Uncertainty: Difference between revisions
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=== Measure-theoretic interpretation === | === Measure-theoretic interpretation === | ||
For my purposes in this section, by a measure on <math>\mathbb{R}</math> I mean one for which all points have measure 0. <math>\Gamma</math> is a countable cover of <math>\mathbb{R}</math> by measurable sets. Any measure <math>\mu</math> on <math>\mathbb{R}</math> restricts to a measure <math>\mu |_\Gamma</math> on <math>\Gamma</math>, where <math display="inline">\mu|_{\Gamma}(S) := \mu \left( \bigcup_{s\in S} s \right) = \sum_{s\in S} \mu(s). </math> In particular, if <math>\mu</math> is a probability measure then so is <math>\mu |_\Gamma</math>. | For my purposes in this section, by a measure on <math>\mathbb{R}</math> I mean one for which all points have measure 0. <math>\Gamma</math> is a countable cover of <math>\mathbb{R}</math> by measurable sets. Any measure <math>\mu</math> on <math>\mathbb{R}</math> restricts to a measure <math>\mu |_\Gamma</math> on <math>\Gamma</math>, where <math display="inline">\mu|_{\Gamma}(S) := \mu \left( \bigcup_{s\in S} s \right) = \sum_{s\in S} \mu(s). </math> In particular, if <math>\mu</math> is a probability measure then so is <math>\mu |_\Gamma</math>. | ||
For an interval <math>I_n</math> of <math>\Gamma</math>, let <math>\mu_n</math> be the probability measure defined by <math display="inline">\mu_n(A) := \frac{\text{Vol}(A \cap I_n)}{\text{Vol}(I_n) } </math>. Let <math>f : \mathbb{R} \rightarrow \mathbb{R} </math> be any measurable function. Then <math>f </math> induces a probability measure <math>f_* \mu_n </math> on <math>\mathbb{R} </math>, and hence a probability measure <math>f_* \mu_n |_\Gamma </math> on <math>\Gamma </math>. In this language, <math>\widetilde{f} </math> is simply the function sending <math display="block">\begin{align} | For an interval <math>I_n</math> of <math>\Gamma</math>, let <math>\mu_n</math> be the probability measure defined by <math display="inline">\mu_n(A) := \frac{\text{Vol}(A \cap I_n)}{\text{Vol}(I_n) } </math>. Let <math>f : \mathbb{R} \rightarrow \mathbb{R} </math> be any measurable function. Then <math>f </math> induces a probability measure <math>f_* \mu_n </math> on <math>\mathbb{R} </math>, and hence a probability measure <math>f_* \mu_n |_\Gamma </math> on <math>\Gamma </math>. In this language, <math>\widetilde{f} </math> is simply the function sending <math display="block">\begin{align} | ||
\widetilde{f} : \Gamma & \rightarrow \text{Prob}(\Gamma) \\ | \widetilde{f} : \Gamma & \rightarrow \text{Prob}(\Gamma) \\ | ||
I_n & \mapsto (f_* \mu_n) |_\Gamma | |||
\end{align} </math> | \end{align} </math> | ||
Revision as of 23:24, 22 October 2024
How do we measure continuous quantities?
First, set up some sort of "triangulation" of the space in which the continuous quantity can take values. For example, a ruler is a "triangulated" line. Typically some sort of quantitative system is chosen to keep track of the cells; e.g. for a ruler we count how many ticks there are.
A measurement of a continuous quantity is simply an identification of which one of those intervals the continuous quantity lies inside. It's a declaration "this quantity lives inside this interval."
How do we apply functions to continuous quantities?
Motivating example
Let's suppose that we are measuring lengths with a meter stick, which goes down to the nearest 2 millimeters. That is, the ticks on the stick appear at 0mm, 2mm, 4mm, 6mm, etc.
Suppose that we measure a certain length, then, by some process, we double the length. That is, we apply the "continuum" function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = 2x} .
That is what does to length itself, but what does do to measurements of length?
Let denote a length lying somewhere in the range millimeters (a minimal interval on the meter stick). Then lies somewhere in the range , which is the union of two minimal intervals of the meter stick. So maybe we should say that induces a multi-valued function?
![{\displaystyle [2k-2,2k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73fba3d12aedbe918012527ee101ead154bb21d6)
![{\displaystyle [4k-4,4k]=[4k-4,4k-2]\cup [4k-2,4k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73588e6c369190bbd94f848afbf9252544419dec)
Hmm, but what would be the induced function of ? It would send the interval to the interval , which is not necessarily a union of two minimal intervals.
![{\displaystyle [3k-3,3k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8adfce2e22c10cc71a73ae8a1025801be32ec71d)
I think it would be best to treat the induced function in a probabilistic manner: If the length is equally likely to be anywhere in the range , then
- it has a 1/2 chance of being in , and a 1/2 chance of being in .
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3x}
has,
- if is even, a 2/3 chance of being in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [3k-2,3k]} , and a 1/3 chance of being in .
- if is odd, a 2/3 chance of being in , and a 1/3 chance of being in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [3(k+1)-4,3(k+1)-2]} .
