Probability - Revision history

Lfox: /* The traditional concept */

2024-07-03T16:19:34Z

The traditional concept

← Older revision		Revision as of 16:19, 3 July 2024
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	Source: <nowiki>https://twitter.com/DavidDeutschOxf/status/1692513674798252452</nowiki>		Source: <nowiki>https://twitter.com/DavidDeutschOxf/status/1692513674798252452</nowiki>

	[TODO elaborate more based on his actual writings]		[TODO elaborate more based on his actual writings] It's not ''completely'' clear that his view contradicts mine, because the many worlds interpretation also says that probability is epistemic; it's your uncertainty about which branch of the universal wave function you're living on.

	== Specification ==		== Specification ==

Lfox at 04:41, 20 May 2024

2024-05-20T04:41:17Z

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	[TODO yuck]		[TODO yuck]

	[Old definition: Probability is a quantitative method of exploiting the [[symmetry]] of a system to reach, ''with certainty'', conclusions about aggregates of outcomes, even when each individual outcome is uncertain.]		[Old definition, delete later: Probability is a quantitative method of exploiting the [[symmetry]] of a system to reach, ''with certainty'', conclusions about aggregates of outcomes, even when each individual outcome is uncertain.]

	The idea behind probability is that when faced with uncertainty, we quantify the outcomes which ''could be'' the case, based on our present knowledge. In some cases, this quantity is discrete, such as all the possible configurations of a deck of cards. In other cases, this quantity is continuous, such as all the possible places on the dart board where a dart could land.		The idea behind probability is that when faced with uncertainty, we quantify the outcomes which ''could be'' the case, based on our present knowledge. In some cases, this quantity is discrete, such as all the possible configurations of a deck of cards. In other cases, this quantity is continuous, such as all the possible places on the dart board where a dart could land.
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	Probability is related to [[symmetry]], in the sense that there is a symmetry among any set of states of affairs which could be the case: they are by definition indistinguishable in the respect that we do not know which of them is the case.		Probability is related to [[symmetry]], in the sense that there is a symmetry among any set of states of affairs which could be the case: they are by definition indistinguishable in the respect that we do not know which of them is the case.

	The ~~way that Probability~~ is ~~for reaching outcomes with~~ certainty [TODO]		The point of probability is to achieve certainty. [TODO elaborate]

	== Examples ==		== Examples ==

Lfox at 04:29, 20 May 2024

2024-05-20T04:29:31Z

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	'''~~Probability~~''' is [~~TODO defn~~]		The '''probability''' of a [[proposition]] is a quantification of all [[possible]] states of the world in which the proposition would be true, given one's present state of uncertainty.

	~~the '''probability''' of a proposition is a measure~~		[TODO yuck]

	~~of the ratio of~~

	[Old definition: Probability is a quantitative method of exploiting the [[symmetry]] of a system to reach, ''with certainty'', conclusions about aggregates of outcomes, even when each individual outcome is uncertain.]		[Old definition: Probability is a quantitative method of exploiting the [[symmetry]] of a system to reach, ''with certainty'', conclusions about aggregates of outcomes, even when each individual outcome is uncertain.]

	The idea behind probability is that when faced with uncertainty, we ''quantify'' the outcomes which could be the case, based on our present knowledge. In some cases, this quantity is discrete, such as all the possible configurations of a deck of cards. In other cases, this quantity is continuous, such as all the possible places on the dart board where a dart could land. ~~We then~~		The idea behind probability is that when faced with uncertainty, we quantify the outcomes which ''could be'' the case, based on our present knowledge. In some cases, this quantity is discrete, such as all the possible configurations of a deck of cards. In other cases, this quantity is continuous, such as all the possible places on the dart board where a dart could land.

	The purpose of quantifying outcomes like this is that it makes them amenable to measurement, to the identification of quantitative relationships between uncertain situations.		The purpose of quantifying outcomes like this is that it makes them amenable to measurement, to the identification of quantitative relationships between uncertain situations.

