It is March 26, 2026, and every single time your
spam filter traps a phishing email or a doctor
predicts a patient's survival rate. Or an AI
parses your voice command, yeah. Right. They
are all relying on a piece of math born in the
1830s. It's pretty well to think about. It is.
So welcome to today's deep dive. We are tackling
logistic regression. And our mission today is
to, well, basically shortcut you past all the
dense mathematical jargon to understand this
hidden engine. The engine that's secretly predicting
our behavior. Exactly. Because we live in a world
defined by absolute outcomes, like you pass a
test or you fail. is fraudulent or it's totally
legitimate. Right, we're surrounded by these
binary zeros and ones. But the standard math
most of us learned in high school, it fundamentally
breaks down when you try to predict those absolutes.
Oh, it completely shatters. I mean, if we want
to understand why standard linear regression
fails so spectacularly here, we can just look
at a classic statistical scenario. OK, laid on
me. So imagine 20 students, and they each spend
anywhere from, say, zero to six hours studying
for a final exam. Sure. Now, we aren't trying
to predict their exact numerical grade, like
a 0 to 100. We only care about the binary outcome.
They either pass, which is a 1, or they fail,
which is a 0. And if you try to use standard
linear regression for that, like if you literally
take a ruler and draw a straight sloped line
through a scatter plot of those ones and zeros,
your line is just going to shoot straight past
the data. Right. Trying to predict a pass or
fail with a straight line is like saying a student
who studies for 10 hours has 120 % chance of
passing. Yeah. Or someone who doesn't study at
all has a negative 20 % chance. It's mathematical
nonsense. You can't have a negative probability.
You really can't. A straight line simply doesn't
understand boundaries. It extends infinitely
in both directions, you know? Yeah. But probability,
by its very definition, has to remain strictly
confined between 0 and 1. You are either 0 %
likely to do something, 100 % likely, or somewhere
within that absolute range. There's no such thing
as being 120 % certain. Exactly. So the straight
line has to be abandoned. We need a completely
different geometric shape. Enter the standard
logistic function. Or the sigmoid function. Right.
So what does that actually look like? Well, imagine
taking that rigid straight line on your graph
and just physically bending the ends. OK. You
pull the top end down so it flattens out. never
quite touches the ceiling of 100 % and then you
pull the bottom end up so it flattens out and
never drops below 0%. Leaving you with this like
elegant S -shaped curve. Exactly and this specific
function can take any real number input from
negative infinity to positive infinity and gracefully
squash it down so the output is forever trapped
between 0 and 1. OK, let's unpack this, because
getting from a straight infinite line to a bounded
s -curve requires a bridge. You can't just plug
standard data into an s -curve and expect it
to work without some kind of mathematical translation.
And that translation mechanism is a concept called
the logit, right? The logistic unit, yes. The
logit basically transforms the log odds of an
event into a clean probability. But before we
even get to the logarithm part, we really have
to talk about odds. Yeah, because odds and probability
are frequently conflated. but they measure entirely
different relationships. Break that down for
us. Sure. So probability is the ratio of success
to the total number of attempts. Like if you
roll a standard six -sided die, your probability
of rolling a four is one out of six. Odds, on
the other hand, are the ratio of success to failure.
So the odds of rolling that four are one to five.
One success for every five failures. So if you
take a coin flip, The probability of getting
heads is 50%, or .5, but the odds are one to
one. One success for every one failure. Perfect
example. And if you have an 80 % probability
of passing that exam we talked about, your odds
are four to one. Four successes for every one
failure. Now we apply the logarithm. The logit
function basically takes the natural logarithm
of those odds. And why do we do that? Because
odds have an asymmetrical limit. They can go
all the way up to positive infinity, but they
can't drop below zero. I mean, you can't have
negative odds. Right. That wouldn't make sense.
But when you take the natural logarithm of the
odds, you magically stretch that scale out in
both directions. Oh, wow. Yeah. A 1 to 1 odds
ratio, a 50 % probability becomes exactly 0.
Anything less than 50 % becomes a negative number
stretching down to negative infinity. And anything
more than 50 % becomes a positive number stretching
to positive infinity. So we've essentially tricked
the math. We really have. By calculating the
log odds, we've created a perfectly infinite
symmetrical scale that a computer can just run
a standard linear equation on under the hood.