![{\displaystyle [4k-4,4k-2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f716d64af25afdba755bbf9ba91ceb1a15049f58)
![{\displaystyle [4k-2,4k]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/137838c9bf2a36e5e357ae24facb868e6f579835)
![{\displaystyle [3k-4,3k-2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f622d9560a0a94070d52b58314f223ea1caaca23)
![{\displaystyle [3(k+1)-6,3(k+1)-4]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78af294e5299637888c9076d1a8ba8e01fa8c390)
The general principle is that you have some sort of probability measure on the triangulation, and then you push it forward to another one.
The "problem" with this is that the true probability distributions are "continuum," as the following example demonstrates.
Let the grid on one axis consist of intervals Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [n\pi /6,(n+1)\pi /6]} , let the grid on the other axis consist of intervals Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [0.1n,0.1(n+1)]} where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n\in \mathbb {Z} } , and let the continuum function . Then sends the interval Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [0,\pi /6]} to the interval Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [{\sqrt {3}}/2,1]=[0.866\cdots ,1]\subset [0.8,0.9]\cup [0.9,1.0]} So after applying to a length in that internal, we will get a length which is in the interval Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [0.9,1.0]} with probability Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\textstyle {\frac {1}{10-5{\sqrt {3}}}}\approx 0.75} , and in the interval Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [0.8,0.9]} with probability Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\textstyle {\frac {9-5{\sqrt {3}}}{10-5{\sqrt {3}}}}\approx 0.25} .
General formulation
Let be a continuous function (which is also a continuum function), and suppose we have a grid consisting of intervals Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [a_{n},b_{n}]} where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a_{n},b_{n}} are real numbers such that . We refer to the ordered set of these intervals as "the grid" and denote it as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Gamma } .
Definition. A probability distribution on a set is a function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p:S\rightarrow [0,1]} , such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p_{s}=0} for all but finitely many Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle s\in S} , and . Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{Prob}}(S)} denote the set of probability distributions on .
Since is continuous, it sends intervals to intervals. If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I=[a,b]} is any interval, define its volume as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{Vol}}(I):=b-a} . Then induces a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widetilde{f} : \Gamma \rightarrow \text{Prob}(\Gamma) } , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widetilde{f}(n)_m := \frac{\text{Vol}([a_n, b_n] \cap f^{-1}([a_m,b_m]) ) }{\text{Vol}([a_n,b_n])} }
The interpretation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widetilde{f}} is completely clear: If the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is applied to a quantity that was measured as being in the grid interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a_n, b_n]} , then the probability that the resulting quantity will be measured as being in the grid interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a_m, b_m]} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widetilde{f}(n)_m} .
We can say that "a measurement of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} relative to the triangulation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} " is a field of random variables, one at each triangle, where the random variable at triangle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is drawn from the probability distribution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widetilde{f}(n)} .
Measure-theoretic interpretation
For my purposes in this section, by a measure on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} I mean one for which all points have measure 0. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} is a countable cover of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} by measurable sets. Any measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} restricts to a measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu |_\Gamma} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mu|_{\Gamma}(S) := \mu \left( \bigcup_{s\in S} s \right) = \sum_{s\in S} \mu(s). } In particular, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is a probability measure then so is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu |_\Gamma} .
For an interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_n} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} , let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_n} be the probability measure defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mu_n(A) := \frac{\text{Vol}(A \cap I_n)}{\text{Vol}(I_n) } } . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \mathbb{R} \rightarrow \mathbb{R} } be any measurable function. Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f } induces a probability measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_* \mu_n } on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R} } , and hence a probability measure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_* \mu_n |_\Gamma } on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma } . In this language, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widetilde{f} } is simply the function sending Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \widetilde{f} : \Gamma & \rightarrow \text{Prob}(\Gamma) \\ I_n & \mapsto (f_* \mu_n) |_\Gamma \end{align} }
Renormalization
Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \mathbb{R} \rightarrow \mathbb{R}} be some continuum function. There is a relationship between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} as measured on a fine triangulation, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} as measured on a coarse triangulation.
Motivating example
Let's suppose that we are measuring lengths with two meter sticks, one which goes down to the nearest 2 millimeters, and one which goes down to the nearest 1 millimeter. That is, the ticks on the first stick appear at 0mm, 2mm, 4mm, 6mm, etc., and the ticks on the second stick appear at 0mm, 1mm, 2mm, 3mm, etc. The former stick defines a triangulation which we will call Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} , and the latter defines a triangulation which we will call Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma'} .
There is a (discontinuous!!) function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i: \Gamma' \rightarrow \Gamma} , sending Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,1] \mapsto [0,2]} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [1,2] \mapsto [0,2]} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [2,3] \mapsto [2,4] } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [3,4] \mapsto [2,4]} , etc. This function is the answer to the question: "If a quantity is measured in a given grid cell of the fine triangulation, which grid cell of the course triangulation will it be measured in?"
More generally, the grid cells of the fine triangulation might not be contained inside the grid cells of the coarse triangulation, so a quantity with a given fine-scale measurement might have multiple possible coarse-scale measurements. So really the answer to the italicized question above should not be a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma' \rightarrow \Gamma } , but rather should be a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma' \rightarrow \text{Prob}(\Gamma) } .