	Probability is related to [[symmetry]], in the sense that there is a symmetry among any set of states of affairs which could be the case: they are by definition indistinguishable in the respect that we do not know which of them is the case.		Probability is related to [[symmetry]], in the sense that there is a symmetry among any set of states of affairs which could be the case: they are by definition indistinguishable in the respect that we do not know which of them is the case.

	Probability is for reaching outcomes with certainty [TODO]		The way that Probability is for reaching outcomes with certainty [TODO]

	== Examples ==		== Examples ==

Lfox at 21:09, 19 May 2024

2024-05-19T21:09:35Z

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			'''Probability''' is [TODO defn]

			the '''probability''' of a proposition is a measure

			of the ratio of

	[Old definition: Probability is a quantitative method of exploiting the [[symmetry]] of a system to reach, ''with certainty'', conclusions about aggregates of outcomes, even when each individual outcome is uncertain.]		[Old definition: Probability is a quantitative method of exploiting the [[symmetry]] of a system to reach, ''with certainty'', conclusions about aggregates of outcomes, even when each individual outcome is uncertain.]

	The idea behind probability is that when faced with uncertainty, we ''quantify'' the outcomes which could be the case, based on our present knowledge. In some cases, this quantity is discrete, such as all the possible configurations of a deck of cards. In other cases, this quantity is continuous, such as all the possible places on the dart board where a dart could land.		The idea behind probability is that when faced with uncertainty, we ''quantify'' the outcomes which could be the case, based on our present knowledge. In some cases, this quantity is discrete, such as all the possible configurations of a deck of cards. In other cases, this quantity is continuous, such as all the possible places on the dart board where a dart could land. We then

			The purpose of quantifying outcomes like this is that it makes them amenable to measurement, to the identification of quantitative relationships between uncertain situations.

	~~The purpose~~ of ~~quantifying outcomes like this is~~ that ~~it makes~~ them ~~amenable to measurement~~.		Probability is related to [[symmetry]], in the sense that there is a symmetry among any set of states of affairs which could be the case: they are by definition indistinguishable in the respect that we do not know which of them is the case.

	Probability is ~~related to [~~[~~symmetry]~~] ~~in the sense that there is a symmetry among any set of outcomes which could be the case: they are by definition indistinguishable in the respect that we do not know which of them is the case.~~		Probability is for reaching outcomes with certainty [TODO]

	== Examples ==		== Examples ==

Lfox at 03:24, 19 May 2024

2024-05-19T03:24:44Z

← Older revision		Revision as of 03:24, 19 May 2024
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	'''Probability''' is a quantitative method of exploiting the [[symmetry]] of a system to reach, ''with certainty'', conclusions about aggregates of outcomes, even when each individual outcome is uncertain.

	The idea is that every context determines some small number <math>\epsilon > 0</math>, such that any event which will only happen with probability less than <math>\epsilon</math> is deemed ''impossible'' (IN THAT CONTEXT). [TODO really this should be a statement about propositions.]

	~~There's an objection that it's "possible" to flip~~ a ~~coin 100000x and get heads every time. This is mis-using~~ the ~~concept~~ of ~~"possible." We have concepts of false~~, ~~unlikely~~, ~~possible~~, ~~probable~~, ~~true, arbitrary, etc.~~ ''~~because we need these categories cognitively~~''. ~~If you categorize~~ a ~~coin landing heads 100000x in~~ a ~~row as something~~ that~~'s "possible," then~~ it ~~destroys your cognitive categories and renders your mind impotent~~. ~~More about this~~ in ~~section below~~.		[Old definition: Probability is a quantitative method of exploiting the [[symmetry]] of a system to reach, ''with certainty'', conclusions about aggregates of outcomes, even when each individual outcome is uncertain.]