Exactly. But the output we actually look at remains
perfectly squashed into that beautiful s curve
of probability. It's brilliant because it allows
us to use the predictable, easily calculable
mechanics of linear math while still honoring
the strict boundaries of the real world. So the
math gives us a way to compress infinity down
into a neat percentage. But I mean, if I'm a
disaster planner or a doctor looking at those
numbers, how do I actually translate that percentage
into a concrete decision? Like, what does this
engine actually spit out? It spits out coefficients.
OK. In this mathematical context, a coefficient
is essentially a multiplier or a weight assigned
to a specific variable. It dictates exactly how
much influence one piece of data has on the final
outcome. Can we look at the odds ratio to see
this in action? Definitely. Let's say we have
a model analyzing building evacuations during
a hurricane, and we're working in base 10. The
model has a variable for number of evacuation
warnings received. If the coefficient for that
specific variable is 2, it means that receiving
just one additional warning increases the odds
of a person evacuating by a factor of 10 squared.
10 to the power of 2, so an increase of 100 times.
An increase in the odds by a factor of 100 purely
from nudging that one variable up by a single
unit. Wait, wait. I want to pause here to make
sure we don't fall into the very trap we just
discussed. So, multiplying the odds by 100 absolutely
does not mean the actual probability multiplies
by 100. Right, exactly. Because if you have a
10 % probability of evacuating, you can't multiply
that by 100 and have a 1000 % probability. The
S -curve prevents that. The S -curve is the great
equalizer here. If you are sitting right in the
middle of the curve, like, at a 50 % probability,
a massive jump in odds will drastically shoot
your probability upward. OK. But if you are already
near the top of that S curve, say, you already
have a 95 % probability of evacuating because
you live right on the coast, your odds can still
multiply by 100. But your actual probability
might only nudge up to like, 98 or 99 percent.
Ah, the closer you get to absolute certainty,
the harder it is to move the needle on the probability
scale. The S -curve basically compresses those
massive skyrocketing jumps and odds into tiny
fractional adjustments at the extreme ends. It
perfectly mimics diminishing returns. And what's
fascinating here is how this exact mathematical
behavior mirrors human decision making across
wildly different environments. Right. Medical
professionals rely on this structure to calculate
the tris, the trauma, and injury severity score.
Oh, I heard of that. Yeah. It takes variables
like patient age, blood pressure, and respiratory
rate, assigns coefficients to each, and maps
them onto the S curve to predict patient mortality.
It literally gives emergency rooms a concrete
probability to guide triage. And it scales to
completely different disciplines without changing
the underlying math. Yeah. Like political scientists
use it to model voter behavior. If you are predicting
whether a Nepalese voter will choose the Nepali
Congress or the Communist Party of Nepal, you
just map demographic variables onto the curve
to find the probability of a vote. That political
example actually highlights an important expansion
of the model. We've mostly been talking about
binary outcomes, pass or fail, live or die. But
the voter has more than two choices. Right. There
are multiple parties. Exactly. This is where
multinomial logistic regression comes into play.
It handles situations with multiple unranked
categories. The math calculates the odds of choosing
one specific party over a baseline reference
party. Oh, interesting. And if the categories
do have a strict order, like A patient rating
their pain on a scale from 1 to 10 statisticians
use ordinal logistic regression. So the S -curve
structure can really stretch to fit whatever
categorical reality we need it to. But this leaves
a massive mechanical gap in our understanding.
How so? Well, we know what the S -curve does.
We know how to read the odds and coefficients
it produces. But how does a computer actually
find the perfect S -curve for a messy set of
raw data? Ah. OK. Well, if we were using linear
regression, the computer would just minimize
the squared error. Right. It would literally
measure the physical distance between the data
points on the graph and the straight line, and
adjust the line until that total distance is
as small as possible. You plug the numbers into
a closed form equation, and you get an immediate
perfect answer. But we don't have a straight
line. We have an S -curve, and the data points
are all clustered at the absolute top and absolute
bottom of the graph. Measuring physical distance
doesn't really work the same way here. It doesn't.
It requires an entirely different approach called
maximum likelihood estimation or MLE. Instead
of minimizing physical distance, MLE calculates
something known as log loss or cross entropy.
Right. And the concept of log loss is often described
using the word surprisell, which I love. It's
literally measuring how shocked the mathematical
model is by reality. Yes. Think back to our students.
Let's say the computer draws a tentative S -curve.
And based on that curve, it predicts a student
who barely studied has a 99 % probability of
failing. OK. But then we look at the actual data,
and that student passed. The model's surprisell
approaches infinity. It is incredibly shocked.
