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functional-estimating.md

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+# Functional Estimating
+
+Suppose a function
+
+$$
+f: ℝ^n \to ℝ
+$$
+
+That is, $f$ takes $n$ real parameters (an $n$-tuple) and returns a single real number.
+
+Suppose we want to determine the number of specified parameters necessary to
+reduce uncertainty of the output to some acceptable range.
+
+There are multiple possible approaches with varying levels of appropriateness.
+
+### [Multisets](https://en.wikipedia.org/wiki/Multiset)
+
+Focused on statistics, we determine we want a _population_ of all values of $f$.
+A true _set_ is insufficient for this use case as duplicate values are not respected.
+
+$$
+g: ℝ^m \to (\text{multisets of real numbers}) \\
+
+g(x_1,...,x_m) = \{\!\{ f(x_1,...,x_m,y_m+1,...y_n) | (y_m+1,...y_n) \in ℝ^{n-m} \}\!\}
+$$
+
+$g(x_1,...,x_m)$ is the uncountably infinite length _multiset_ of all possible
+outputs of $f(x_1,...,x_n)$
+
+### [Partial Application](https://en.wikipedia.org/wiki/Partial_application)
+
+It seems that the more common way of doing what I'm trying to would be to
+partially apply $(x_1,...,x_m)$ to $f(x_1,...,x_n)$
+
+$$
+f: ( X × Y × Z ) \to N \\
+
+\text{partial}(f): (Y × Z) \to N
+$$
+
+then work with the level set of $f_{\text{partial}}$
+
+<https://en.wikipedia.org/wiki/Currying>
+
+
+### [Pushforward Measure](https://en.wikipedia.org/wiki/Pushforward_measure)
+
+Start with a function $f:X \to Y$.
+Think of $X$ as the domain and $Y$ as the codomain.
+
+Pick a [measure](https://en.wikipedia.org/wiki/Measure_(mathematics)) $\mu$ on $X$.
+In general, a measure $\mu$ is a systematic way of assigning a "weight" to subsets of $X$.
+
+In continuous scenarios,
+$\mu$ might be something like the length/area/volume measure (Lebesgue measure)
+or a probability distribution
+(which also assigns a total measure of 1 to the entire domain).
+
+In discrete scenarios, $\mu$ is a count measure:
+each point in $X$ has weight 1,
+so the measure of a finite set is just how many points it has.
+
+Let $f : X \to Y$ be a measurable function and $\mu$ a measure on $X$.
+
+The pushforward measure $f_{\ast}\mu$ is the measure on $Y$ defined by
+
+$$
+\forall B \subseteq Y \text{(measurable)}, \quad
+
+(f_{\ast}\mu)(B) = \mu(\{x \in X : f(x) \in B\})
+$$
+
+In other words, "The measure of a subset $B$ of the codomain ($Y$)
+equals the measure (in $X$) of all points that map into $B$."
+
+$\mu(f^{−1}(\{y\})) =$ the weight (in this case count) assigned to $Y$
+
+If the domain $X$ is finite and $\mu$ is just counting measure:
+
+The pushforward measure $f_{\ast}\mu$ on the codomain $Y$ literally says something like:
+$(f_{\ast}\mu)(\{y\}) =$ the number of times $Y$ appears as an output.
+Equivalent to the multiset $\{\!\{ f(x): x \in X \}\!\}$
+
+When $X$ is infinite (countably or uncountably so),
+we move beyond plain
+"multisets" to a more general concept of "measure on $Y$."
+But it's the same intuitive idea:
+
+Instead of a simple integer count of duplicates, we have a "weight"
+(which could be infinite or fractional, depending on the measure).
+$μ(f−1({y}))$ is how "large" or "frequent" the set of points mapping to $Y$ is,
+according to $\mu$.
+
+## Abstractions
+
+There are two competing abstractions for measure analysis of the project price function:
+
+### Ignoring Aleatory Uncertainty
+
+We assume that price is certain for any defined scope, that _any_ variation in
+price reflects influence of a measurable parameter.
+
+```none
+   f(a_1,...,*,...,a_n)
+ _ ├                
+ │ ├              / 
+ █ ├       ______/  
+ █ ├    __/         
+ │ ├  _/            
+ ▔ ├                
+   └──┴──┴──┴──┴──┴─x_k
+```
+
+$$
+f: ℝ^n \to ℝ
+$$
+
+**Pros:**
+
+* In theory far simpler.
+
+### Accounting for Aleatory Uncertainty
+
+We assume that price is subject to some inherent randomness
+
+```none
+   f(a_1,...,*,...,a_n)
+ ┬ ├           ┬  ┬ 
+ │ ├        ┬  │  █ 
+ █ ├     ┬  █  █  │ 
+ █ ├  ▁  ┿  │  ┴  ┴ 
+ │ ├  ┿  ┴  ┴       
+ ▔ ├  ▔             
+   └──┴──┴──┴──┴──┴─x_k
+```
+
+**Pros:**
+
+* Likely more accurate