Series on Max Fierro

Poem 17. "Presence"

Fri, 20 Mar 2026 00:00:00 +0000

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 Everywhere,
your presence is replicated
in iridescent bubbles, floating
 full of reflections:

 I find you in explorations
 of the frontier, lost incursions
 of language into concepts.

 Those thoughts
 form a massive tree:
 from its sprouting branches bloom
 flowers of hands whose fingers unfold,
 touching you for the first time once again:
 exploring your surface
 scattered across the world of ideas,
 in textures of endless definition:
 with ten thousand hands I hold you,
 I spill my consciousness thin
 over your infinite face,
 iridescing in the sun:

 when you look at me
 I dissolve in the intensity
 of a single word:

 my world is in your eyes,
 so full of reflections.

 Always,
 time flows towards you until it reaches you:
 a pool of water at your feet
 will soon be an ocean of time.

 When I am away from you,
 the fruit of the tree slowly
 loses its vibrant signal,
 carried in tears of time
 running down my face:

 so I fight for you,
 my senses heighten to extremes
 forging thought into your form,
 to arrive at you in my mind,
 to be with you for an instant
 that pops like a bubble:
 leaving behind images of you
 that reflect over the water:
 they echo in my mind:
 resonating signals,
 compounding beauty.

 Time will flow until I find you
 in the next instant, once again:
 together eternally,
 floating tenacious.

 Everywhere and always,
I am completely in love with you,
 Julia.

Poem 16. "Complete"

Mon, 12 Jan 2026 00:00:00 +0000

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turbulent flames over the water,
soothing stillness hugs me fixed-
come evening, you dress me
in ever more radiant skin,
 now warm evermore:
of all the earthly pillars built from rock,
infinite is the throne in which we now lay;
 high
waterfall of light, one endless flow,
it washes through insufficient hands:
 always insufficient
 yet now complete
 and forever more:
we phosphoresce in our presence,
 glowing evermore:
everything shines over the water-
set ablaze and held calmly still:

Poem 15. "Harmonic"

Wed, 07 Jan 2026 00:00:00 +0000

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rivers of ice, pillars of glass,
like water, light runs through crystal;
thoughts travel inside your forehead:
in my mind, to think is now to reason
the precious movements of your will:
across the earth
 glacial penmanship,
 beautiful writing

springs of magma, columns of light,
deep in those channels, molten flows;
warm blood filters through your dermis:
on my chest, to feel is now to witness
waves of sun washing over your skin:
across the earth
 volcanic flows,
 solar embrace

Poem 14. "Temescal"

Tue, 30 Dec 2025 00:00:00 +0000

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i enter and hear hopeful tones
from aspects of you composed;
with restraint chosen out of many,
allowed to exist by your hands

in another room of your house
your words become tulip petals;
lightly into the air you pronounce
your voice in a violet spectrum

outside, the clouds always wander;
in an ephemeral december window,
your image trembles through water
of raindrops chosen by your words

i don't spill these thoughts, i hold
a sunday evening in crystal fixed
refracting your luminous warmth:
an original gift of your goodness

Poem 13. "Carlos"

Sun, 28 Dec 2025 00:00:00 +0000

Dedicated to my dear friend, Carlos Rodríguez.

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piedra angular sin esquinas
tu videncia es abundante
 y de color azul:

en tierras lejanas desmigajaste
la esencia de un reino podrido;
 eres cierto, tenaz,
 y súmamente nítido:

a cuero y tezontle, vital como la ceiba
 hundes raíces por donde vas pisando:

tuerces los manantiales enterrados
 de agua que mana por tu mente,
 y que es de color azul:

contra el horizonte de tu luz admiro
 el fruto esperanzado de tus manos:

Poem 12. "Templo"

Sat, 20 Dec 2025 00:00:00 +0000

I was thinking about the nature of art, and concluded that my artistic vision with regards to any particular medium should stem from a single total transmedial aesthetic that is consistent with and inspired by my values. I wanted to write something that would prove that, if abstraction is morally accepted (which I personally do), such a transmedial aesthetic could be allowed to live in full breadth even in single-medium works. My success is unkown, as usual.

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veo las columnas de un sonido diáfano
instituyéndose al profundo respirar
 del órgano:

desde los balcones de mi mente,
veo el espectro de tu voz
de innumerables tonos hecho:

veo los arcos, arcadas sobrepuestas,
componiéndose con cada palabra escrita
 por mi mano:

desde las palabras con mi puño hechas
veo la presencia de tu nombre
reverberando por este templo:

Poem 11. "Tabachín"

Fri, 19 Dec 2025 00:00:00 +0000

Dedicated to a person-like concept I considered while meditating as a recipient to feelings of love.

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tu piel se irisaba en la luz
y el aire trepidaba al tocarte,
todo se reflejaba en tus ojos
que surtía el sol de imágenes:
entre las flamas del horizonte,
te vi sentada bajo un tabachín.

Poem 10. "Zopilotes"

Sun, 16 Nov 2025 00:00:00 +0000

I am on the more depressive side of personhood.

