Bayes’ Theorem - How probabilities change with new information

Introduction

Imagine this: You wake up, look out the window and see the street is wet. What do you think? “It rained.” But then you remember: your neighbour mentioned buying a new sprinkler.
And you saw a news story recently about faulty water pipes in your area.

What happens now?
You adjust your belief – maybe it wasn’t rain after all. Maybe the sprinkler or a broken pipe?

That’s Bayesian thinking in action: “How does the probability of a cause change when I get new evidence?”

In our daily lives, we constantly make assumptions based on observations. Whether its about the weather, our health or checking our inbox, we’re always (often unconsciously) asking:

“How likely is Cause A, given that I observe Evidence B?”

That’s the essence of Bayesian thinking. And although Bayes’ Theorem is over 250 years old, it’s more relevant than ever – from medical diagnostics to artificial intelligence.

So what is Bayes' Theorem, Really?

This formula answers the question:

“What is the probability of A being true, given that B has happened?”

Where:

• P(A): Your prior belief (initial probability of A)

• P(B∣A): The likelihood (how likely B is if A is true)

• P(B): The overall probability of B happening (normalization factor)

1. Start with a prior belief P(A)

You begin with an initial assumption – for example:

“1 out of 100 people has this disease.”

This is your prior probability.

Represented by the growing chart: Based on experience or general data, you start with an initial idea of how likely A is.

2. Throw in new evidence – B happens

Now some new information appears — event B occurs.
For example:

“The test came back positive.”

Represented by a bar chart on a screen: You receive new data (evidence) that influences your belief.

3. Update belief P(A∣B)

Now, using Bayes’ Theorem, you update your belief:

“Given the positive test result, how likely is it now that the person is actually sick?”

Shown by the red (sick) and blue (healthy) figures:

• You start with a guess (e.g., person might be sick),

• You update your belief after the evidence (maybe more or less likely).

This is a recalculation of probability based on new information.

Example: Disease Test Calculation

Let's say:

• 1% of people have a disease → P(Sick) = 0.01

• The test is 99% accurate:

  • If you're sick, it gives a positive result 99% of the time → P(Positive∣Sick) = 0.99

  • If you're healthy, it still gives a false positive 5% of the time → P(Positive∣Healthy) = 0.05

Now someone tests positive. What’s the probability that they’re actually sick?

Means we want to find P(Sick∣Positive)

Bayes' Theorem tells us:

Result:

Even though the test is 99% accurate, the probability that a person actually has the disease given a positive test result is only about 16,67%!

Why?

Because the disease is so rare, the chance that a positive test result is a false alarm is actually much higher than most people intuitively expect.

Why Bayes’ Theorem Is So Powerful?

We’ve now seen how Bayes’ Theorem works—and how it can update our beliefs when new evidence comes in. But why is this so important?

Classic vs. Bayesian Thinking

Traditional (or frequentist) statistics treats probability as something fixed and objective. For example: "This test is 99% accurate."

Bayesian thinking is different. It treats probability as a degree of belief that can change as we learn more about the background.

Instead of asking, "What's the fixed probability that someone is sick?", we ask: "Given what I now observe, how likely is it that this person is sick?"

This allows us to update our beliefs, rather than rely on static assumptions.

Learning from Experience

Bayesian models are like (some) people: they learn over time.

Let's say we test 10 people with a medical test and refine our belief based on the results. Then we test 100 people—and adjust again. This incremental updating is what makes Bayesian inference so useful in practice.

The more evidence we get, the better our estimates become.

A Note on Controversy

Bayes' Theorem wasn't always accepted.

In fact, for over 100 years, it was mostly ignored—because it challenged the dominant "objective" view of probability.

Frequentists saw probability s fixed and repeatable. Bayesian's, on the other hand, argued that probability could reflect personal belief—as long as it's updated rigorously when new evidence appears.

This sparked a philosophical debate that still exists today. But in many practical fields—from medicine to AI—the Bayesian view is increasingly winning out.

Bayes in Everyday Life – Real-Life Examples

As we have already seen in the calculation example, bayesian thinking plays an important role in medicine in the assessment of sick or healthy. Apart from this, however, there are also other examples in which Bayes occurs in everyday life.

E-Mail - Is This Spam?

Your email program classifies messages as spam based on words and senders.

If the subject line includes “Free”, “Bitcoin”, and “Urgent”, the chance of it being spam increases.
But legitimate emails can include those words, too.
Spam filters learn from experience and get better over time.

→ Use case: Bayesian spam filters.

Weather - Will It Really Rain Today?

The forecast says: 70% chance of rain.
Does that mean it will rain 70% of the day?

Not exactly. It means that on 7 out of 10 days with similar conditions, it has rained in the past.

→ Bayes helps combine prediction and prior data.

Netflix - What Should You Watch Next?

You watched a sci-fi movie yesterday and a documentary today.
Netflix “guesses” what you’ll want to watch next, based on the probability inferred from your watch history and based on what other users with a similar history have watched.

→ Use case: Bayesian recommendation systems.

Crime Solving - Who's the Culprit?

Sherlock Holmes is the ultimate Bayesian thinker:

“The dog didn’t bark, even though someone entered the house – so it must have been someone the dog knew.” (A.C. Doyle - Silver Blaze)

He updates his hypotheses with every new clue – just like Bayes.

Bayes' Theorem is more than just a mathematical formula — it's a powerful way of thinking. In a world full of uncertainty, it gives us a structured method to update our beliefs as new information comes in. Whether we're evaluating a medical test result, making weather predictions, or solving mysteries like Sherlock Holmes, Bayesian reasoning helps us think more clearly and adapt more intelligently.

In essence, it teaches us this:

“Don’t just believe — revise your belief when the evidence calls for it.”

Bayesian thinking isn’t limited to statisticians or data scientists. It's a mindset we all use, often without realizing it. Learning to apply it consciously can make us better decision-makers — and maybe even a bit more like Sherlock Holmes.

So next time you’re faced with new information, ask yourself:
“How does this change what I thought I knew?”
That’s the Bayesian way.

Conclusion