			The idea behind probability is that when faced with uncertainty, we ''quantify'' the outcomes which could be the case, based on our present knowledge. In some cases, this quantity is discrete, such as all the possible configurations of a deck of cards. In other cases, this quantity is continuous, such as all the possible places on the dart board where a dart could land.

			The purpose of quantifying outcomes like this is that it makes them amenable to measurement.

			Probability is related to [[symmetry]] in the sense that there is a symmetry among any set of outcomes which could be the case: they are by definition indistinguishable in the respect that we do not know which of them is the case.

	== Examples ==		== Examples ==
	The following are examples of things that occur with numerical probability:		The following are examples of things that occur with numerical probability:

	- weather events ~~[TODO questionable]~~		- weather events

	- dice rolls, coin flips, drawing a card from a shuffled deck, lotteries, etc		- dice rolls, coin flips, drawing a card from a shuffled deck, lotteries, etc

	- pseudoRNGs		- pseudoRNGs. [Sometimes people ask: Are pseudoRNGs ''really'' random? The answer is ''yes'', because a good pRNG is precisely one whose outcomes are not something we know how to predict. TODO move]

	- the direction of cosmic rays		- the direction of cosmic rays
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	[TODO I really don't like this example. Part of the reason not to be concerned about heart attacks is that you can only do so much to prevent them. Also, the heart has an identity, and it will definitely fail or not fail based on nature. It's not like the universe is a place where anything at all is possible. We might not know all of the reasons for heart failure (or we might). When we do, we can get rid of probability.]		[TODO I really don't like this example. Part of the reason not to be concerned about heart attacks is that you can only do so much to prevent them. Also, the heart has an identity, and it will definitely fail or not fail based on nature. It's not like the universe is a place where anything at all is possible. We might not know all of the reasons for heart failure (or we might). When we do, we can get rid of probability.]

			The idea is that every context determines some small number <math>\epsilon > 0</math>, such that any event which will only happen with probability less than <math>\epsilon</math> is deemed ''impossible'' (IN THAT CONTEXT). [TODO really this should be a statement about propositions.] There's an objection that it's "possible" to flip a coin 100000x and get heads every time. This is mis-using the concept of "possible." We have concepts of false, unlikely, possible, probable, true, arbitrary, etc. ''because we need these categories cognitively''. If you categorize a coin landing heads 100000x in a row as something that's "possible," then it destroys your cognitive categories and renders your mind impotent. More about this in section below.[TODO does this rly belong here?]

	== The traditional concept ==		== The traditional concept ==

Lfox: /* Specification */

2024-04-30T00:04:23Z

Specification

← Older revision		Revision as of 00:04, 30 April 2024
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	== Specification ==		== Specification ==
	'''~~Principle~~ of probability.''' We should assign number <math>1/k</math> to each of the <math>k</math> possible states of system with a <math>k</math>-fold symmetry. This number is called the probability of the state.		'''Specification of probability.''' We should assign number <math>1/k</math> to each of the <math>k</math> possible states of system with a <math>k</math>-fold symmetry. This number is called the ''probability'' of the state.

	~~Consider~~ a ~~two-sided coin~~. The ~~coin has~~ a ~~twofold~~ symmetry. ~~It follows that two distinguishable coins have a fourfold symmetry, three distinguishable coins~~ have ~~an eightfold symmetry~~, and ~~<math>n</math> distinguishable coins have a <math>2^n</math>-fold symmetry. "Distinguishable coins" could mean the same coin at different periods in time (say after different flips)~~.		Sometimes this specification is called a principle (see e.g. the "fundamental assumption of statistical mechanics"). The principle say something like "any state in a system with a symmetry is equally likely to occur." That approach assumes we have some prior concept of probability, and then tries to equate it with multiplicity.