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 días de nada,
semanas perdidas
como perros en la calle:
 los relojes del invierno
 son noches sin cuerda,
 estos días
 estoy hecho de aire, o de polvo:
 mis palabras se vuelan
 y todo lo que pienso
 se vuelve pétalos de loto,
 escuelas de peces traslúcidos
 que pronto se desvanecen:
 voy absorto por el mundo
 que me da vértigo la nada,
 entonces pienso
 para tener mi ración
 de paraíso terrestre,
 que aquellos zopilotes
 van bajando del cielo:
 buscan horas huecas,
 comen huesos vacíos:
 desperdicios
 días de nada,

Poem 9. "Sugerencias"

Mon, 13 Oct 2025 00:00:00 +0000

Mostly nonsense.

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son soledades entre columnas de piedra
vacíos entre las personas del mundo
las cumbres de la tierra, sus árboles
venas de fuego que mueren en el mar
un silencio eterno en la cima del cielo
pensamientos perdidos bajo palabras
instantes que tocan la piel del futuro
otras manos, sentidas por última vez
las nubes se desmoronan y son lluvia
espacios vivos en volúmenes sin nada
bosques de noche, un verde sin ocaso
la luz de una estrella que ya no existe
verdades absolutas entre los detalles
las miradas y su amor incandescente
desvíos de vida, la soledad inminente
ley del sol en el desierto, y un charco
libre misericordia de un océano cruel
el fruto sanguíneo de labios efímeros
sueño fugaz, los ríos ya desembocan
las hojas mojadas se surten de brillo
el suelo se nutre y las raíces respiran
la gente vive y se abraza con fuerza
ceñido, todo está aquí y es completo
de día se siente en todos lados calor
ya con compañía, fosforece la noche
nuestra piel está cubierta de energía
somos indistinguibles entre nosotros

Poem 8. "Cáliz"

Sun, 21 Sep 2025 00:00:00 +0000

Dedicated to a person-like concept I considered while meditating as a recipient to feelings of loss and affection.

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te recuerdo en mi sangre,
calidez que fluye por mi cuello
 derriba mi cabeza;
te recuerdo en mis lágrimas,
 cierro mis ojos, y te veo
 entre formas familiares;
siento que te abrazo,
 existes entre mis brazos
 abrazándose a sí mismos,
 siento
 que te cubro de lágrimas,
 siento la calidez
 de tu sangre;
en mi mente
 te recojo y te defiendo
 del tiempo,
 te junto en un cáliz
 hecho con mis manos,
 que poco a poco
 te derrama:

Poem 7. "Absoluto"

Mon, 15 Sep 2025 00:00:00 +0000

Thoughts after the night sky of Joshua Tree National Park, which I visited with my sister Mara.

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desde ya distantes astros
por un volumen transparente
 se derrama
 un cuerpo que mi visión sostiene;
mi mirada atraviesa el cielo
 y se filtra por un medio invisible;
se somete a su estructura,
 y se concreta,
 se cristaliza:

 si lo busco,
todo existe en ese cuerpo;
 cada cosa
 es una de sus caras,
 cada perspectiva
 es una de sus esquinas,
 y dentro de su totalidad
se encuentra a si misma
 mi mirada:

ser que se yergue
 entre patrones que yo elijo,
 tu cara lo es todo
 y existe en la nada,
 tu mirada me ve
 y me concreta,
 me cristaliza:

Poem 6. "Lena"

Mon, 08 Sep 2025 00:00:00 +0000

Dedicated to Lena, who passed away of old age. (I love you, darling.)

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entrevista criatura echada en la baldosa,
la luz penetraba el follaje
 del helecho donde te resguardabas:
en ese entonces, los dos éramos niños y
 nos ceñía la juventud de nuestra familia,
ese hogar llevaba tu color,
 y en ese jardín
 nos vestía el espacio de una piel
 más radiante y transparente

criatura entrevista entre las décadas,
 tiempo tenaz que fluye a mi costado:
 todavía soy un niño, pero sin ti
 ya no soy joven,
 mi familia ya no es joven
 ahora que tú no estás,
 te resguardo dentro de mí,
 y en ti siempre se encontrarán
 aquellas joyas de nuestra juventud
 dentro de las cuales centellea
la luz que compartimos

Poem 5. "Zoe"

Mon, 28 Jul 2025 00:00:00 +0000

Dedicated to Zoe, with whom I shared a relationship.

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caudaloso río de la memoria;
sacrificio, derrota clavada,
memoria total, pensamiento
que me abruma,
 te vas
para regresar una vez más,
siempre una vez más, nunca
te vas,
 pero tampoco llegas;
ya te fuiste, y aún sigues
aquí, sigues en el instante,
la hora te produce siempre,
ya no sueño, recordar tu luz
me despierta, me carcome,
lo pagué todo para pensarte,
símbolo del olvido,
 cara
atorada siempre en el pasado,
sólo te pienso en el pasado,
palabras que escribo pensando
en el pasado;
 ahí perteneces,
que el instante es una nada;
el futuro toca ya la puerta
de mi alma, y no lo escucho;
 silencio que me abruma,
pensamientos que me gritan,
 memorias que me ahogan
en su caudal, río perdido
 que te sigue buscando 

Poem 4. "Consuelo"

Sun, 27 Jul 2025 00:00:00 +0000

Words given to me by my mother over the phone.

