Bayes: meningitis and stiff neck

(The word in the question is read as "Bayesian (belief)" network.)

Bayesian network

A Bayesian network (belief network, causal network) is a probabilistic graphical model: a directed acyclic graph whose nodes are random variables and whose arcs show direct dependence (cause → effect). Each node has a conditional probability table P(node | parents), and the joint distribution is P(X1,...,Xn) = Π P(Xi | Parents(Xi)). It is used for reasoning under uncertainty (medical diagnosis, spam filtering, fault finding).

Types

1. Based on structure
   • Singly connected network (polytree): only one undirected path between any two nodes; inference is efficient (linear time).
   • Multiply connected network: more than one undirected path between some nodes (loops); inference is NP-hard in general.
2. Based on time
   • Static Bayesian network: variables at one point in time.
   • Dynamic Bayesian network (DBN): models how variables change over time (hidden Markov models are a special case).
3. Types of reasoning it supports: causal (cause → effect), diagnostic (effect → cause) and intercausal (explaining away).

Numerical problem

Let M = patient has meningitis, S = patient has a stiff neck.

Given: P(S | M) = 0.5, P(M) = 1/50000, P(S) = 1/20.

Bayes' theorem: P(M | S) = P(S | M) · P(M) / P(S)

= 0.5 × (1/50000) / (1/20)

= (0.5 / 50000) × 20

= 0.00001 × 20

= 0.0002 = 1/5000

Answer: the probability that a patient with a stiff neck has meningitis is 0.0002 (1 in 5000, or 0.02 %). It is small because meningitis is very rare, even though it often causes a stiff neck.