Interfacing biology, category theory and mathematical statistics

Because this paper lies at the interface between different mathematical specialties, the present section summarizes its contents in straight text. To begin with, the MP is a property that a category may satisfy when it involves structurally different diagrams sharing the same cocones. To state our main results, it will not be necessary to consider the general MP though. In fact, the particular case of preordered sets will suffice, in which case the MP reduces to Proposition 1.
Second, in statistical hypothesis testing, a hypothesis can be seen as a predicate, of which we can aim at determining the truth value by using statistical decisions. There exist many optimality criteria to devise a decision to test a given hypothesis. In non-Bayesian approaches, which will be our focus below, such criteria are specified through the notions of size and power.

The size is the least upper bound for the probability of rejecting the hypothesis when this one is actually true. We generally want this size to remain below a certain value called level, because the hypothesis to test mostly represents the standard situation. For instance, planes in the sky are rare events, after all, and the standard hypothesis is "there is no plane", which represents the nominal situation. A too large level may result in an intolerable cluttering of a radar screen.

We do not want to be bothered by too many alarms. In contrast, when the hypothesis is false, we want to reject it with the highest possible confidence. The probability that a decision rejects the hypothesis when this one is actually false is called the power of the decision. For a given testing problem, we thus look for decisions with maximal power within the set of those decisions that have a size less than or equal to a a specified level. This defines a preorder. A maximal element in this preorder is said to be optimal.

Different hypotheses to test may thus require different criteria, specified through different notions of size and different notions of power. This is what we exploit below to exhibit two sets of "structurally different" decisions that satisfy the MP.

To carry out this construction, we consider the detection of a signal in independent standard gaussian noise, a classical problem in many applications. This is an hypothesis testing problem for which there exists an optimality criterion where the size is the so-called probability of false alarm and the power is the so-called probability of detection. This criterion has a solution, the Neyman-Pearson (NP) decision, which is thus the maximal element of a certain preorder. We can consider a second class of decisions, namely, the RDT decisions. These decisions are aimed at detecting deviations of a signal with respect to a known deterministic model in presence of independent standard gaussian noise. This problem is rotationally invariant and the RDT decisions are optimal with respect to a specific criterion defined through suitable notions of size and power. They are maximal elements of another preordered set. Although not dedicated to signal detection, these decisions can be used as surrogates to NP decisions to detect a signal. It turns out that the family of RDT decisions and that of NP decisions satisfy the MP as stated in Theorem 4. This is because the more data we have, the closer to perfection both decisions are.