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Ay, mijo. Que te bendiga Dios.
No eres de ule ni de fierro.
Te amo con todo mi ser.
Eres mi corazón.

Poem 3. "Lucy"

Sat, 12 Jul 2025 00:00:00 +0000

Dedicated to my dear friend, Lucy Revina.

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Marsupial madre de un amor tenaz
que se resguarda entre mis costillas;
me llena de un sentimiento líquido
que se construye al cerrar mis ojos.

Llueves gotas de ideas inmarcesibles.
Toda la noche llueves sobre mi rostro;
gotas de lluvia que veo ya sobre mi piel
son transparentes, por tu transparencia.

Cómo te extraño, querida amiga;
que convertiste nuestra vivienda en hogar,
y cuidas siempre de mí, así cómo yo de ti.
Pronto nos veremos otra vez.

Poem 2. "Occidental"

Wed, 30 Apr 2025 00:00:00 +0000

Failed attempt at processing a concept that I still can’t quite explain.

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Son trayectorias que trazan entre los átomos,
son curvas que desnudan misterios paralelos.
Es hipótesis de tiempo escrita en el espacio,
información que se filtra entre la materia.

The Elo Rating System through Likelihood Gradient Ascent

Wed, 30 Apr 2025 00:00:00 +0000

Abstract

Probability and optimization are strong monsters. The Elo rating system, used to estimate performance in competitive chess, online dating, and AI agents, is an under-the-hood reminder of this fact that operates within many of the systems that need to establish comparative metrics. This piece is my contribution to the endless pile of explainers on the topic. I exercise an emphasis on bayesian statistics and optimization that should ring a bell for anyone familiar with the basics of machine learning.

Background

Mathematical Orderings

At the risk of including a needless dependency on the topic of this piece, I introduce you to the idea of an ordering. Colloquially, we take this to mean an arrangement (i.e., a permutation) of a set of things. We will replace that with its formal meaning, which is a specific kind of binary relation.

Definition

A binary relation $R$ from a set $X$ to another $Y$ is a subset of $X \times Y$, where it is possible that $X = Y$.

This should seem odd, as a subset is in no obvious way reminiscent of a permutation. But introducing some new syntax to indicate membership in a relation,

$$ (x, y) \in R \vdash xRy, $$

we are an example away from making sense. In particular, consider $R = \, \leq$ (less-than). When we say things like “$x \leq y$,” we are in fact using syntactic sugar for “$(x, y) \in \, \leq$.” With this in mind, we can take a look at partial orders.

Definition

A partial order $R$ is a binary relation over a set $X$ and itself which satisfies the following:

Reflexivity. This means that $\forall x \in X, \, xRx$.
Antisymmetry. This means that $\forall (x, y) \in X^2, \, xRy \wedge yRx \implies x = y$.
Transitivity. This means that $\forall (x, y, z) \in X^3, \, xRy \wedge yRz \implies xRz$.

The canonical example of a partial order is $\subseteq$ over $\mathcal P(S)$. Importantly, a partial order over a set does not imply a permutation over it, because of the possibility for two elements $x$ and $y$ to be unrelated, or in other words, for $\neg xRy$ and $\neg yRx$. In a total order, we simply do not allow this.

Definition

A total order $R$ is a partial order that is also total, which means that $\forall (x, y) \in X^2, \, xRy \vee yRx$.

The canonical example of a total order is $\leq$ over $\mathbb{R}$. With a total order, there is a single valid ordering $\bold{x}$ (i.e. arrangement or permutation) over its set $X$ such that $x_iRx_{i + 1}$ for all $i = 0, \ldots, |X| - 1$. One more variation we can make on the idea of an order is that of a weak order.

Definition

A weak order $R$ is a total order that is not necessarily antisymmetric.

In other words, it is possible that for distinct elements $x$ and $y$, both $xRy$ and $yRx$. This is conceptually aligned with allowing “ties” in any resulting ordering, potentially sacrificing their uniqueness.

Elo Ratings and Updates

We can take a look at the question that Arpad Elo (kind of) answered: How can you compare the skill level of two chess players?

Arpad Elo (August 25, 1903 – November 5, 1992)

His proposed procedure is straightforward. Each player $i \in N$ will have a real-valued rating $r_i$, which will be a proxy for their skill level. These ratings will be initialized at some predetermined value for all players. Then, when there is a match between player $i$ and $j$, the following updates are made:

$$ \begin{align*} &r_j \gets r_j + k(s_j - e_j), \\ &r_i \gets r_i + k(s_i - e_i), \end{align*} $$

where¹,

$$ e_p = \frac{1}{1 + e^{-(r_p - r_{\text{other}})}}, \;\;\;\; s_p = \begin{cases} 1 &\text{if $p$ wins},\\ 0.5 &\text{if draw},\\ 0 &\text{if $p$ loses},\\ \end{cases} $$

and $k$ is a constant chosen arbitrarily. So, as players accrue matches with other players, their ratings are updated according to the above rules with the hope that they will eventually stabilize. Now, the difference between players’ ratings can be used to compare their skill levels via the ordering $\leq$ on $\mathbb{R}$.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a method used to fit distribution parameters to samples. The setup for MLE is a random variable $Y$ of known distribution $\mathcal{D_\theta}$ (parameterized by $\theta$), with access to IID samples $\langle y_i \rangle \sim \mathcal{D}_\theta$. The objective is to estimate $\hat\theta$ such that the likelihood function $\mathcal{L}$ is maximized,

$$ \begin{equation} \hat\theta_\mathrm{MLE} = \argmax_\theta \, \mathcal{L}(\theta; \langle y_i \rangle) = \argmax_\theta \, \prod_i \mathbb{P}_\theta[Y = y_i]. \end{equation} $$

