Music, Math, Medicine
The Question of the Constant
Why Is the Bias Not Time-Varying?
Every model begins with an assumption so quiet it pretends not to exist.
In the General Linear Model, that assumption is the constant term.
$$ y_t = W x_t + b + \epsilon_t $$But here the unease begins: why is $b$ not indexed by time?
If everything else breathes, adapts, deforms—why does the bias pretend to be eternal?
Option I: $b_t$ — The Bias as a Time-Series
If the system is alive, then the baseline must drift.
$$ y_t = W x_t + b_t + \epsilon_t $$- Physiology: Homeostasis is not fixed. Mean arterial pressure rises with fear, falls with rest.
- Performance: Your left hand tightens when the room is cold, loosens when the groove locks.
- Inference: The model learns a moving intercept—adaptive, reactive, fragile.
Here, $b_t$ is mood. Fatigue. Context. The rent. The audience. The night.
This is survival mode: the baseline as a function of circumstance.
But Then a Problem Appears
If the bias is always drifting, nothing is ever learned.
The model chases noise. The groove never settles.
- A constantly adapting $b_t$ erases memory.
- Every night becomes the first night.
- There is no idiom—only reaction.
This is anxiety disguised as flexibility.
A system too plastic cannot accumulate identity.
Option II: $b$ as Idiom
Now consider the opposite heresy.
$$ y_t = W x_t + b_{\text{idiom}} + \epsilon_t $$Here, the bias is not learned online.
It is memorized offline.
- The Blues: I–IV–V is not recalculated. It is known.
- R&B: The turnaround lives in the hands before thought.
- Bach: The Goldberg Variations assume a ground bass that never asks permission.
This $b$ is cultural preload.
An inherited intercept.
The Bias as Chord-Progression
A chord progression is not data.
It is a coordinate system.
- You do not “decide” the blues.
- You enter it.
- The song unfolds because the bias is stable enough to permit play.
Improvisation does not float freely.
It orbits a remembered gravity.
Memory Without Time Index
The most powerful priors are not time-stamped.
They persist across generations.
- Church cadences learned before literacy
- Walk-ups encoded before theory
- Resolutions felt before explanation
This is not $b_t$.
This is $b$ as scar.
The Hybrid Resolution
The real system cheats.
$$ y_t = W x_t + b_{\text{idiom}} + \delta b_t + \epsilon_t $$- $b_{\text{idiom}}$: What you know by heart.
- $\delta b_t$: What tonight does to it.
The idiom anchors.
The deviation expresses.
This is why the blues sounds the same—and never the same.
Why This Matters
If you treat the bias as fully time-varying, you will never sound like yourself.
If you freeze it completely, you will fossilize.
Style lives in the compromise:
- Enough rigidity to be recognizable
- Enough drift to remain alive
The constant term is not constant.
But neither is it up for debate every night.
Coda: Knowing It by Heart
You do not calculate the blues.
You recall it.
The bias is what remains when the clock is removed.
When time breaks, the idiom holds.
The most important parameter is the one you stop estimating.