Numeric uncertainty - Redigeerimiste ajalugu

Tammet: /* Fuzzy measures */

2020-05-15T06:32:32Z

Fuzzy measures

←Vanem redaktsioon		Redaktsioon: 15. mai 2020, kell 08:32
124. rida:		124. rida:
	The standard example picture is [https://upload.wikimedia.org/wikipedia/commons/thumb/6/61/Fuzzy_logic_temperature_en.svg/660px-Fuzzy_logic_temperature_en.svg.png like this].		The standard example picture is [https://upload.wikimedia.org/wikipedia/commons/thumb/6/61/Fuzzy_logic_temperature_en.svg/660px-Fuzzy_logic_temperature_en.svg.png like this].

	[https://en.wikipedia.org/wiki/Fuzzy_logic Fuzzy logic] was built to handle such cases. The fuzzy logic approach works quite well in specific		[https://en.wikipedia.org/wiki/Fuzzy_logic Fuzzy logic] <font color="red">E2</font> was built to handle such cases. The fuzzy logic approach works quite well in specific
	measurement cases with rules, and is thus actually used in practice.		measurement cases with rules, and is thus actually used in practice.

Tammet – 15. mai 2020, kell 06:28

2020-05-15T06:28:09Z

←Vanem redaktsioon		Redaktsioon: 15. mai 2020, kell 08:28
79. rida:		79. rida:

	Two main camps for handling more-classical kinds of probabilities are		Two main camps for handling more-classical kinds of probabilities are
	* [https://en.wikipedia.org/wiki/Frequentist_probability Frequentists]: the core intuition is treating probabilities as limit values of statistical experiments, when more and more experiments are made. Dice-kinds of examples fit this camp very well.		* [https://en.wikipedia.org/wiki/Frequentist_probability Frequentists]: <font color="red">E2</font> the core intuition is treating probabilities as limit values of statistical experiments, when more and more experiments are made. Dice-kinds of examples fit this camp very well.
	* [https://en.wikipedia.org/wiki/Bayesian_probability Bayesianists] are more subjective, interpreting probability as "reasonable expectation" (a probability is assigned to a hypothesis), yet basing their calculations on the famous [https://en.wikipedia.org/wiki/Bayes%27_theorem Bayes theorem] of [https://en.wikipedia.org/wiki/Conditional_probability Conditional probability]		* [https://en.wikipedia.org/wiki/Bayesian_probability Bayesianists] <font color="red">E2</font> are more subjective, interpreting probability as "reasonable expectation" (a probability is assigned to a hypothesis), yet basing their calculations on the famous [https://en.wikipedia.org/wiki/Bayes%27_theorem Bayes theorem] of [https://en.wikipedia.org/wiki/Conditional_probability Conditional probability]

	=== Subjective probability or confidence ===		=== Subjective probability or confidence ===
105. rida:		105. rida:
	Have a brief look at the following main important lines of reasoning/camps-of-thought, just to understand that there are really		Have a brief look at the following main important lines of reasoning/camps-of-thought, just to understand that there are really
	many different approaches. Most of the following are somewhat similar to [https://en.wikipedia.org/wiki/Bayesian_probability Bayesian probability], sometimes		many different approaches. Most of the following are somewhat similar to [https://en.wikipedia.org/wiki/Bayesian_probability Bayesian probability], sometimes
	using also some form of [https://en.wikipedia.org/wiki/Imprecise_probability Imprecise probability]:		using also some form of [https://en.wikipedia.org/wiki/Imprecise_probability Imprecise probability]: <font color="red">E2</font>
	* [https://en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory Dempster-Shafer_theory]		* [https://en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory Dempster-Shafer_theory]
	* [https://en.wikipedia.org/wiki/Probabilistic_logic Probabilistic logic]		* [https://en.wikipedia.org/wiki/Probabilistic_logic Probabilistic logic]