In other words, MLE is the optimization procedure associated with finding the distribution parameters that were most likely to generate observed data, provided that we know or assume its distribution.

MAP Estimation

When there is access to a (known or assumed) prior $p(\theta)$ on the distribution of parameters, we can fold it into our optimization process by doing MLE on the posterior distribution, which by Bayes’ theorem,

$$ \begin{align*} p(\theta \mid \langle y_i\rangle) &= \frac{p(\langle y_i \rangle \mid \theta) p(\theta)}{p(\langle y_i \rangle)} \; \propto \; \underbrace{p(\langle y_i \rangle \mid \theta)}_{\displaystyle{\mathcal{L}(\theta; \langle y_i \rangle)}} p(\theta). \end{align*} $$

The resulting parameters $\hat\theta_\mathrm{MAP}$ are then a maximum a posteriori (MAP) estimate,

$$ \begin{align*} \hat\theta_{\mathrm{MAP}} &= \argmax_\theta\;p\bigl(\theta \mid \langle y_i\rangle\bigr) \\ &= \argmax_\theta\;\Bigl[\mathcal{L}\bigl(\theta;\langle y_i\rangle\bigr)\,p(\theta)\Bigr] \\ &= \argmax_\theta\;\Bigl[\prod_{i=1}^n \mathbb{P}_\theta\bigl[Y=y_i\bigr] \times p(\theta)\Bigr]. \end{align*} $$

Optimization

Sometimes, it is possible to find closed-form solutions for $\hat\theta_\mathrm{MAP}$ and $\hat\theta_\mathrm{MLE}$ through convex optimization. For example, samples with gaussian noise lead to the closed-form solution of the OLS problem through the process of MLE.

However, most of the time the resulting optimization objective of MLE (and hence also MAP estimation) is not convex. Here, gradient-based approaches (along with all other non-convex optimization techniques) are helpful for finding local maxima of the likelihood objective.

Derivation

The derivation presented here will depart from the usual in hopes of contributing some kind of novelty. We begin with the game-theory-native idea of payoff, which we will take to be a numeric value representing a player’s utility differential with respect to the start of a game $G$,

$$ \text{payoff of player i} = p_i. $$

Next, we will consider player performance. Just as Elo, we take the performance of a player $i$ on a game $G$ to be a real-valued random variable $X_i$, independent to other players.

Observation

Perhaps Elo motivated this decision after noticing the variance of his own performance over the chess board.

Then, we will expand our setup by allowing players to outperform others, which we will present through the difference between the performance of two players during $G$ (which is another RV),

$$ \delta_{i, \, j} = X_i - X_j. $$

We will also establish a relationship between $\delta_{i , \, j}$ and $p_i$. For this purpose, we introduce a game-specific mapping $g$ with a noise term $\epsilon \sim \mathcal{N}(0, \sigma_\epsilon^2)$, which together form the generative process of payoffs:

$$ p_i = g(\delta_{i, \, j}) + \epsilon. $$

Finally, we will assert a prior on the distribution of $X_i$, which we will refer to as $\mathcal{D}(\theta_i)$ without yet deciding on a particular distribution (just that it is parameterized by $\theta_i$). This prior $\pi(x)$ will be global for all players, and its distribution parameters will be $\theta_\pi$.

Note

While this is a global prior, notice that none of the following breaks if it were player-specific from the start.

So far, none of this has helped us answer the question that Elo answered. For that, we will introduce one last artifact on top of our setup; each player $i$ will have a “rating” $r_i$, which we will ultimately use to order players by skill in our system or organization:

$$ r_i = \mathbb{E}[X_i]. $$

MAP Estimation

Clearly, since our goal is to know players’ ratings, the only additional information we will need to get them are the distribution parameters $\theta_i$. Of course, at a lack of observations, we can assert from our prior

$$ r_i = \mathbb{E}_{X \, \sim \, \pi}[X]. $$

But what if at the end of a game $G$ between players $i$ and $j$, we observe WLOG the payoff $p_i$? Here, we will be wishing that $g$ is neatly invertible. Assuming it is, we arrive at the following MLE for their difference in performance via application of $(1)$:

$$ \begin{equation} \hat\delta_{i, \, j} = \argmax_{\delta} \exp\!\Bigl(-\frac{(p_i-g(\delta))^2}{2\sigma_\epsilon^2}\Bigr) = g^{-1}(p_i). \end{equation} $$

Note

Notice we did not use a prior when estimating $\hat\delta_{i, \, j}$. This assumption is due to Elo; we will not use players’ history when calculating their performance for a single game. This is the design decision that, by omission, accounts for sudden changes in player skill (as a result of learning, etc.).