	So we see that a history of <math>n</math> coin flips belongs to a system with a <math>2^n</math>-fold symmetry. So by the ~~principle~~ of probability, it occurs with probability <math>2^{-n}</math>.		Consider a two-sided coin. The coin has a twofold symmetry. It follows that two distinguishable coins have a fourfold symmetry, three distinguishable coins have an eightfold symmetry, and <math>n</math> distinguishable coins have a <math>2^n</math>-fold symmetry. "Distinguishable coins" could mean the same coin at different periods in time (say after different flips). So we see that a history of <math>n</math> coin flips belongs to a system with a <math>2^n</math>-fold symmetry. So by the specification of probability, it occurs with probability <math>2^{-n}</math>.

			This principle generalizes of course.

Lfox: /* Specification */

2024-04-29T23:20:24Z

Specification

← Older revision		Revision as of 23:20, 29 April 2024
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	Consider a two-sided coin. The coin has a twofold symmetry. It follows that two distinguishable coins have a fourfold symmetry, three distinguishable coins have an eightfold symmetry, and <math>n</math> distinguishable coins have a <math>2^n</math>-fold symmetry. "Distinguishable coins" could mean the same coin at different periods in time (say after different flips).		Consider a two-sided coin. The coin has a twofold symmetry. It follows that two distinguishable coins have a fourfold symmetry, three distinguishable coins have an eightfold symmetry, and <math>n</math> distinguishable coins have a <math>2^n</math>-fold symmetry. "Distinguishable coins" could mean the same coin at different periods in time (say after different flips).

			So we see that a history of <math>n</math> coin flips belongs to a system with a <math>2^n</math>-fold symmetry. So by the principle of probability, it occurs with probability <math>2^{-n}</math>.

Lfox at 21:16, 29 April 2024

2024-04-29T21:16:11Z

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	'''Probability''' is a quantitative method of exploiting the [[symmetry]] of a system to reach, with certainty, conclusions about aggregates of outcomes, even when each individual outcome is uncertain.		'''Probability''' is a quantitative method of exploiting the [[symmetry]] of a system to reach, ''with certainty'', conclusions about aggregates of outcomes, even when each individual outcome is uncertain.

	The idea is that every context determines some small number <math>\epsilon > 0</math>, such that ~~anything~~ which will only happen with probability less than <math>\epsilon</math> is deemed ''impossible'' (IN THAT CONTEXT).		The idea is that every context determines some small number <math>\epsilon > 0</math>, such that any event which will only happen with probability less than <math>\epsilon</math> is deemed ''impossible'' (IN THAT CONTEXT). [TODO really this should be a statement about propositions.]

	There's an objection that it's "possible" to flip a coin 100000x and get heads every time. This is mis-using the concept of "possible." We have concepts of false, unlikely, possible, probable, true, arbitrary, etc. ''because we need these categories cognitively''. If you categorize a coin landing heads 100000x in a row as something that's "possible," then it destroys your cognitive categories and renders your mind impotent. More about this in section below.		There's an objection that it's "possible" to flip a coin 100000x and get heads every time. This is mis-using the concept of "possible." We have concepts of false, unlikely, possible, probable, true, arbitrary, etc. ''because we need these categories cognitively''. If you categorize a coin landing heads 100000x in a row as something that's "possible," then it destroys your cognitive categories and renders your mind impotent. More about this in section below.