Given $q\in[0,\infty[$, $\mathcal{B}_{\infty}(q)$ is the set of all real random variables $\Delta\in\mathcal{M}(\Omega,\mathbb{R})$ such that ${\left|\Delta\right|_{\_}{\infty}}\leqslant q$. As usual, we write $X\thicksim\mathcal{N}(0,1)$ to mean that $X\in\mathcal{M}(\Omega,\mathbb{R})$ is standard normal. Given a sequence $(X_{\_}n)_{\_}{n\in\mathbb{N}}\in\mathcal{M}(\Omega,\mathbb{R})^{\mathbb{N}}$ of real random variables, we write $X_{\_}1,X_{\_}2,\ldots\stackrel{{\scriptstyle\text{iid}}}{{\thicksim}}\mathcal{N}(0,1)$ to mean that $X_{\_}1,X_{\_}2,\ldots$ are independent and identically distributed with common distribution $\mathcal{N}(0,1)$.
Decisions et Observations. Throughout, $\mathcal{M}\!\left(\{0,1\}\!\times\!\Omega,\big{\{}0,1\big{\}}\right)$ designates the set of all measurable functions ${D}:\big{\{}0,1\big{\}}\times\Omega\to\big{\{}0,1\big{\}}$. Any element of $\mathcal{M}\!\left(\{0,1\}\!\times\!\Omega,\big{\{}0,1\big{\}}\right)$ is called a decision for obvious reasons given below. If ${D}\in\mathcal{M}\!\left(\{0,1\}\!\times\!\Omega,\big{\{}0,1\big{\}}\right)$ then, for any $\varepsilon\in\big{\{}0,1\big{\}}$, ${D}(\varepsilon)$ denotes the Bernoulli-distributed random variable ${D}(\varepsilon):\Omega\to\big{\{}0,1\big{\}}$ defined for any given $\omega\in\Omega$ by ${D}(\varepsilon)(\omega)={D}(\varepsilon,\omega)$. An $n$-dimensional test is hereafter any measurable function $f:\mathbb{R}^{n}\to\{0,1\}$ and $\mathcal{M}\left(\mathbb{R}^{n},\{0,1\}\right)$ stands for the set of all $n$-dimensional tests. A measurable function $X:\left\{0,1\right\}\times\Omega\to\mathbb{R}^{n}$ is hereafter called an observation and $\mathcal{M}\left(\{0,1\}\times\Omega,\mathbb{R}^{n}\right)$ denotes the set of all these observations. Given a test $f\in\mathcal{M}\left(\mathbb{R}^{n},\{0,1\}\right)$ and $X\in\mathcal{M}\left(\{0,1\}\times\Omega,\mathbb{R}^{n}\right)$, ${D}=f(X)$ is trivially a decision: ${D}\in\mathcal{M}\!\left(\{0,1\}\!\times\!\Omega,\big{\{}0,1\big{\}}\right)$. If $X\in\mathcal{M}\left(\{0,1\}\times\Omega,\mathbb{R}^{n}\right)$ then, for any $\varepsilon\in\{0,1\}$, $X(\varepsilon)=X(\varepsilon,\cdot)\in\mathcal{M}(\Omega,\mathbb{R}^{n})$ is defined for every $\omega\in\Omega$ by $X(\varepsilon)(\omega)=X(\varepsilon,\omega)$.
Empirical means. We define the empirical mean of a given sequence $y=(y_{\_}n)_{\_}{n\in\mathbb{N}}$ of real values as the sequence $\left(\langle{y}\rangle_{n}\right)_{\_}{n\in\mathbb{N}}$ of real values such that, $\forall n\in\mathbb{N},\langle{y}\rangle_{n}\,\,:=\,\,\frac{1}{n}\sum_{\_}{i=1}^{n}y_{\_}i$. By extension, the empirical mean of a sequence $Y=(Y_{\_}n)_{\_}{n\in\mathbb{N}}$ of random variables where each $Y_{\_}n\in\mathcal{M}(\Omega,\mathbb{R})$ is the sequence $\left(\langle{Y}\rangle_{n}\right)_{\_}{n\in\mathbb{N}}$ of random variables where, for each $n\in\mathbb{N}$, $\langle{Y}\rangle_{n}\in\mathcal{M}(\Omega,\mathbb{R})$ is defined by $\langle{Y}\rangle_{n}\,\,:=\,\,\frac{1}{n}\sum_{\_}{i=1}^{n}Y_{\_}i$. Therefore, for any $\omega\in\Omega$, $\langle{Y}\rangle_{n}(\omega)\,\,:=\,\,\langle{Y(\omega)}\rangle_{n}$ with $Y(\omega)=(Y_{\_}n(\omega))_{\_}{n\in\mathbb{N}}$. If $Y=(Y_{\_}n)_{\_}{n\in\mathbb{N}}$ is a sequence of observations ($\forall\,n\in\mathbb{N}$, $Y_{\_}n\in\mathcal{M}\left(\{0,1\}\times\Omega,\mathbb{R}\right)$), we define the empirical mean of $Y$ as the sequence $\left(\langle{Y}\rangle_{n}\right)_{\_}{n\in\mathbb{N}}$ of observations such that, for $\varepsilon\in\{0,1\}$, $\langle{Y}\rangle_{n}\in\mathcal{M}\left(\{0,1\}\times\Omega,\mathbb{R}\right)$ with $\langle{Y}\rangle_{n}(\varepsilon)=\langle{Y(\varepsilon)}\rangle_{n}$ and $Y(\varepsilon)=(Y_{\_}n(\varepsilon))_{\_}{n\in\mathbb{N}}$.
Preordered sets. Given a preordered set $(E,\preceq)$ and $A\subset E$, the set of maximal elements of $A$ is denoted by $\max\left(A,(E,\preceq)\right)$, the set of upper bounds of $A$ is denoted by $\textup{upper}\left(A,(E,\preceq)\right)$ and the set of least upper bounds of $A$ in $(E,\preceq)$ is denoted by $\sup\left(A,(E,\preceq)\right)$.

