyway.
====Goats and sportscars====
====Goats and sportscars====
{{Drop|T|he neatest illustration}} of how Bayesian priors are meant to work is the “Monty Hall” problem, named for the ghost of the gameshow ''Deal or No Deal'' and famously articulated in a letter to ''Parade'' Magazine as follows:
{{Drop|T|he neatest illustration}} of how “Bayesian priors” work is the “Monty Hall” problem, named for the ghost of the gameshow ''Deal or No Deal'':
{{quote|
{{quote|
Suppose you're on a game show, and youʼre given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows whatʼs behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?}}
A game show contestant is asked to choose a prize from behind one of three doors. She is told one door conceals s a sports car and the other two goats. [''Why goats? — Ed'']


If you have not seen it before, intuitively you may say, ''no'': each door carried an equal probability before the host opened a door — 1/3 — and each carries an equal probability afterward — 1/2. While the odds are ''better'' now, they’re still even between each remaining door. So it should not matter whether you switch or not.
When the contestant has chosen, the host theatrically opens one of doors she did not choose, to reveal a goat.  


Bayesian probability shows this intuition to be ''wrong''. The host will never open the door you chose, nor the one concealing the car, so while this new information tells you nothing about your original choice: you already knew at least one of the other doors (and possibly both of them) didn’t contain the car. It does, however, tell you something about the ''remaining'' choice. The odds as between the other two doors change, from 1/3 each to 0/3 for the open door — we now know it definitely doesnʼt hold the car — and 2/3 for the closed one, which still might.
“Knowing what you know now, would you reconsider?”}}


So you ''should'' switch doors. You exchange a 1/3 chance of being ''right'' for a 1/3 risk of being wrong.  
If you have not seen it before, intuitively you may say, well, at the beginning each door carried an equal probability — 1/3 — and the remaining doors still do after the reveal — 1/2 — so while the player’s odds have ''improved'', either choice remains even. It diesn’t matter whether she sticks or twists, so she should be indifferent.


This proposal outrages some people, at first. Apparently, even statisticians. But it is true. It is easier to see if you imagine instead there are ''one thousand'' doors, not three, and after your first pick the host opens 998 of the other doors.  
Bayesian probability theory shows this intuition to be ''wrong''.
 
Staying put is to commit to choice you made then the odds were worse. So its odds remain the same. You have no more information about your original choice: you already knew it may or may not contain the car. You do, however, know something new about one of the doors you ''didn’t'' choose. The odds as between the other two doors change, from 1/3 each to 0/3 for the open door — it definitely ''doesnʼt'' hold the car — and 2/3 for the closed one, which still might.
 
The probabilities for the remaining options are therefore 1/3, for your original choice, and 2/3 for the other remaining door.
 
Oddly, a new person who now arrives and is presented the choice without that prior information, would calculate the probability at 50:50. The probabilities are a calculation based upon what you know. The calculation would be wrong because an important assumption in calculating probabilities - that the car and goat were randomly, normally distributed between two doors - is wrong. A third door has been unrandomly eliminated.
 
So you ''should'' switch doors. You exchange a 1/3 chance of being ''right'' for a 1/3 risk of being wrong. This proposal outrages some people, at first. Apparently, even statisticians. But it is true.  
 
It is easier to see if instead there are ''one thousand'' doors, not three, and after your first pick the host opens 998 of the other doors.  


Here you know you were almost certainly wrong first time, so if every possible wrong answer but one is revealed to you it stands more obviously to reason that the other door which accounts for 999/1000 of the original options, is the one holding the car.
Here you know you were almost certainly wrong first time, so if every possible wrong answer but one is revealed to you it stands more obviously to reason that the other door which accounts for 999/1000 of the original options, is the one holding the car.