Lfox at 20:41, 25 April 2024

2024-04-25T20:41:34Z

← Older revision		Revision as of 20:41, 25 April 2024
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	Let's say that you don't take this advice, and instead think of the heart attack as "possible." It will wreak havoc in your mind. For example, it's also possible that this afternoon, your boss will want to talk to you about how your business report report is coming along. How could you concentrate on preparing for a potential meeting with your boss, when it's "possible" that you're going to get a heart attack today?? You couldn't. Or for example, by the same standard by which you accepted that it's possible you'll get a heart attack while climbing the stairs, you are also logically compelled to accept that it's possible you'll die in an elevator-related accident. How should you get to the 5th floor of your office building, when it's "possible" you'll die no matter which way you do it?? You better just stay on the 1st floor and not risk it. Etc.		Let's say that you don't take this advice, and instead think of the heart attack as "possible." It will wreak havoc in your mind. For example, it's also possible that this afternoon, your boss will want to talk to you about how your business report report is coming along. How could you concentrate on preparing for a potential meeting with your boss, when it's "possible" that you're going to get a heart attack today?? You couldn't. Or for example, by the same standard by which you accepted that it's possible you'll get a heart attack while climbing the stairs, you are also logically compelled to accept that it's possible you'll die in an elevator-related accident. How should you get to the 5th floor of your office building, when it's "possible" you'll die no matter which way you do it?? You better just stay on the 1st floor and not risk it. Etc.

	And note that this prospect of a "possible" heart attack is just one of an endless list of "possible" ways that a healthy 30-year-old "could" die today. I chose it arbitrarily, and there are many other things I could have added to the list.		And note that this prospect of a "possible" heart attack is just one of an endless list of "possible" ways that a healthy 30-year-old "could" die today. I chose the possibility of a heart attack rather arbitrarily, and there are many other things I could have added to the list.

			[TODO I really don't like this example. Part of the reason not to be concerned about heart attacks is that you can only do so much to prevent them. Also, the heart has an identity, and it will definitely fail or not fail based on nature. It's not like the universe is a place where anything at all is possible. We might not know all of the reasons for heart failure (or we might). When we do, we can get rid of probability.]

	== The traditional concept ==		== The traditional concept ==
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	[TODO elaborate more based on his actual writings]		[TODO elaborate more based on his actual writings]

			== Specification ==
			'''Principle of probability.''' We should assign number <math>1/k</math> to each of the <math>k</math> possible states of system with a <math>k</math>-fold symmetry. This number is called the probability of the state.

			Consider a two-sided coin. The coin has a twofold symmetry. It follows that two distinguishable coins have a fourfold symmetry, three distinguishable coins have an eightfold symmetry, and <math>n</math> distinguishable coins have a <math>2^n</math>-fold symmetry. "Distinguishable coins" could mean the same coin at different periods in time (say after different flips).

Lfox at 20:25, 25 April 2024

2024-04-25T20:25:03Z

← Older revision		Revision as of 20:25, 25 April 2024
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	Let's say that you don't take this advice, and instead think of the heart attack as "possible." It will wreak havoc in your mind. For example, it's also possible that this afternoon, your boss will want to talk to you about how your business report report is coming along. How could you concentrate on preparing for a potential meeting with your boss, when it's "possible" that you're going to get a heart attack today?? You couldn't. Or for example, by the same standard by which you accepted that it's possible you'll get a heart attack while climbing the stairs, you are also logically compelled to accept that it's possible you'll die in an elevator-related accident. How should you get to the 5th floor of your office building, when it's "possible" you'll die no matter which way you do it?? You better just stay on the 1st floor and not risk it. Etc.		Let's say that you don't take this advice, and instead think of the heart attack as "possible." It will wreak havoc in your mind. For example, it's also possible that this afternoon, your boss will want to talk to you about how your business report report is coming along. How could you concentrate on preparing for a potential meeting with your boss, when it's "possible" that you're going to get a heart attack today?? You couldn't. Or for example, by the same standard by which you accepted that it's possible you'll get a heart attack while climbing the stairs, you are also logically compelled to accept that it's possible you'll die in an elevator-related accident. How should you get to the 5th floor of your office building, when it's "possible" you'll die no matter which way you do it?? You better just stay on the 1st floor and not risk it. Etc.

	And note that this prospect of a "possible" heart attack is just one of an endless list of "possible" ways that a healthy 30-year-old "could" die today.		And note that this prospect of a "possible" heart attack is just one of an endless list of "possible" ways that a healthy 30-year-old "could" die today. I chose it arbitrarily, and there are many other things I could have added to the list.

	== The traditional concept ==		== The traditional concept ==