Exactly. The algorithm's sole objective is to
minimize its own surprise. It wants its predictions
to align as closely as possible with the actual
observed event. But because there is no simple
close -form equation to solve for the perfect
S -curve all at once, the computer has to guess
and check. It uses iterative numerical methods
to basically find the bottom of the error curve.
A common analogy for this is Newton's method,
like imagine you're blindfolded standing on the
side of a mountain and your goal is to reach
the lowest point in the valley. the point of
minimum surprise. You can't see the valley. All
you can do is feel the slope of the ground right
beneath your feet. You figure out which direction
goes down, and you take a step. And then you
feel the slope again, calculate the new downward
trajectory, and take another step. The computer
calculates a tentative curve, measures the surprizal,
calculates the derivative, which is the slope,
to find the direction of less surprise, adjusts
the coefficients slightly, and repeats the process.
Step by step, it slowly crawls down the mountain
until the ground levels out and it can't minimize
the surprise any further. And that is when the
model has converged. Yes. But that blindfolded
walk down the mountain can go completely wrong
if the terrain is deceptive. The mathematical
crawling process can fail. Wait, really? How?
Well, the model might never converge. It might
just keep calculating forever. One primary reason
this happens is a lack of data. There's a general
heuristic called the rule of 10, which basically
states you need at least 10 actual events per
explanatory variable to get stable coefficients.
If your model has five variables, you need at
least 50 occurrences of the outcome you are tracking.
Otherwise, the math doesn't have enough terrain
to find the valley. And another fatal trap is
a concept called complete separation, right?
Yes. Complete separation occurs when your predictors
perfectly predict the outcome. OK, wait. On the
surface, perfectly predicting the outcome sounds
like the ultimate goal. If students studies for
exactly six hours, they pass 100 % of the time.
Why is perfection breaking the math? We have
to remember that the algorithm's goal, minimizing
surprise. If six hours of studying guarantees
a pass, the only way the model can experience
absolute zero surprise is to turn the S -curve
into an infinitely steep vertical wall right
at the six hour mark. Oh, I see. It wants the
probability to instantly snap from zero to one.
But the formula relies on real numbers, and you
cannot graph a perfectly vertical line with real
numbers in this function. The algorithm just
keeps stepping down the mountain, pushing the
coefficient higher and higher, trying to build
that vertical wall. It pushes toward infinity,
the math panics, and the entire model crashes.
So what does this all mean? Like, why should
anyone outside of a data science lab care about
maximum likelihood estimation or the algorithm
minimizing its surprisingly? Because it's everywhere.
Exactly, because this isn't just a statistical
quirk. This process of iterative Guessing, measuring
shock, and adjusting coefficients is the fundamental
mechanism of supervised machine learning. It
really is. When a tech company announces they
have trained a new AI to filter your inbox or
diagnose a lung scan, this is what that training
looks like under the hood. The AI is iteratively
adjusting its internal S -curves to minimize
its surprise when tested against real -world
training data. It is actively minimizing the
cross entropy loss function. This specific mathematical
process is literally the bedrock upon which modern
artificial intelligence is built. Which makes
the timeline of this math almost unbelievable.
We are talking about the engine of modern AI
but the origins of this formula predate the American
Civil War. Yeah, they have absolutely nothing
to do with computers or machine learning. Right.
The history of logistic regression is honestly
a testament to the universality of mathematics.
It begins in the 1830s with a Belgian mathematician
named Pierre Francois Verhulst. And he wasn't
looking at data sets to predict behavior. He
was looking at biological population growth.
Because at the time, the assumption was that
populations just grew exponentially, right? But
Verhost looked at the reality of resources and
realized populations eventually hit a ceiling.
Yes. He introduced the concept of carrying capacity.
A population grows rapidly at first, but as food
and space become scarce, that growth slows down
and eventually levels off. And when he modeled
this constraint mathematically, he produced the
very first logistic function. He gave the world
the S curve. He did. And the scientific process
that followed wasn't a straight line of development.
It was chaotic, isolated and incredibly cross
-disciplinary. It wasn't invented by a tech bro
in a garage. It was built by chemists, biologists
and medical researchers over a century. Absolutely.
In 1883, a chemist named Wilhelm Ostwald was
studying autocatalysis situations where a chemical
reaction creates a product that basically accelerates
the reaction itself. But eventually the raw materials
run out. So he independently discovered the exact
same S -curve to model chemical constraints?