The multiplicity principle (MP) comes from [5]. It proposes a categorical approach to the biological degeneracy principle, which ensures a kind of flexible redundancy. Roughly, MP, in a category $\mathscr{C}$, ensures the existence of structurally non isomorphic diagrams with same colimit. A formal definition relies on the notion of a cluster between diagrams in a category $\mathscr{C}$.

For instance, a connected cone from $c$ to $D$ can be seen as a cluster from the constant functor $\Delta(c)$ to $D$; and any cocone from $E$ to $c$ is a cluster $E\to\Delta(c)$.

A cluster $G:D\to E$ defines a functor $\Omega G:\mathbf{Cocones}\left(E\right)\to\mathbf{Cocones}\left(D\right)$ mapping a cocone $\alpha$ to the cocone $\alpha\circ G$ (composite of $\alpha$, seen as a cluster, and $G$, which is a cluster).

$D$ and $E$ having the same cocones translates the property of both systems to accomplish the same function. The absence of clusters between $D$ and $E$ that define an isomorphism, reflects the structural difference between $D$ and $E$, which is key to robustness and adaptability: if the system described by $E$ fails, then $D$ may replace it.

The main purpose of this paper is to find a meaningful instance of the MP in some preorder. In the following, we do not distinguish between a preorder and its associated category.

Albeit trivial, the following lemma will be helpful.

Let $\varepsilon\in\{0,1\}$ be the unknown indicator value on whether a certain physical phenomenon has occurred ($\varepsilon=1$) or not ($\varepsilon=0$). We aim at determining this value. It is desirable to resort to something more evolved than tossing a coin to estimate $\varepsilon$. However, whatever ${D}$, the decision is erroneous for any $\omega\in\Omega$ such that ${D}(\varepsilon,\omega)\neq\varepsilon$. We thus have two distinct cases.
False alarm probability: If $\varepsilon=0$ and ${D}(0,\omega)=1$, we commit a false alarm or error of the $1^{\text{st}}$ kind, since we have erroneously decided that the phenomenon has occurred while nothing actually happened. We thus define the false alarm probability (aka size, aka error probability of the $1^{\text{st}}$ kind) of ${D}$ as:

Among all possible decisions, the omniscient oracle ${D}^{*}\in\mathcal{M}\!\left(\{0,1\}\!\times\!\Omega,\big{\{}0,1\big{\}}\right)$ is defined for any pair $(\varepsilon,\omega)\in\big{\{}0,1\big{\}}\times\Omega$ by setting ${D}^{*}(\varepsilon,\omega)=\varepsilon$. Its probability of false alarm is $0$ and its probability of detection is $1$: $\mathbb{P}_{\textsc{fa}}\left[{{D}^{*}}\right]=0$ et $\mathbb{P}_{\textsc{det}}\left[{{D}^{*}}\right]=1$. This omniscient oracle has no practical interest since it knows $\varepsilon$. That’s not really fair! Since it is not possible in practice to guarantee a null false alarm probability, we focus on decisions whose false alarm probabilities are upper-bounded by a real number $\gamma\in(0,1)$ called level. We state the following definition.

We can easily prove the existence of an infinite number of elements in $\mathrm{Dec}_{\gamma}$ that all have a detection probability equal to $1$. Whence the following definition.

Oracles with level $\gamma$ have no practical interest either since they require prior knowledge of $\varepsilon$! Therefore, we restrict our attention to decisions in $\mathrm{Dec}_{\gamma}$ that "approximate" at best the oracles with level $\gamma$, without prior knowledge of $\varepsilon$, of course. To this end, we must preorder decisions.