Tammet: /* A motivating example */

2020-05-08T07:57:28Z

A motivating example

←Vanem redaktsioon		Redaktsioon: 8. mai 2020, kell 09:57
6. rida:		6. rida:
	I have an intuitive rule which says that birds can fly with the probability 90%. Let us formalize this as:		I have an intuitive rule which says that birds can fly with the probability 90%. Let us formalize this as:

	* bird(object1) : 0.8 : John		* bird(object1) : 0.8 :: John
	* bird(object1) : 0.7 : Mike		* bird(object1) : 0.7 :: Mike
	* [All X. bird(X) => canfly(X)] : 0.9 : me		* [All X. bird(X) => canfly(X)] : 0.9 :: me

	What can I derive from here? A simple idea is to combine the bird observation probabilities to a stronger one, using a standard probability calculation rule P(a v b) = P(a) + P(b) - (P(a) * P(b)),		What can I derive from here? A simple idea is to combine the bird observation probabilities to a stronger one, using a standard probability calculation rule P(a v b) = P(a) + P(b) - (P(a) * P(b)),
	which holds in case a and b are independent observations. We get:		which holds in case a and b are independent observations. We get:

	* bird(object1) : 0.8+0.7-(0.9*0.7) = 0.87 : John, Mike		* bird(object1) : 0.8+0.7-(0.9*0.7) = 0.87 :: John, Mike

	Second, using the derivation rule, decreasing the probability according to a probability rule P(a & b) = P(a) * P(b), again in case when a and b are independent:		Second, using the derivation rule, decreasing the probability according to a probability rule P(a & b) = P(a) * P(b), again in case when a and b are independent:

	* canfly(object1) : 0.87 * 0.9 = 0.783 : John,Mike,me		* canfly(object1) : 0.87 * 0.9 = 0.783 :: John,Mike,me

	Now, what can go wrong with such calculations? A lot: there are nontrivial nuances. For example, does		Now, what can go wrong with such calculations? A lot: there are nontrivial nuances. For example, does

Tammet: /* A motivating example */

2020-05-08T07:56:09Z

A motivating example

←Vanem redaktsioon		Redaktsioon: 8. mai 2020, kell 09:56
10. rida:		10. rida:
	* [All X. bird(X) => canfly(X)] : 0.9 : me		* [All X. bird(X) => canfly(X)] : 0.9 : me

	What can I derive from here? A simple idea is to combine the bird observation probabilities to a stronger one, using ~~the~~ rule P(a v b) = P(a) + P(b) - (P(a) * P(b)),		What can I derive from here? A simple idea is to combine the bird observation probabilities to a stronger one, using a standard probability calculation rule P(a v b) = P(a) + P(b) - (P(a) * P(b)),
	which ~~stems from the Bayes rule~~ in case a and b are independent observations. We get:		which holds in case a and b are independent observations. We get:

	* bird(object1) : 0.8+0.7-(0.9*0.7) = 0.87 : John, Mike		* bird(object1) : 0.8+0.7-(0.9*0.7) = 0.87 : John, Mike

	Second, using the derivation rule, decreasing the probability according to a rule P(a & b) = P(a) * P(b), again in case when a and b are independent:		Second, using the derivation rule, decreasing the probability according to a probability rule P(a & b) = P(a) * P(b), again in case when a and b are independent:

	* canfly(object1) : 0.87 * 0.9 = 0.783 : John,Mike,me		* canfly(object1) : 0.87 * 0.9 = 0.783 : John,Mike,me

Tammet: /* A motivating example */

2020-05-08T07:51:51Z

A motivating example

←Vanem redaktsioon		Redaktsioon: 8. mai 2020, kell 09:51
31. rida:		31. rida:
	in several ways.		in several ways.

	~~Also:~~		What more could go wrong or require extra care?
	* Are the calculation rules we used really true or applicable? In which context are they applicable? Can they lead		* Are the calculation rules we used really true or applicable? In which context are they applicable? Can they lead to incorrect or unintuitive results in other cases?
	to incorrect or unintuitive results in other cases?
	* What about "independent": are these statements independent and to what degree?		* What about "independent": are these statements independent and to what degree?
	* Are we sure our procedure does not use same input several times to "over-strengthen" the probabilities?		* Are we sure our procedure does not use same input several times to "over-strengthen" the probabilities?