Note

We just estimated the difference between the performance of the players from the payoff of a single player. The invertibility of $g$ has the hidden implication that it is strictly monotonic; no two differences in performance lead to the same payoff, and the greater the difference, the greater the payoff for the outperforming player.

Knowing this, we can perform a bayesian update to our prior through MAP estimation. Writing down the joint posterior of the parameters of $X_i$ and $X_j$,

$$ \begin{equation} p(\theta_i, \, \theta_j \mid \hat\delta_{i, \, j}) \propto \underbrace{ p(\hat\delta_{i, \, j} \mid \theta_i, \, \theta_j) }_{ \displaystyle{p(\hat\delta_{i, \, j} \mid X_i - X_j)} } p(\theta_i)p(\theta_j). \end{equation} $$

Notice that we already have access to priors $p(\theta_i)$ and $p(\theta_j)$; those are quite simply $\pi(\theta_i)$ and $\pi(\theta_j)$, which we assume per our initial setup.

Gaussian Performance

We proceed by considering the case where $X_i \sim \mathcal{N}(r_i, \, \sigma_i^2)$, such that $\theta_i = (r_i, \, \sigma_i^2)$. That is, player performance is gaussian-distributed,

$$ \begin{equation} p_{X_i}(x_i \mid \theta_i) = \frac{1}{\sqrt{2\pi} \, \sigma_i} \exp\!\Bigl(-\frac{(x_i - r_i)^2}{2\,\sigma_i^2}\Bigr). \end{equation} $$

Our next goal is to set up an analytic function for the likelihood $p(\hat\delta_{i, \, j} \mid \theta_i, \, \theta_j)$. We observe that we have access to the conditional density of $\hat\delta_{i, \, j}$

$$ \begin{equation} p(\hat\delta_{i, \, j} \mid x_i, \, x_j) = \frac{1}{\sqrt{2\pi}\,\sigma_\varepsilon} \exp\!\Bigl(-\frac{\bigl(g(\hat\delta_{i, \, j}) - (x_i - x_j)\bigr)^2}{2\,\sigma_\varepsilon^2}\Bigr) \;|g'(\hat\delta_{i, \, j})| \end{equation} $$

by using $(2)$ implicitly through the change of variables

$$ p(\hat\delta_{i, \, j} \mid x_i, \, x_j) = p_{p_i}\bigl(g(\hat\delta_{i, \, j}) \mid x_i, \, x_j \bigr) \;\Bigl|\frac{d}{d\hat\delta_{i, \, j}}\,g(\hat\delta_{i, \, j})\Bigr|, $$

where we take

$$ p\bigl(p_i \mid x_i, \, x_j \bigr) = \frac{1}{\sqrt{2\pi}\,\sigma_\varepsilon} \exp\!\Bigl(-\frac{\bigl(p_i - g(x_i - x_j)\bigr)^2}{2\,\sigma_\varepsilon^2}\Bigr). $$

Now, we can use $(4)$ and $(5)$ to derive the desired likelihood by marginalizing,

$$ \begin{align*} p\bigl(\hat\delta_{i, \, j} \mid \theta_i,\theta_j\bigr) &= \iint p\bigl(\hat\delta_{i, \, j}\mid x_i,x_j\bigr)\; p_{X_i}(x_i \mid \theta_i)\;p_{X_j}(x_j \mid \theta_j)\, dx_i\,dx_j \\ &= \iint \frac{1}{\sqrt{2\pi}\,\sigma_\varepsilon} \exp\!\Bigl(-\frac{\bigl(g(\hat\delta_{i, \, j}) - (x_i - x_j)\bigr)^2} {2\,\sigma_\varepsilon^2}\Bigr)\, \bigl|g'(\hat\delta_{i, \, j})\bigr| \\[-2pt] &\quad\;\times\, \frac{1}{\sqrt{2\pi}\,\sigma_i} \exp\!\Bigl(-\frac{(x_i - r_i)^2}{2\,\sigma_i^2}\Bigr) \;\frac{1}{\sqrt{2\pi}\,\sigma_j} \exp\!\Bigl(-\frac{(x_j - r_j)^2}{2\,\sigma_j^2}\Bigr) \,dx_i\,dx_j. \\[6pt] \end{align*} $$