In practice, observations help us decide whether the phenomenon has occurred or not. By collecting a certain number of them, we can expect to make a decision. Hereafter, observations are assumed to be elements of $\mathcal{M}\left(\{0,1\}\times\Omega,\mathbb{R}\right)$ and corrupted versions of $\varepsilon$. We suppose that we have a sequence $(Y_{\_}n)_{\_}{n\in\mathbb{N}}$ of such random variables. As a first standard model, we could assume that, for any $n\in\mathbb{N}$ and any $(\varepsilon,\omega)\in\{0,1\}\times\Omega$, $Y_{\_}n(\varepsilon,\omega)=\varepsilon+X_{\_}n(\omega)$ with $X_{\_}1,X_{\_}2,\ldots,X_{\_}n,\ldots\stackrel{{\scriptstyle\text{iid}}}{{\thicksim}}\mathcal{N}(0,1)$. In this additive model, $X_{\_}n$ models noise on the $n$th observation. We could make this model more complicated and realistic by considering random vectors instead of variables. However, with respect to our purpose, the significant improvement we can bring to the model is elsewhere. Indeed, we have assumed above that the signal, regardless of noise, is $\varepsilon$. However, from a practical point of view, it is more realistic to assume that the $n$th observation $Y_{\_}n$ captures $\varepsilon$ in presence of some interference $\Delta_{\_}n$, independent of $X_{\_}n$. In practice, the probability distribution of $\Delta_{\_}n$ will hardly be known and, as a means to compensate for this lack of knowledge, we assume the existence of a uniform bound on the amplitude of all possible interferences. Therefore, we assume that, for all $(\varepsilon,\omega)\in\{0,1\}\times\Omega$, $Y_{\_}n(\varepsilon,\omega)=\varepsilon+X_{\_}n(\omega)+\Delta_{\_}n(\omega)$ and the existence of $q\in[0,\infty)$ such that $\Delta_{\_}n\in\mathcal{B}_{\infty}(q)$. After all, this model is standard in time series analysis: $\varepsilon$ plays the role of a trend, $\Delta_{\_}n$ is the seasonal variation and $X_{\_}n$ is the measurement noise.

For each $q\in[0,\infty)$, $\textup{Seq}_{\,q}$ henceforth designates the set of all the sequences:

such that, $\forall n\in\mathbb{N}$ and $\forall(\varepsilon,\omega)\in\{0,1\}\times\Omega$, $Y_{\_}n(\varepsilon,\omega)=\varepsilon+\Delta_{\_}n(\omega)+X_{\_}n(\omega)$, where $\Delta_{\_}n\in\mathcal{B}_{\infty}(q)$ and $X_{\_}n\thicksim\mathcal{N}(0,1)$ are independant. Therefore, for all $n\in\mathbb{N}$ and all $\varepsilon\in\{0,1\}$, $Y_{\_}n(\varepsilon)=\varepsilon+\Delta_{\_}n+X_{\_}n$, with $X_{\_}1,X_{\_}2,\ldots,X_{\_}n,\ldots\stackrel{{\scriptstyle\text{iid}}}{{\thicksim}}\mathcal{N}(0,1)$.

In this respect, suppose that $Z=\Theta+W\in\mathcal{M}(\Omega,\mathbb{R}^{n})$, where $\Theta$ and $W$ are independent elements of $\mathcal{M}(\Omega,\mathbb{R}^{n})$. In the sequel, we assume that $W\thicksim\mathcal{N}(0,\mathbf{I}_{n})$, $\mathbf{I}_{n}$ being the $n\times n$ identity matrix, and consider the mean testing problem of deciding on whether $|\langle{\Theta}\rangle_{n}(\omega)|\leqslant\tau$ (null hypothesis $\mathcal{H}_{\_}0$) or $|\langle{\Theta}\rangle_{n}(\omega)|>\tau$ (alternative hypothesis $\mathcal{H}_{\_}1$), when we are given $Z(\omega)=\Theta(\omega)+W(\omega)$, for $\omega\in\Omega$. The idea is that $\Theta$ oscillates uncontrollably around $0$ and that only sufficient large deviations of the norm should be detected. This is a particular Block-RDT problem, following the terminology and definition given in [11]. This problem is summarized by dropping $\omega$, as usual, and writing:

Standard likelihood theory [9, 2, 3] does not make it possible to solve this problem. Fortunately, this problem can be solved as follows via the Random Distortion Testing (RDT) framework.
Size and power of tests for mean testing. We seek tests with guaranteed size and optimal power, in the sense specified below.

