2026-03-24 16:58 Tags:

1. Problem Setup

We are solving a binary classification problem:

$$[y\in 0,1]$$

Goal:

$$[P(y=1\mid X)]$$

We want a model that outputs a probability between 0 and 1.

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2. Why Linear Regression Fails

Linear regression assumes:

$$[y={\beta }_0+{\beta }_1{x}_1+\cdots +{\beta }_p{x}_p]$$

Problems:

1. Output is unbounded:

2. [ (-\infty, +\infty) ] $$→ invalid for probabilities

3. No probabilistic interpretation

4. Not suitable for classification boundaries

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You are already very close. The missing piece is:

why do we go from probability to odds, and then from odds to log-odds?

Let’s do only that piece.

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Step 1: What are we trying to build?

We want a model that takes in:

$$[z={\beta }_0+{\beta }_1{x}_1+\cdots +{\beta }_p{x}_p]$$

This (z) can be anything:

• very negative

• zero

• very positive

$$[z\in (-\infty ,+\infty )]$$

But we want the final output to be a probability:

$$[P(y=1\mid X)]$$

and probability must be:

$$[0\le P\le 1]$$

So yes: real-number input, probability output.

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Step 2: Why not just force z to equal probability?

Suppose we say:

$$[P=z]$$

Then if (z=3), probability is 3.

Impossible.

If (z=-2), probability is -2.

Also impossible.

So we need some transformation:

$$[z\to P]$$

that turns any real number into something between 0 and 1.

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Step 3: What kind of transformation do we need?

We need a function that does this:

• input: any real number

• output: between 0 and 1

That is the actual problem.

Now here is the key:

there are many functions that can do this.

So the real question is not:

why must it be odds?

It is:

why is odds/log-odds a convenient bridge?

That is the part we now unpack.

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Step 4: Start from probability

Probability is bounded:

$$[P\in (0,1)]$$

This bounded interval is annoying for linear modeling.

Why?

Because linear models naturally live on the whole real line:

$$[-\infty \text{ to }+\infty ]$$

So instead of trying to make linear models live inside ((0,1)), we do the opposite:

take probability, and transform it into something unbounded.

This is the important move.

We are not yet choosing odds because of magic.

We are choosing a way to “unbound” probability.

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Step 5: First transformation: probability to odds

Define:

$$[\text{odds}=\frac{P}{1-P}]$$

Let’s see what this does.

If (P=0.5):

$$[\text{odds}=\frac{0.5}{0.5}=1]$$

If (P=0.8):

$$[\text{odds}=\frac{0.8}{0.2}=4]$$

If (P=0.2):

$$[\text{odds}=\frac{0.2}{0.8}=0.25]$$

So now odds lives in:

$$[(0,+\infty )]$$

Good news:

• no longer capped at 1

Bad news:

• still cannot be negative

• still not the full real line

So odds gets us part of the way, but not all the way.

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Step 6: Second transformation: odds to log-odds

Now take log:

$$[\log \left(\frac{P}{1-P}\right)]$$

What happens now?

• if odds is very small, log is very negative

• if odds = 1, log = 0

• if odds is very large, log is very positive

Now the range becomes:

$$[(-\infty ,+\infty )]$$

Perfect.

This is exactly the same range as the linear predictor:

$$[z={\beta }_0+{\beta }_1{x}_1+\cdots +{\beta }_p{x}_p]$$

So now we can say:

$$[\log \left(\frac{P}{1-P}\right)=z]$$

and that is legal, natural, and mathematically clean.

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Step 7: So why “go to odds” first?

Because log cannot be taken directly on probability in the right way.

Let’s compare.

If you take just:

$$[\log (P)]$$

then since (P\in(0,1)),

$$[\log (P)\in (-\infty ,0)]$$

This only gives you negative numbers.

Not enough.

If you take:

$$[\log (1-P)]$$

same problem.

So we need a transformation of probability that:

• respects the fact that probability has two sides: event and non-event

• gives something that can go from very small to very large

• then log makes it span all real numbers

That is why:

$$[\frac{P}{1-P}]$$

is so useful.

It compares:

probability of event / probability of non-event

That is the odds.

Then log turns that positive ratio into the whole real line.

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Step 8: Intuition in plain language

Probability alone says:

• chance event happens

Odds says:

• how much more likely the event is than the non-event

Log-odds says:

• a version of that comparison that can be modeled linearly

So the path is:

1. probability is bounded

2. odds removes the upper bound

3. log-odds removes the lower bound too

4. now it matches a linear expression

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Step 9: The key bridge

You were asking:

how did we suddenly turn to odds ratio?

The answer is:

we did not jump there randomly.

We needed to transform probability from a bounded scale to an unbounded scale.

The odds is an intermediate step that compares event vs non-event:

$$[\frac{P}{1-P}]$$

Then the log of that gives a quantity on the full real line:

$$[\log \left(\frac{P}{1-P}\right)]$$

That makes it compatible with:

$$[{\beta }_0+{\beta }_1{x}_1+\cdots +{\beta }_p{x}_p]$$

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Step 10: One sentence summary

We go to odds, then log-odds, because probability is bounded in ((0,1)), while linear models live on $((-\infty ,+\infty ))$, and log-odds is the simplest transformation that connects those two worlds.

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