Bayes’ Theorem Explained: How to Compare Hypotheses and Update Your Beliefs

When something happens, how do you decide what caused it?

A friend does not return your message. A project performs worse than expected. An AI system gives you an uncertain answer. You notice an unusual pattern in some data.

Most of us quickly settle on one explanation:

“They must be annoyed with me.”

“The strategy failed.”

“The AI probably knows what it is talking about.”

“This pattern cannot be a coincidence.”

But a single observation can usually be explained by several different hypotheses. The first explanation that enters your mind is not necessarily the most probable one.

This guide explains Bayes’ theorem in plain English and shows how Bayesian reasoning can be used to compare competing explanations, combine prior knowledge with new evidence, and update your confidence more coherently.

You will learn:

• what priors, likelihoods and posterior probabilities mean;
• why the normaliser or denominator is necessary;
• how to calculate Bayes’ theorem across several hypotheses;
• how Bayesian reasoning helps counter one-shot thinking;
• how to use the Bayesian Hypothesis Comparator GPT to work through your own examples.

Try it yourself: Open the Bayesian Hypothesis Comparator GPT

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What is Bayesian reasoning?

Bayesian reasoning is a method for updating the probability of a hypothesis when new evidence becomes available.

In its simplest form, it asks three questions:

1. How plausible was this hypothesis before I saw the new evidence?
2. How likely would the evidence be if this hypothesis were true?
3. After considering the competing hypotheses, how confident should I now be?

Bayesian reasoning does not demand absolute certainty. It allows beliefs to be represented as degrees of confidence that change when new information arrives.

This is useful because most real-world reasoning occurs under uncertainty. We rarely possess complete evidence, perfectly reliable measurements or an exhaustive understanding of every possible cause.

The goal is therefore not to become certain about everything. It is to update your confidence in a disciplined and transparent way.

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Thomas Bayes and the origins of Bayes’ theorem

Thomas Bayes was an eighteenth-century English minister and mathematician whose work helped establish a method for reasoning backwards from observed events to their possible causes.

Bayes died in 1761 without publishing the work for which he is now best known. His friend Richard Price examined and edited Bayes’ papers and communicated the resulting essay to the Royal Society. It was published posthumously in 1763 as An Essay towards Solving a Problem in the Doctrine of Chances.

The underlying problem remains highly relevant:

Given that we have observed some evidence, how should that evidence change our confidence in the possible explanations?

More than 250 years later, Bayesian methods are used throughout statistics, scientific research, machine learning, medical reasoning and decision analysis. In 2018, the University of Edinburgh officially opened its £45 million Bayes Centre for data science and artificial intelligence—an indication of the continuing influence of Bayes’ ideas.

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The problem with one-shot thinking

Suppose that you perform badly in a job interview and immediately conclude:

“I am not confident or authentic enough to compete.”

That is one possible explanation. It is not the only one.

Other hypotheses might include:

• the successful candidate had more directly relevant experience;
• the interview questions disproportionately covered an unfamiliar area;
• the panel already had a strong internal candidate;
• you communicated well overall but gave one weak technical answer;
• the organisation was looking for a different professional profile.

One-shot thinking occurs when we accept the first plausible story without deliberately generating and comparing alternatives.

The danger is not simply that the first explanation might be wrong. It is that once an explanation has been adopted, it can influence which evidence we notice, remember and seek next.

A better reasoning habit is:

Do not ask only, “Could this explain what happened?”

Ask:

“How does this explanation compare with the most plausible alternatives?”

Research on debiasing suggests that deliberately considering an alternative or considering the opposite can reduce premature fixation on an initial interpretation.

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Three biases that interfere with belief updating

Anchoring

Anchoring occurs when the first number, interpretation or possibility we encounter exerts too much influence over later judgement.

For example, if a salary of £30,000 is mentioned first, later offers may feel high or low relative to that figure—even when the realistic market range is £35,000 to £45,000.

In hypothesis comparison, the first explanation can act as an anchor. Later evidence is then interpreted relative to it rather than used to compare the full range of possibilities.

Availability

The availability heuristic occurs when we judge an event as more probable because examples come readily to mind.

