Bayesian statistics is a probabilistic approach that combines prior beliefs with new data to quantify uncertainty. It’s used for parameter estimation, decision-making under uncertainty, and in various fields like machine learning and Bayesian networks, where modeling complex, uncertain relationships is crucial. Bayesian statistics is a versatile tool for managing uncertainty and making informed decisions.
2023-11-07
What is Bayesian Statistics?
Bayes’ Theorem
The fundamental equation of Bayesian statistics is Bayes’ theorem:
\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]
Where:
P(A|B) is the probability of the hypothesis being true, given the evidence.
P(B|A) is the probability of the evidence occurring if the hypothesis is true.
P(A) is the prior probability of the hypothesis being true.
P(B) is the total probability of the evidence.
Dataset US Arrests
data(USArrests) head(USArrests)
Murder Assault UrbanPop Rape Alabama 13.2 236 58 21.2 Alaska 10.0 263 48 44.5 Arizona 8.1 294 80 31.0 Arkansas 8.8 190 50 19.5 California 9.0 276 91 40.6 Colorado 7.9 204 78 38.7
US Arrests for Murder
Before employing Bayesian statistics, it’s beneficial to create data visualizations to explore and gain insights into the various types of US arrests across different states. This approach allows for a comprehensive examination of the information pertaining to US arrest statistics and their regional distribution.
US Arrests for Assault
Comparison of UrbanPop against Murder, Rape, and Assault
Higher UrbanPop correlates positively with Murder, Assault, and Rape, indicating that more urbanized areas typically exhibit increased crime rates.
Using Bayes Theorem to predict future murders in US States
By using Bayes’ theorem, future murders can be predicted by utilizing the “USArrests” dataset. This will require installing “rstan” or “brms” packages. Stan is a programming language for Bayesian data analysis.
\[ P(\text{Murder}|\text{UrbanPop, Assault, Rape}) = \frac{P(\text{UrbanPop, Assault, Rape}|\text{Murder}) \cdot P(\text{Murder})}{P(\text{UrbanPop, Assault, Rape})} \]
Code for the Bayes Theorem to plot the USArrests
data(USArrests) murderdata <- USArrests$Murder urbanpopdata <- USArrests$UrbanPop assaultdata <- USArrests$Assault rapedata <- USArrests$Rape bayestheorem <- sum(murderdata == "Murder" & urbanpopdata * assaultdata * rapedata) / sum(urbanpopdata * assaultdata * rapedata)
Plot for the Bayes Theorem
hist(bayestheorem, main = "Histogram of the Bayes Theorem", xlab = "P(Murder | UrbanPop, Assault, Rape)", col = "pink")