Yes. And decades later, in 1920, Raymond Pearl
and Lowell Reed rediscovered it yet again, completely
unaware of Reholst and applied it back to population
growth. It proves that this mathematical shape
is an intrinsic property of the natural world,
governing everything from yeast in a petri dish
to chemical reactions. It does. But convincing
the broader statistical community to adopt it
was a massive hurdle. In the 1930s, the battleground
was bioassays, testing the lethal potency of
various drugs and toxins. Okay. The dominant
model of the era was the probit model, championed
by heavyweights like Chester Bliss and Ronald
Fisher. The probit model used a normal distribution
curve. It worked, but the math was dense and
computationally heavy. Right. Here's where it
gets really interesting. In 1944, a researcher
named Joseph Berkson steps into the fray. He
takes Reholt's population growth curve, recognizes
its mathematical elegance, and coins the term
logit as a direct punchy alternative to the probit
model. Yeah, and he spent the next several decades
relentlessly arguing that his logit model was
superior. And Berkson eventually won the war.
By the 1970s, the logit model reached parity
with the probit model and then largely surpassed
it in widespread use. The deciding factor was
really just computational simplicity. The math
of the logit model, particularly its use of logarithms
and odds, was significantly easier for the computers
of that era to process. But predicting how much
toxin kills a beetle is a long way from predicting
human behavior. How did Berkson's bioassay math
become the foundation for predicting voter choice
and consumer habits? That leap occurred in 1973,
driven by an economist named Daniel McFadden.
Okay. He achieved a massive theoretical breakthrough
by linking multinomial logistic regression directly
to the economic theory of discrete choice. McFadden
basically proved that you could use this exact
S -curve math to model how a rational human actor
chooses the option that provides them with the
greatest utility. Oh, wow. Yeah. He moved the
math from measuring physical limits like carrying
capacity to measuring abstract human preference.
That foundation basically exploded the use of
logistic regression across the social sciences,
economics, and marketing. From yeast, to chemicals,
to beetles, to the stock market. Exactly. And
if we connect this to the bigger picture, the
convergence becomes even more profound. How so?
The formula Berson was fighting for in the 1940s,
the equation that maps inputs through a sigmoid
function to generate a probability between 0
and 1, is functionally identical to a single
layer perceptron in a modern artificial neural
network. That is wild. The exact same mathematical
logic that predicted how a population of animals
reaches its physical limit is currently computing
the very first layer of an artificial intelligence's
thought process. A single layer neural network
computes its output using that S -curve. And
crucially, the mathematical derivative of the
sigmoid function is incredibly easy for a computer
to calculate. Right. That ease of calculation
is what makes the back propagation process possible.
It is what allows modern neural networks to adjust
their weights, step down the mountain, and actually
learn. Okay, let's bring you back to the surface.
We have covered an immense expanse of theory
today. We started by dismantling the straight,
rigid line of linear regression and learned why
mapping binary absolutes requires bending reality
into an elegant S -curve. We broke down the mechanics
of the logit, exploring how converting probabilities
into log odds allows us to trick the math into
using linear equations for bounded outcomes.
Right, and we examined how coefficients act as
weights, multiplying odds, while the S -curve
naturally compresses those massive jumps as we
approach absolute certainty. We looked under
the hood of machine learning to see maximum likelihood
estimation in action, watching a computer blindly
step down a mountain to minimize its surprizal.
We explored why perfect predictions break the
model by demanding infinitely steep walls. And
we traced a chaotic 200 -year history watching
a simple equation for population constraints
evolve into the discrete choice models of modern
economics and the neural pathways of artificial
intelligence. It really is the hidden architecture
of decision -making. It truly is. Which brings
me to a final slightly mind -bending thought
from the research to leave you with. Oh, this
is a good one. There's a specific mathematical
interpretation of logistic regression called
the latent variable model. This interpretation
assumes that there isn't actually a hard binary
in nature. Right. It assumes there is an unobserved
hidden continuous variable, a hidden spectrum
of utility or desire or intent paired with the
random noise of the universe. Under this theory,
the binary outcome the 1 or the 0, the yes or
the no, is simply an indicator of whether that
hidden internal variable has finally crossed
a specific, invisible threshold. Think about
the last absolute choice you made today, yes
or no, buy or don't buy, click or don't click.
We view our daily choices as definitive absolutes.
But if the latent variable model is right, Underneath,
your simple yes or no is a hidden, swirling,
continuous spectrum of utility, constantly battling
against random noise. It's crazy thought. It
is. Next time you make a definitive choice, ask
yourself, what does your hidden continuous variable
look like right now? And how much of your final
decision was just the math of the standard logistic
distribution?