Set $\mathcal{S}=\big{\{}\textrm{id},-\textrm{id}\big{\}}$ where id is the identity of $\mathbb{R}$. Endowed with the usual composition law $\circ$ of functions, $(\mathcal{S},\circ)$ is a group. Let $\mathcal{A}$ be the group action that associates to each given ${s}\in\mathcal{S}$ the map $\mathcal{A}_{\_}{s}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ defined for every $x=(x_{\_}1,x_{\_}2,\ldots,x_{\_}n)\in\mathbb{R}^{n}$ by $\mathcal{A}_{\_}{s}(x)=({s}(x_{\_}1),{s}(x_{\_}2),\ldots,{s}(x_{\_}n)).$ Readily, the mean testing problem is invariant under the action of $\mathcal{A}$ in that $\mathcal{A}_{\_}{s}(Z)=\mathcal{A}_{\_}{s}(\Theta)+W^{\prime}$ where $W^{\prime}=(W^{\prime}_{\_}1,W^{\prime}_{\_}2,\ldots,W^{\prime}_{\_}n)\thicksim\mathcal{N}(0,\mathbf{I}_{\_}n)$ is independent of $\mathcal{A}_{\_}{s}(\Theta)$. Therefore, $\mathcal{A}_{\_}{s}(Z)$ satisfies the same hypotheses as $Z$. We also have $|\langle{\mathcal{A}_{\_}{s}(\Theta)}\rangle_{n}|=|\langle{\Theta}\rangle_{n}|$. Hence, the mean testing problem remains unchanged by substituting $\mathcal{A}_{\_}{s}(\Theta)$ for $\Theta$ and $W^{\prime}$ for $W$. It is thus natural to seek $\mathcal{A}$-invariant tests, that is, tests $f\in\mathcal{M}\left(\mathbb{R}^{n},\{0,1\}\right)$ such that $f(\mathcal{A}_{\_}{s}(x))=f(x)$ for any ${s}\in\mathcal{S}$ and any $x\in\mathbb{R}^{n}$.

On the other hand, since we can reduce the noise variance by averaging observations, we consider $\mathcal{A}$-invariant integrator tests, that is, $\mathcal{A}$-invariant tests $f\in\mathcal{M}\left(\mathbb{R}^{n},\{0,1\}\right)$ for which exists $\overline{f}\in\mathcal{M}\left(\mathbb{R}^{1},\{0,1\}\right)$, henceforth called the reduced form of $f$, such that $f(\bm{x})=\overline{f}(\langle{\bm{x}}\rangle_{n})$ for any $\bm{x}\in\mathbb{R}^{n}$. Reduced forms of $\mathcal{A}$-invariant integrator tests are also $\mathcal{A}$-invariant: $\forall x\in\mathbb{R}$, $\forall{s}\in\mathcal{A}$, $\overline{f}({s}(x))=\overline{f}(x)$. We thus define $\mathfrak{C}^{[n]}_{\gamma}\subset\mathrm{Tests}^{[{n}]}_{\gamma}$ as the class of all $\mathcal{A}$-invariant integrator tests with level $\gamma$. We thus have $f\in\mathfrak{C}^{[n]}_{\gamma}$ if:
[Size]: $\alpha^{[n]}(f)\leqslant\gamma$;
[$\mathcal{A}$-invariance]: $\forall\,({s},x)\in\mathcal{S}\times\mathbb{R}^{n}$, $f\left(\mathcal{A}_{\_}{s}(x)\right)=f(x)$;
[Integration]: $\exists\,\overline{f}\in\mathcal{M}\left(\mathbb{R}^{1},\{0,1\}\right)$, $\forall\,x\in\mathbb{R}^{n}$, $f(x)=\overline{f}(\langle{x}\rangle_{n})$.
The following result derives from the foregoing and [10, 11].