After seeing extensive coverage of a plane crash, for instance, a person may temporarily overestimate the probability of an aviation accident.

What is memorable is not necessarily what is statistically common.

Confirmation bias

Confirmation bias involves searching for, interpreting or remembering information in ways that support an existing belief.

Once you believe that a colleague dislikes you, you may focus on one curt reply while overlooking several friendly conversations.

Bayesian reasoning cannot automatically eliminate these biases. It can, however, make important assumptions visible:

• What hypothesis did I begin with?
• What alternatives have I omitted?
• Am I giving favourable evidence too much weight?
• Would this evidence also be likely under another explanation?

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Bayesian reasoning in online information environments

Online information environments can make one-shot thinking more difficult to escape.

The content people encounter is shaped by a combination of:

• whom they follow;
• what their contacts share;
• what they previously clicked;
• what ranking systems predict will maintain their attention;
• what they actively choose to read or ignore.

This can create feedback loops in which familiar interpretations become easier to encounter and competing interpretations become less cognitively available.

The evidence around “filter bubbles” is more nuanced than the claim that algorithms alone trap everyone inside a personalised worldview. Research indicates that social networks, personal selection and algorithmic ranking can all influence exposure, with their relative effects varying across platforms, users and contexts.

The practical lesson is straightforward:

Repeated exposure to a claim is not the same as strong evidence for that claim.

A Bayesian approach encourages us to step outside the information stream and ask:

• What are the competing explanations?
• What evidence would each explanation predict?
• What evidence am I not being shown?
• What observation would meaningfully change my mind?

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Bayes’ theorem in plain English

For a hypothesis (H) and some evidence (E), Bayes’ theorem is:

P(H | E) = [P(E | H) × P(H)] / P(E)

In words:

posterior = (likelihood × prior) / normaliser

The four important components are the prior, likelihood, normaliser and posterior.

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The prior probability: (P(H))

The prior probability represents how plausible the hypothesis was before the new evidence was considered.

It may be informed by:

• established background rates;
• previous observations;
• historical data;
• scientific knowledge;
• relevant experience;
• a deliberately cautious estimate when firm data are unavailable.

For example, imagine that a weather forecast gives a 70% probability of rain.

Your initial hypotheses might be:

P(rain) = 0.70

P(no rain) = 0.30

These are the prior probabilities.

Priors should not be treated as arbitrary hunches when relevant data are available. At the same time, pretending that you have no prior assumptions does not remove them. It merely leaves them implicit.

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The likelihood: (P(E | H))

The likelihood asks:

How probable would this evidence be if the hypothesis were true?

This is written as:

P(E | H)

It is important not to confuse the likelihood with the posterior.

Consider the difference:

P(E | H)

means:

How likely is the evidence if the hypothesis is true?

Whereas:

P(H | E)

means:

How likely is the hypothesis after seeing the evidence?

These probabilities are not interchangeable.

For example:

The probability of a positive test given that a condition is present

is not automatically the same as:

The probability that the condition is present given a positive test.

The second probability also depends on how common the condition was before the test result—the prior probability. This distinction is one reason Bayes’ rule is important in diagnostic reasoning.

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The unscaled score: (P(H)P(E | H))

For each hypothesis, multiply the prior probability by the likelihood of the evidence:

unscaled score = P(H) × P(E | H)

This score combines:

• how plausible the hypothesis was initially; and
• how well the hypothesis predicts the new evidence.

It is called “unscaled” because the resulting scores will not necessarily add up to 1, or 100%.

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The normaliser: (P(E))

The normaliser—also called the denominator or marginal probability of the evidence—is the combined probability of observing the evidence across all the hypotheses being compared.

When comparing several mutually exclusive hypotheses:

P(E) = Σ_i P(H_i) × P(E | H_i)

For three hypotheses, this becomes:

P(E) = P(H_1)P(E | H_1) + P(H_2)P(E | H_2) + P(H_3)P(E | H_3)

The normaliser converts the unscaled scores into posterior probabilities that add up to 1.

