Bayes Theorem
A statistical method used in spatial analysis to update the probability of a hypothesis based on observed data, relevant for spatial modelling and prediction.
What is Bayes Theorem?
Bayes' Theorem is a fundamental concept in probability theory and statistics that describes how to update the probability of a hypothesis based on new evidence or information.
The theorem is expressed as:
P(A∣B)=P(B∣A)⋅P(A)P(B)P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}P(A∣B)=P(B)P(B∣A)⋅P(A)
Where:
P(A∣B)P(A|B)P(A∣B) is the posterior probability: the probability of event A given that B has occurred.
P(B∣A)P(B|A)P(B∣A) is the likelihood: the probability of observing event B given that A is true.
P(A)P(A)P(A) is the prior probability of A (before considering B).
P(B)P(B)P(B) is the total probability of B occurring.
In simple terms:
Bayes' Theorem allows us to revise existing predictions or beliefs based on new data. It's widely used in machine learning, medical diagnosis, spam filtering, and risk assessment, where decision-making is based on probabilities that get refined as more information becomes available.
Related Keywords
In machine learning, the Bayes Theorem modifies a hypothesis's probability in response to fresh data. It is essential for applications like spam detection and medical diagnosis in models like Naive Bayes.
Suppose 2% of people have a disease. A test correctly detects it 95% of the time but gives a false positive 5% of the time. If someone tests positive, what’s the chance they really have it?
Using Bayes’ Theorem:
𝑃
(
Have Disease | Positive
)
=
0.95
×
0.02
(
0.95
×
0.02
)
+
(
0.05
×
0.98
)
≈
0.28
P(Have Disease | Positive)=
(0.95×0.02)+(0.05×0.98)
0.95×0.02
≈0.28
So even with a positive test, there’s only a 28% chance the person actually has the disease.
When new information becomes available, probabilities are updated using Bayes' Theorem. It is used in financial risk assessment, spam filtering, medical diagnosis, and machine learning to make decisions in the face of ambiguity.
The Bayes Theorem adjusts an event's probability in light of fresh data. Similar to changing your guess when you discover new information, it integrates past knowledge with fresh data to provide a more accurate chance.