After another cup of coffee, we arrive at the following version of our joint likelihood $p\bigl(\hat\delta_{i, , j} \mid \theta_i,\theta_j\bigr)$,

$$ \begin{aligned} &= \int \frac{1}{\sqrt{2\pi}\,\sigma_\varepsilon} \exp\!\Bigl(-\frac{\bigl(g(\hat\delta_{i, \, j}) - d\bigr)^2}{2\,\sigma_\varepsilon^2}\Bigr) \;|g'(\hat\delta_{i, \, j})|\; \\ &\quad\;\quad\;\times \frac{1}{\sqrt{2\pi(\sigma_i^2+\sigma_j^2)}} \exp\!\Bigl(-\frac{(d - (r_i - r_j))^2}{2(\sigma_i^2+\sigma_j^2)}\Bigr) \,dd \end{aligned} $$

where $d = x_i - x_j$ (hinted at in equation $(3)$) is possible because $X_i - X_j \sim \mathcal{N}(r_i - r_j, \sigma_i^2 + \sigma_j^2)$. Finally, we obtain the following after remembering an important fact from signal processing,

$$ \begin{align} \mathcal{J}_\mathrm{MLE}(\theta_i, \, \theta_j; \, \hat\delta_{i, \, j}) = \frac{|g'(\hat\delta_{i, \, j})|}{\sqrt{2\pi\,\bigl(\sigma_\varepsilon^2 + \sigma_i^2 + \sigma_j^2\bigr)}}\, \exp\!\Biggl(-\frac{\bigl(g(\hat\delta_{i, \, j}) - (r_i - r_j)\bigr)^2} {2\,\bigl(\sigma_\varepsilon^2 + \sigma_i^2 + \sigma_j^2\bigr)}\Biggr). \end{align} $$

Wonderful. We then attend to the reflexes drilled into our brains from machine learning, and find the gradient of the log-likelihood with respect to learned… ahem, the ratings $\bold{r} = [r_i, \, r_j]^\top$:

$$ \begin{equation} \nabla_\bold{r}\log\mathcal{J}_\mathrm{MLE}(\theta_i, \, \theta_j; \, \hat\delta_{i, \, j}) = \begin{bmatrix} \displaystyle\frac{g(\hat\delta_{i, \, j}) - (r_i - r_j)}{\sigma_\varepsilon^2 + \sigma_i^2 + \sigma_j^2} \\[8pt] \\ \displaystyle-\frac{g(\hat\delta_{i, \, j}) - (r_i - r_j)}{\sigma_\varepsilon^2 + \sigma_i^2 + \sigma_j^2} \end{bmatrix}. \end{equation} $$

Note

By taking the $\log$ of the joint likelihood we achieve nothing, but we respect a very important tradition².

Using $\nabla_\bold{r}\log\mathcal{J}(\theta_i, \, \theta_j; \, \hat\delta_{i, \, j})$ as it stands to adjust $\bold{r}$ would be tantamount to MLE on $\bold{r}$. To turn this into a proper MAP estimate we must also fold in our prior terms into $(6)$, which we assume to be gaussian:

$$ \begin{aligned} \mathcal{J}_\mathrm{MAP}(\theta_i, \, \theta_j; \, \hat\delta_{i, \, j}) &= \frac{\lvert g'(\hat\delta_{i,j})\rvert} {\sqrt{2\pi\,\bigl(\sigma_\varepsilon^2 + \sigma_i^2 + \sigma_j^2\bigr)}} \exp\!\Bigl(-\frac{\bigl(g(\hat\delta_{i, \, j}) - (r_i - r_j)\bigr)^2} {2\,\bigl(\sigma_\varepsilon^2 + \sigma_i^2 + \sigma_j^2\bigr)}\Bigr)\\ &\quad\;\times\; \frac{1}{\sqrt{2\pi}\,\sigma_\pi} \exp\!\Bigl(-\frac{(r_i - r_\pi)^2}{2\,\sigma_\pi^2}\Bigr) \;\times\; \frac{1}{\sqrt{2\pi}\,\sigma_\pi} \exp\!\Bigl(-\frac{(r_j - r_\pi)^2}{2\,\sigma_\pi^2}\Bigr). \end{aligned} $$

Being again unable to ignore our instincts,

$$ \begin{equation} \nabla_{\mathbf r}\log \mathcal{J}_{\mathrm{MAP}}(\theta_i,\theta_j;\hat\delta_{i, \, j}) = \begin{bmatrix} \displaystyle \frac{g(\hat\delta_{i, \, j}) - (r_i - r_j)}{\sigma_\varepsilon^2 + \sigma_i^2 + \sigma_j^2} \\\\ \displaystyle -\frac{g(\hat\delta_{i, \, j}) - (r_i - r_j)}{\sigma_\varepsilon^2 + \sigma_i^2 + \sigma_j^2} \end{bmatrix} - \begin{bmatrix} \displaystyle \frac{r_i - r_\pi}{\sigma_\pi^2} \\\\ \displaystyle \frac{r_j - r_\pi}{\sigma_\pi^2} \end{bmatrix}. \end{equation} $$

Checkpoint

Let us take a step back for a second, and roughly see what is on the table. Intuitively, we are:

Observing a materialized payoff $p_i$.
Inverting $g$ to recover the latent skill gap $\hat\delta_{i, \, j}$ that was most likely to produce $p_i$.
Comparing that inferred gap to our current belief of the skill gap $r_i - r_j$.
Deriving the change to $r_i$ and $r_j$ would bring our belief closer to $\hat\delta_{i, \, j}$.

Then, the gradient-ascent update with step size $k$,

$$ \begin{equation} \bold{r}_{t + 1} \gets \bold{r}_{t} + k\nabla_{\bold{r}}\log\mathcal{J}_\mathrm{MAP}(\theta_i, \, \theta_j; \, \hat\delta_{i, \, j}), \end{equation} $$

offers a complete recovery (and generalization) of the Elo update after an observed payoff $p_i$.