This is the step that is often omitted from informal explanations of Bayesian reasoning. It is nevertheless essential when calculating the posterior probability of every hypothesis.

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The posterior probability: (P(H | E))

The posterior probability is your updated level of confidence after taking the evidence into account.

For each hypothesis:

P(H_i | E) = [P(H_i)P(E | H_i)] / [Σ_j P(H_j)P(E | H_j)]

In plain English:

Multiply the prior by the likelihood, then divide by the combined score for all the competing hypotheses.

The result is a set of coherent updated probabilities.

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A Bayesian inference example: will it rain in Reykjavík?

Imagine that you are visiting Reykjavík.

The night before a planned hike, the forecast gives a 70% probability of rain.

For this simplified teaching example, we will compare two hypotheses:

• (H_1): It will rain later.
• (H_2): It will not rain later.

The prior probabilities are:

P(H_1)=0.70

P(H_2)=0.30

When you wake up, you see a bright blue sky with very little cloud.

This is your new evidence:

E = clear morning

The following likelihoods are illustrative estimates rather than meteorological measurements:

P(E | H_1)=0.20

A clear morning would be possible if rain were coming later, but we are estimating that it would be relatively unusual.

P(E | H_2)=0.90

A clear morning would be considerably more likely if the day were going to remain dry.

Step 1: Calculate the unscaled scores

For rain:

0.70 × 0.20 = 0.14

For no rain:

0.30 × 0.90 = 0.27

Step 2: Calculate the normaliser

Add the unscaled scores:

P(E)=0.14+0.27=0.41

Step 3: Calculate the posterior probability of rain

P(H_1 | E)=0.14 / 0.41 ≈ 0.34

The updated probability of rain is approximately 34%.

Step 4: Calculate the posterior probability of no rain

P(H_2 | E)=0.27 / 0.41 ≈ 0.66

The updated probability of no rain is approximately 66%.

Step 5: Check that the probabilities add up

0.34+0.66=1.00

The posterior probabilities therefore add up to 100%.

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What did the Bayesian calculation tell us?

Before looking outside, rain was considered more probable than no rain:

70% > 30%

After seeing the clear morning, the ranking reversed:

34% < 66%

The new evidence was much more compatible with the no-rain hypothesis. It was sufficiently diagnostic to overcome the higher prior probability assigned to rain.

This does not mean that rain has become impossible. Nor does the calculation itself tell you whether to go hiking.

A decision also depends on factors such as:

• the cost of getting wet;
• the availability of waterproof clothing;
• the danger of the proposed route;
• how easily the plan can be changed;
• additional weather information.

Bayesian inference updates beliefs. Decision-making combines those beliefs with consequences, values and available actions.

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Why compare every hypothesis?

Suppose you calculated only the rain score:

0.70 × 0.20 = 0.14

On its own, 0.14 is difficult to interpret. It is not the posterior probability of rain.

It becomes meaningful only when compared with the scores for the alternative hypotheses.

The normaliser performs that comparison:

0.14+0.27=0.41

Dividing each score by 0.41 tells us how the available probability should be distributed across the two explanations.

This is why Bayesian hypothesis comparison should calculate a posterior for every hypothesis, not merely the preferred one.

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Comparing three or more hypotheses

The same process works with three or more explanations.

Suppose an important email has not received a response.

You might compare:

• (H_1): The recipient has been too busy to reply.
• (H_2): The message was overlooked or filtered.
• (H_3): The recipient is deliberately avoiding the request.

You would then:

1. assign prior probabilities to the three hypotheses;
2. estimate how likely the non-response would be under each;
3. multiply every prior by its corresponding likelihood;
4. add the three unscaled scores to obtain the normaliser;
5. divide each score by the normaliser;
6. compare the posterior probabilities.

The arithmetic is straightforward. The more difficult—and intellectually useful—part is deciding which hypotheses and likelihood estimates are defensible.

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A five-step Bayesian mindware routine

You do not always need to perform a full numerical calculation. Bayesian reasoning can also function as a reusable thinking routine.

1. State the observation

Describe what actually happened without building an explanation into it.

Weak:

“My colleague ignored me.”