Tammet: /* A motivating example */

2020-05-08T07:50:57Z

A motivating example

←Vanem redaktsioon		Redaktsioon: 8. mai 2020, kell 09:50
31. rida:		31. rida:
	in several ways.		in several ways.

	Also~~, are~~ the calculation rules we used really true or applicable? In which context are they applicable? Can they lead		Also:
	to incorrect or unintuitive results in other cases? ~~And what aboout~~ "independent": are these statements independent and		* Are the calculation rules we used really true or applicable? In which context are they applicable? Can they lead
	to what degree? ~~Also, what~~ is the "formalism" we used, ie how to encode that a rule/fact has a probability and sources?		to incorrect or unintuitive results in other cases?
	Can we use quantifiers also there, like quantifying over sources or something?		* What about "independent": are these statements independent and to what degree?
			* Are we sure our procedure does not use same input several times to "over-strengthen" the probabilities?
			* What is the "formalism" we used, ie how to encode that a rule/fact has a probability and sources? Can we use quantifiers also there, like quantifying over sources or something?

	== Different types of uncertainties ==		== Different types of uncertainties ==

Tammet: /* A motivating example */

2020-05-08T07:37:10Z

A motivating example

←Vanem redaktsioon		Redaktsioon: 8. mai 2020, kell 09:37
33. rida:		33. rida:
	Also, are the calculation rules we used really true or applicable? In which context are they applicable? Can they lead		Also, are the calculation rules we used really true or applicable? In which context are they applicable? Can they lead
	to incorrect or unintuitive results in other cases? And what aboout "independent": are these statements independent and		to incorrect or unintuitive results in other cases? And what aboout "independent": are these statements independent and
	to what degree?		to what degree? Also, what is the "formalism" we used, ie how to encode that a rule/fact has a probability and sources?
			Can we use quantifiers also there, like quantifying over sources or something?

	== Different types of uncertainties ==		== Different types of uncertainties ==

Tammet: /* A motivating example */

2020-05-08T07:17:54Z

A motivating example

←Vanem redaktsioon		Redaktsioon: 8. mai 2020, kell 09:17
3. rida:		3. rida:

	John observes something in the field which looks like a bird and estimates the probability of it being a bird as 80%.		John observes something in the field which looks like a bird and estimates the probability of it being a bird as 80%.
	Mike observes the same object, but ~~has~~ estimates the probability as 70%.		Mike observes the same object, but estimates the probability as 70%.
	I have an intuitive rule which says that birds can fly with the probability 90%. Let us formalize this as:		I have an intuitive rule which says that birds can fly with the probability 90%. Let us formalize this as:

Tammet: /* A motivating example */

2020-05-08T07:10:01Z

A motivating example

←Vanem redaktsioon		Redaktsioon: 8. mai 2020, kell 09:10
32. rida:		32. rida:

	Also, are the calculation rules we used really true or applicable? In which context are they applicable? Can they lead		Also, are the calculation rules we used really true or applicable? In which context are they applicable? Can they lead
	to incorrect or unintuitive results in other cases?		to incorrect or unintuitive results in other cases? And what aboout "independent": are these statements independent and
			to what degree?

	== Different types of uncertainties ==		== Different types of uncertainties ==

Tammet: /* A motivating example */

2020-05-08T07:09:23Z

A motivating example

←Vanem redaktsioon		Redaktsioon: 8. mai 2020, kell 09:09
30. rida:		30. rida:
	For this particular question, we have to be careful to indicate what the probability number means: it can be interpreted		For this particular question, we have to be careful to indicate what the probability number means: it can be interpreted
	in several ways.		in several ways.

			Also, are the calculation rules we used really true or applicable? In which context are they applicable? Can they lead
			to incorrect or unintuitive results in other cases?

	== Different types of uncertainties ==		== Different types of uncertainties ==