Discussion

Procedural Discrepancy

Usually, implementations of Elo updates do not consider a prior. Instead, they simply initialize parameters at some default amount, then do MLE (as opposed to MAP estimation) to produce gradient updates. I decided to display the full MAP estimate because I think it is more principled; if you believe that ratings “start off” at some amount, that constitutes a bayesian prior in my eyes.

Distribution Discrepancy

The Elo rating system assumes a logistic distribution on player performance, not gaussian. However, the above procedure will invariantly recover Elo updates as presented in the background section with both distributions (at least in form). I thought it would be somewhat interesting to make it gaussian.

Fixed Parameters

In theory, one could estimate the variance parameters using the exact same procedure, by taking the gradient of the joint likelihood with respect to them in addition to the means (the ratings). Surprisingly, people do things similar to this – although not in this particular way. See the Glicko rating system.

Redundancy with $g$

You may have noticed that throughout our derivations (most notably in equations $(7)$ and $(8)$) there are $g(\hat\delta_{i, \, j})$ terms that can be safely replaced with $p_i$ by definition, and can be therefore seen as redundant. This is a completely accurate observation.

I decided to make $g$ explicit to make the fundamental link between payoffs and performance differentials also explicit, which is something I consider to be a lot more principled. In fact, $g$ does not need to be strictly monotonic, as we never explicitly evaluate $g^{-1}(\small\bullet)$. However, not satisfying this property may result in a lack of parameter identifiability, which is easy to forget if you discard the symbol early on.

Weak Ordering

It is important to acknowledge that mapping player skill to $\mathbb{R}$ and then using $\leq$ to order players is a fundamentally misguided approach to how the world works. In doing so, we establish a weak ordering among players, but completely ignore that some players have qualities that make them strong against some players and weak against others (in a manner that is potentially cyclic).

Example

To illustrate this, consider three players of rock-paper-scissors. One always plays rock, one always plays paper, and the other scissors. You will find that there is no way of assigning them a real number such that the player with the highest number beats both of the other players in expectation.

Still, sometimes we are forced to make rankings which make sense in expectation. In the real world, there is sufficient variance in player attributes that there are actors that can consistently beat some others. Here, systems such as Elo’s bring real utility. But as a human, you should trust your intuition more than some potentially senseless number.

Outcome Prediciton

Further expanding on the inadequacy of ranking players via weak order, consider the very plausible machine learning task of outcome prediction, say, for the game of Basketball.

It is tempting to, for example, train a network $f_\theta : \mathbb{R}^n \to \mathbb{R}$ on $n$-dimensional encodings of teams to predict a scalar value, where you then train in tandem over historic game outcomes $\langle ((a, b), \, y)_i \rangle_{i \in D}$.

Here, teams $a, \, b \in \mathbb{R}^n$ played each other and achieved outcome $y \in \{-1, 1\}$ for each match $i \in D$. One could optimize under the following loss,

$$ \mathcal{L}((a, \, b), \, y; \, \theta) = \log\exp\bigl( 1 + y(f_\theta(a) - f_\theta(b))\bigr) + \lambda(f_\theta(a) + f_\theta(b)), $$

where the regularization term helps with stability. Then, $f_\theta$ would essentially become a rating estimator. Whoever does this, however, will have the same fundamental problem as the Elo system; a weak order cannot capture the potentialy cyclic structure of actors’ dominance on each other.

The solution, of course, is to instead train another model $f_\theta : \mathbb{R}^{2n} \to \mathbb{R}$ that admits pairings as an input via concatentaion, and implements typical binary cross-entropy loss:

$$ \mathcal{L}((a, \, b), \, y; \, \theta) = -\bold{I}_y\,\log\bigl(\sigma(f_\theta(a \Vert b))\bigr) - (1 - \bold{I}_y)\,\log\bigl(1-\sigma(f_\theta(a \Vert b))\bigr). $$

However, there is no free lunch – when training over pairs of teams in $T$, the sample space of the task grows with the size of $T \times T$, naturally increasing the amount of out-of-distribution data for your model quadratically. Of course, this problem was ignored by the first formulation too, just in a different way.

Appendix

Gaussian Convolution

Here, I justify equation $(6)$ by instantiating a proof of the fact that the convolution of two gaussians is another gaussian determined by the parameters of the original gaussians.

