# Bayesian network introduction

Hello this is most definitely a question for dummies i feel. The question concern bayesian network and inference thereof. I've heard about the product rule, bayes theorem and the chain rule. However this bayesian network still keeps eluding me... So i hope someone could give a quick recap on how i should think the following assignment which i have created based on some research.

So giving the following problem statement:

The Bayesian Network LMV below has three nodes for boolean variables, L, M and V. The probabilities for L and M are: P (M = true) = 0.2 and P (L = true) = 0.7. The conditional probabilities for variable V are as shown in the table below:

Question: What value is the value of P(V = false | L = false) ?

How is this computed? This realy confuses me because i dont get how we can isolate P(V = False | L= False) when the image suggest that we only have the conditional probability of V given both L and M? Any help appreciated :)

## 1 Answer

The conditional probability is : \begin{align}\mathsf P(V{=}f\mid L{=}f)~&=~ \dfrac{\mathsf P(V{=}f, L{=}f)}{\mathsf P(L{=}f)}&&\text{by definition}\\[3ex]&=~\dfrac{\displaystyle\sum\limits_{m\in\{f,t\}} \mathsf P(V{=}f,L{=}f,M{=}m)}{\mathsf P(L{=}f)}&&\text{by Law of Total Probability}\\[3ex] &=~\dfrac{\displaystyle\sum\limits_{m\in\{f,t\}}\mathsf P(V{=}f\mid L{=}f,M{=}m)~\mathsf P(L{=}f)~\mathsf P(M{=}m)}{\mathsf P(L{=}f)}&&\text{by DAG Factorisation}\\[1ex] &~~\vdots\end{align}

Well, you can take it from here, I am sure.