What Happens If We Do Not Divide by P(D)in Bayes’ Rule?

Last Updated on April 17, 2026 by Rajeev Bagra

Many learners ask:

If Bayes’ theorem is

then what happens if we do not divide by ?

It is still meaningful, but it gives something different.

If we remove the denominator, we get:

What We Get Without Dividing

This equals:

This is called the joint probability.

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[b]Meaning of Joint Probability[/b]

It means the probability that both events happen together:

[list]
[] is true
[] is observed
[/list]

So it is mathematically valid.

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[b]Why It Is Not Posterior Probability[/b]

Usually Bayes’ theorem asks:

Given that happened, what is the probability that is true?

That is:

To answer this, we only look at cases where D occurred.

So we divide by:

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[b]Example: Medical Test[/b]

Suppose:

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[]
[]
[/list]

Then:

Meaning:

0.9% of all people both have disease and test positive.

But to find probability of disease given positive test:

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[b]Useful in Practice[/b]

Often we write:

This means posterior probability is proportional to prior times likelihood.

Normalization happens later.

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[b]Simple Intuition[/b]

[list]
[]Without dividing by = raw overlap score
[]With dividing by = updated conditional probability
[/list]

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[b]Final Summary[/b]

If we do not divide by , we get:

This is meaningful, but it is not the same as:

which is the updated belief after observing evidence.