Bayesian Belief Networks

Are a representation of the probability of variables and their conditional dependencies through a graphical network.

Where,

• Node: Variable and it’s associated Probability distribution.
• Parent Node: Prior.
• Child Node: Conditional Dependency.

$$\mathrm{P}(e)=\sum _{d\in \mathcal{D}}\mathrm{P}(e\mid d)\cdot \underset{\mathrm{P}(d)}{\underbrace{\sum _{c\in \mathcal{C}}\sum _{a\in \mathcal{A}}\mathrm{P}(d\mid c,a)\cdot \mathrm{P}(a)\cdot \stackrel{\mathrm{P}(c)}{\overbrace{\sum _{b\in \mathcal{B}}\mathrm{P}(c\mid b)\cdot \mathrm{P}(b)}}}}$$

1. Node A: $P(a)$
2. Node B: $P(b)$
3. Node C: $P(c)={\sum }_{b\in B}P(c|b)\cdot P(b)$
4. Node D: $P(d)={\sum }_{a\in A}{\sum }_{c\in C}P(d|a,c)\cdot P(a)\cdot P(c)$
5. Node E: $P(e)={\sum }_{d\in D}P(e|d)\cdot P(d)$

Applications

• Medical Diagnosis: It can be used to model relationships between symptoms, diseases and risk factors.

graph TD
     Diseases
    D1[Heart Disease]
    D2[Diabetes]
    D3[Hypertension]

     Patient Characteristics to Diseases
    C1 --> D1 & D2 & D3
    C2 --> D1 & D2 & D3
    C3 --> D1 & D2 & D3

     Styling
    classDef characteristics fill:stroke-width:2px;
    classDef diseases fill:stroke-width:2px;
    classDef symptoms fill:stroke-width:2px;

    class C1,C2,C3 characteristics;
    class D1,D2,D3 diseases;
    class S1,S2,S3,S4 symptoms;

    %% Title
    title[Medical Diagnosis Bayesian Network]
    style title fill:none,stroke:none