week2

I.

Bayes’ Theorem is a fundamental concept in probability theory that allows us to update our beliefs or probabilities based on new evidence. It provides a way to calculate the probability of a hypothesis or event given prior knowledge and new information. The theorem is stated as: \[P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\] where:

• \(P(A|B)\) represents the probability of event A given event B.

• \(P(B|A)\) is the probability of event B given event A.

• \(P(A)\) is the prior probability of event A.

• \(P(B)\) is the prior probability of event B.

An example of Bayes’ Theorem in action is medical diagnosis. Let’s say we want to determine the probability of a person having a certain disease given that they exhibit a specific set of symptoms. We have the prior probability of the disease occurring in the general population \(P(A)\), the probability of observing those symptoms in people with the disease \(P(B|A)\), and the probability of observing those symptoms in the general population \(P(B)\). By applying Bayes’ Theorem, we can calculate the updated probability \(P(A|B)\) of the person having the disease based on the symptoms they exhibit.

II.

library(graph)

## 载入需要的程辑包：BiocGenerics

## 
## 载入程辑包：'BiocGenerics'

## The following objects are masked from 'package:stats':
## 
##     IQR, mad, sd, var, xtabs

## The following objects are masked from 'package:base':
## 
##     anyDuplicated, aperm, append, as.data.frame, basename, cbind,
##     colnames, dirname, do.call, duplicated, eval, evalq, Filter, Find,
##     get, grep, grepl, intersect, is.unsorted, lapply, Map, mapply,
##     match, mget, order, paste, pmax, pmax.int, pmin, pmin.int,
##     Position, rank, rbind, Reduce, rownames, sapply, setdiff, sort,
##     table, tapply, union, unique, unsplit, which.max, which.min

library(Rgraphviz)

## 载入需要的程辑包：grid

library(graph)
library(Rgraphviz)

# Probability values
probAcademic <- 0.35
probSporting <- 0.20
probNoEvent <- 0.45

probFullGivenAcademic <- 0.25
probFullGivenSporting <- 0.70
probFullGivenNoEvent <- 0.05

probNotAcademic <- 1 - probAcademic
probNotSporting <- 1 - probSporting
probFull <- (probAcademic * probFullGivenAcademic) + 
            (probSporting * probFullGivenSporting) + 
            (probNoEvent * probFullGivenNoEvent)

# Probability of sporting event given full
probSportingGivenFull <- (probSporting * probFullGivenSporting) / probFull
print(probSportingGivenFull)

## [1] 0.56

# Node names
node1 <- "Start"
node2 <- "Academic"
node3 <- "Sporting"
node4 <- "NoEvent"
node5 <- "FullGivenAcademic"
node6 <- "NotFullGivenAcademic"
node7 <- "FullGivenSporting"
node8 <- "NotFullGivenSporting"
node9 <- "FullGivenNoEvent"
node10 <- "NotFullGivenNoEvent"

nodeNames <- c(node1, node2, node3, node4, node5, node6, node7, node8, node9, node10)

rEG <- new("graphNEL", nodes = nodeNames, edgemode = "directed")

rEG <- addEdge(node1, node2, rEG, 1)
rEG <- addEdge(node1, node3, rEG, 1)
rEG <- addEdge(node1, node4, rEG, 1)
rEG <- addEdge(node2, node5, rEG, 1)
rEG <- addEdge(node2, node6, rEG, 1)
rEG <- addEdge(node3, node7, rEG, 1)
rEG <- addEdge(node3, node8, rEG, 1)
rEG <- addEdge(node4, node9, rEG, 1)
rEG <- addEdge(node4, node10, rEG, 1)

eAttrs <- list()
eAttrs$label <- c(toString(probAcademic), toString(probSporting), toString(probNoEvent),
                  toString(probFullGivenAcademic), toString(1 - probFullGivenAcademic),
                  toString(probFullGivenSporting), toString(1 - probFullGivenSporting),
                  toString(probFullGivenNoEvent), toString(1 - probFullGivenNoEvent))

q <- edgeNames(rEG)
names(eAttrs$label) <- c(q[1], q[2], q[3], q[4], q[5], q[6], q[7], q[8], q[9])

# Tree diagram
attributes <- list(
  node = list(fillcolor = "red", fontsize = "20"),
  edge = list(color = "black"),
  graph = list(rankdir = "LR")
)

plot(rEG, edgeAttrs = eAttrs, attrs = attributes)