Better:

“My colleague did not reply to two messages sent over three working days.”

2. Generate competing hypotheses

Aim for several genuinely different explanations rather than minor variations of the same story.

3. Establish the priors

Ask:

• How common is each explanation generally?
• What relevant background information do I possess?
• Am I assigning a high prior merely because an explanation is vivid?

4. Test the diagnosticity of the evidence

For each hypothesis, ask:

If this hypothesis were true, how likely would I be to see this evidence?

Evidence is highly diagnostic when it is considerably more likely under one hypothesis than under the alternatives.

5. Update without overclaiming

Prefer language such as:

• “The evidence currently favours…”
• “This has become more plausible because…”
• “The leading explanation remains uncertain.”
• “The conclusion depends heavily on the estimated prior.”
• “Another observation could reverse the ranking.”

Bayesian reasoning should produce calibrated confidence, not artificial certainty.

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Try the Bayesian Hypothesis Comparator GPT

The Bayesian Hypothesis Comparator GPT guides you through the full comparison.

You provide:

• an observation or piece of evidence;
• two or more possible hypotheses;
• prior probabilities, when known;
• likelihood estimates, when known.

The GPT then helps you:

• clarify the hypotheses;
• estimate or review the priors;
• estimate the likelihood of the evidence under each hypothesis;
• calculate every unscaled score;
• calculate the normaliser;
• calculate every posterior probability;
• identify which hypothesis is currently best supported;
• examine which assumptions are driving the result.

Try the Bayesian Hypothesis Comparator GPT

Watch the tutorial for using the Bayesian Hypothesis Comparator GPT

When firm data are unavailable, the GPT may use illustrative estimates. These should be treated as assumptions for exploration, not as established facts.

A useful prompt is:

“I observed [evidence]. My three possible explanations are [H1], [H2] and [H3]. Help me estimate the priors and likelihoods, calculate the normaliser and compare the posterior probabilities.”

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Bayesian reasoning exercises

Exercise 1: A friend did not return your call

Possible hypotheses:

• They were busy and forgot.
• They are upset after your previous conversation.
• Their phone was unavailable or out of charge.

Questions:

• Which explanation has the highest prior?
• Would one missed call be unusual under any of them?
• What additional evidence would distinguish the hypotheses?

Exercise 2: Your keys are missing

Possible hypotheses:

• You left them in the house.
• They fell from your pocket while you were outside.
• You left them at work.

Questions:

• Where are your keys usually found when misplaced?
• What locations did you visit?
• Which evidence would be most diagnostic?

Exercise 3: You expect a promotion

Possible hypotheses:

• Your performance and experience make promotion likely.
• The organisation rarely promotes more than one person each year.
• A recent reorganisation has delayed promotions.

Questions:

• What is the organisation’s base rate for promotion?
• How predictive is a strong performance review?
• What new evidence would meaningfully change your estimate?

Exercise 4: An AI answer appears convincing

Possible hypotheses:

• The answer is accurate and well supported.
• The answer is partly correct but omits an important qualification.
• The answer is a fluent but fabricated response.

Questions:

• Does the answer provide checkable primary sources?
• Would the same confident wording appear under both correct and incorrect answers?
• What independent evidence would discriminate among the hypotheses?

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Important limits of Bayesian hypothesis comparison

Bayes’ theorem produces a valid calculation from the probabilities supplied to it. It does not guarantee that those probabilities are accurate.

The result depends on the priors

Poorly chosen priors can distort the posterior.

Where possible, use relevant base rates, empirical data or clearly justified background knowledge.

Likelihoods can be difficult to estimate

People often ask:

“How likely is my hypothesis given the evidence?”

But the likelihood requires the reverse question:

“How likely is this evidence if my hypothesis is true?”

Confusing these two quantities can produce serious errors.

The hypothesis set may be incomplete

Posterior probabilities are conditional on the hypotheses included in the model.

If an important explanation has been omitted, the leading listed hypothesis may receive more confidence than it deserves.

The hypotheses should be clearly distinguishable

When hypotheses overlap heavily, assigning priors and likelihoods becomes difficult.