Proof

This was made via ChatGPT with o4-mini-high and adjusted by me, because you can probably find it in a textbook somewhere. Let

$$ f(d)=\frac{1}{\sqrt{2\pi}\,\sigma_1}\exp\!\Bigl(-\frac{(d-\mu_1)^2}{2\,\sigma_1^2}\Bigr), \quad g(d)=\frac{1}{\sqrt{2\pi}\,\sigma_2}\exp\!\Bigl(-\frac{(d-\mu_2)^2}{2\,\sigma_2^2}\Bigr). $$

We wish to show

$$ \int_{-\infty}^{\infty} f(d)\,g(d)\,dd =\frac{1}{\sqrt{2\pi\,(\sigma_1^2+\sigma_2^2)}}\, \exp\!\Bigl(-\frac{(\mu_1-\mu_2)^2}{2\,(\sigma_1^2+\sigma_2^2)}\Bigr). $$

Set $A=\sigma_1^2$ and $B=\sigma_2^2$. Then,

$$ f(d)\,g(d) =\frac{1}{2\pi\sqrt{AB}} \exp\!\Bigl(-\tfrac12\bigl[\tfrac{(d-\mu_1)^2}{A}+\tfrac{(d-\mu_2)^2}{B}\bigr]\Bigr). $$

Combine quadratic terms:

$$ B(d-\mu_1)^2 + A(d-\mu_2)^2 =(A+B)\Bigl(d-\frac{B\mu_1 + A\mu_2}{A+B}\Bigr)^2 +\frac{AB}{A+B}(\mu_1-\mu_2)^2. $$

Define

$$ m=\frac{B\mu_1 + A\mu_2}{A+B}, \quad C=\frac{AB}{A+B}. $$

Then,

$$ \int f(d)\,g(d)\,dd =\frac{1}{2\pi\sqrt{AB}} \int \exp\!\Bigl(-\tfrac12\bigl[\tfrac{(d-m)^2}{C}+\tfrac{(\mu_1-\mu_2)^2}{A+B}\bigr]\Bigr) \,dd. $$

Factor out the constant term and use

$$ \int \exp\Bigl(-\frac{(d-m)^2}{2C}\Bigr) \, dd =\sqrt{2\pi\,C}. $$

Hence,

$$ \begin{aligned} \int f(d)\,g(d)\,dd &=\frac{\sqrt{2\pi\,C}}{2\pi\sqrt{AB}} \exp\!\Bigl(-\frac{(\mu_1-\mu_2)^2}{2\,(A+B)}\Bigr)\\ &=\frac{1}{\sqrt{2\pi\,(A+B)}} \exp\!\Bigl(-\frac{(\mu_1-\mu_2)^2}{2\,(A+B)}\Bigr). \quad \square \end{aligned} $$

Instantiation

Consider the expression which $(6)$ was derived from,

$$ \begin{aligned} &\int \frac{1}{\sqrt{2\pi}\,\sigma_\varepsilon} \exp\! \overbrace{ \Bigl(-\frac{\bigl(g(\hat\delta_{i, \, j}) - d\bigr)^2}{2\,\sigma_\varepsilon^2}\Bigr) }^{ \text{Quadratic term is symmetric.} } \;|g'(\hat\delta_{i, \, j})|\; \\ &\quad\;\quad\;\times \frac{1}{\sqrt{2\pi(\sigma_i^2+\sigma_j^2)}} \exp\!\Bigl(-\frac{(d - (r_i - r_j))^2}{2(\sigma_i^2+\sigma_j^2)}\Bigr) \,dd. \end{aligned} $$

Now, use the substitutions

$$ \mu_1 = g(\hat\delta_{i, \, j}), \quad \mu_2 = r_i - r_j, \quad A = \sigma_\varepsilon^2, \quad B = \sigma_i^2 + \sigma_j^2, $$

and re-attach the Jacobian factor $|g^\prime(\hat\delta_{i, \, j})|$ to recover

$$ \mathcal{J}(\theta_i,\theta_j;\hat\delta_{i, \, j}) =\frac{|g'(\hat\delta_{i, \, j})|} {\sqrt{2\pi\,(\sigma_\varepsilon^2 + \sigma_i^2 + \sigma_j^2)}}\, \exp\!\Bigl(-\frac{(g(\hat\delta_{i, \, j}) - (r_i - r_j))^2} {2\,(\sigma_\varepsilon^2 + \sigma_i^2 + \sigma_j^2)}\Bigr). $$

I modified a term in $e_p$ to exclude scaling factors, to make it look less crazy. These scaling factors make the resulting ratings quite practical by allowing one to make comparisons like “player $i$ is 10x better than $j$ if $i$’s rating is 400 points higher.” ↩︎
Taking the $\log$ makes it easier to deal with multiple samples, as it turns the product in $(1)$ into a sum. But here, we only use one sample, so it is useless. However, tradition is important for learning. ↩︎

Poem 1. "Delia"

Sun, 27 Apr 2025 00:00:00 +0000

Dedicated to my beautiful mother, Delia García.

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Madre que no me suelta el estandarte,
madre de selva que me cubre como yedra,
estás completamente marcada en mí;
me arrullas desde mi penumbra,
me susurras el nombre de Dios.

Ya quítame de aquí que me muero, madre,
dame las palabras que me corresponden,
sacude desde tu sigilo mi sangre
y hazme llegar tu amor.

Soledad eterna y vida corta,
nunca te me vayas a ir.

Madre que a ciegas todo lo ve;
corazón de parota, palabras de luz.
Eres la cumbre de este desierto,
autora del método mío;
eres madre de todo.

Águila de quinientas virtudes,
reguilete infinito de colores,
piedra de orgullo inexorable.

Árbol terrestre que toca el cielo;
sosiego inminente,
calor solar.