---
id:
aliases: []
tags:
  - destiny/uncertain
  - topic/estimating
  - topic/math
  - type/idea
  - authorship/llm
title: Functional Estimating
dg-publish: true
---
# 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