Try to define explanations that are sufficiently clear to make different predictions.

Repeated evidence may not be independent

Five articles repeating the same original claim do not necessarily provide five independent pieces of evidence.

Likewise, several symptoms, measurements or reports may arise from the same underlying source.

A probability is not a decision

The most probable explanation is not automatically the best action.

Decisions also require consideration of:

• costs;
• benefits;
• risks;
• reversibility;
• ethical constraints;
• the consequences of being wrong.

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Safety boundary

The Bayesian Hypothesis Comparator is an educational reasoning tool. It is not a medical, legal, financial or emergency diagnostic system.

Do not use estimated probabilities from the tool to diagnose a health condition, determine whether an emergency is genuine, make an investment decision or replace advice from an appropriately qualified professional.

In high-stakes situations, the comparator can help organise questions and identify assumptions. It should not be treated as an authoritative probability calculator when reliable data and expert assessment are required.

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Frequently asked questions

What is Bayes’ theorem in simple terms?

Bayes’ theorem is a rule for updating the probability of a hypothesis after observing new evidence. It combines your prior probability with how likely the evidence would be if the hypothesis were true.

What is the difference between a prior and a posterior?

The prior is your probability estimate before considering the new evidence. The posterior is your updated probability after incorporating that evidence.

What is a likelihood in Bayes’ theorem?

The likelihood is the probability of observing the evidence assuming that a particular hypothesis is true. It is written as (P(E | H)).

Why is the normaliser necessary?

The normaliser adds together the prior-times-likelihood scores for all the hypotheses. Dividing by it ensures that the resulting posterior probabilities add up to 1, or 100%.

Can Bayes’ theorem compare more than two hypotheses?

Yes. Calculate an unscaled score for every hypothesis, add all the scores to obtain the normaliser, and divide every score by that normaliser.

Do the prior probabilities have to add up to 100%?

When the hypotheses are being treated as mutually exclusive and collectively exhaustive, their prior probabilities should add up to 100%.

When the hypothesis set may be incomplete, the resulting posteriors should be interpreted as probabilities conditional on the explanations included.

Can AI estimate the priors and likelihoods for me?

AI can suggest illustrative estimates and help identify relevant factors. Unless those estimates are grounded in reliable data, they should be treated as assumptions rather than objective probabilities.

Is the hypothesis with the highest posterior definitely true?

No. It is simply the best-supported hypothesis within the specified model, given the chosen priors, likelihoods and evidence.

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From first impressions to disciplined updating

Bayesian reasoning does not require you to abandon intuition.

It asks you to make intuition answerable to comparison.

Instead of accepting the first explanation, you:

state the observation
→ generate alternatives
→ establish the priors
→ examine the evidence under each hypothesis
→ calculate the normaliser
→ update every hypothesis
→ remain open to further evidence

The central question is no longer:

“Can I tell a plausible story about what happened?”

It becomes:

“Which explanation best predicted the evidence, compared with the alternatives?”

That shift—from plausible storytelling to explicit comparison—is the foundation of better reasoning under uncertainty.

Compare your own hypotheses: Open the Bayesian Hypothesis Comparator GPT

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Related IQ Mindware guides

• Relational reasoning: discovering rules, constraints and relationships across changing problems.
• AI workflow cognition: protecting judgement when working with AI-generated information.
• Cognitive control: interrupting automatic responses and selecting a more deliberate reasoning process.
• Far transfer: testing whether a reasoning method remains useful when the surface problem changes.
• Evidence and claims: distinguishing plausible interpretations from conclusions supported by stronger evidence.

Selected references

Bakshy, E., Messing, S., and Adamic, L. A. (2015). Exposure to ideologically diverse news and opinion on Facebook. Science, 348, 1130–1132.

Bayes, T., and Price, R. (1763). An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53, 370–418.

Hirt, E. R., and Markman, K. D. (1995). Multiple explanation: A consider-an-alternative strategy for debiasing judgments. Journal of Personality and Social Psychology, 69, 1069–1086.

Pariser, E. (2011). The Filter Bubble: What the Internet Is Hiding from You. Penguin Press.