Bayesian Network: Inference¶

Table of Content¶

• Introduction
  • Probabilistic Models
  • Conditional Independence
• Introduction to Bayes’ Networks

Introduction¶

• ## Probabilistic Models Suppose insurance wants to insure a personal car forone year Criteria to be considered Individual information

  • machine type
  • Age
  • education
  • History

  • ..

    Based on these criteria, consider how likely a person is to have an accident and examine the amount and cost of the accident.

    We face uncertainty in determining the amount of insurance and we can not model it exactly Now to make a smart decision we need to have a model based on statistics Then, from this statistical modeling, we can draw a statistical conclusion from how much it is probable that the person will suffer damages in the next year and how much it will have to be paid by the insurance company.

    How to build a statistical model ?
    How to use it to make the right decision ?

    Let us first give an example : A Reasoning Scenario

    I'm at work, neighbor John calls to say that my alram is ringing, but neighbor Mary doesn't call. Sometimes it's set off by minor earthquakes. Is there a burglar? To answer this question, we must first make a statistical model and then draw a statistical inference. So we have statistical language:

  • Variables:

    • B: + | - (boolean) (true for exsit burglar).
    • J: + | - (boolean) (true if John call me).
    • M: + | - (boolean) (true if Mery call me).

    • A: + | - (boolean) (true if alarm is ringing).

      P(B=+b , J=+j , M=+m)/p(A) A=( J=+j , M=-n)

      Which is the interpretation of the above speech. Are the variables independent of each other?
      Should John call us randomly, is there any trust in his words?
      So it must be said that these variables will have an interdependence. This is not how the probability can be calculated and we need more information, which we will get acquainted with later, a concept called joint property distribution
      
      Now suppose we know the probability that a series of random variables will occur together, in this example John and Marry and alarms and earthquakes (in this example each of these variables is binary and makes things easier for us). We have five variables that must create two to the power of five states Which creates a 32-rows table whose columns are variables and one column for the probability . Now, in the probability of calculating theft in this example, there is no earthquake in our conditional probability, and the states in which the earthquake occurs and does not occur must also be calculated in the table.
      The difficulties of this can be said:

    • 1. The number of random variables is so much, which causes the number of rows to increase exponentially,
    • 1. The difficulty in the inference section.

  • ### independence • Two variables $X$ and $Y$ are independent if:

    $$ \forall x,y: P(X=x, Y=y) = P(X=x)P(Y=y) $$

    This says that their joint distribution factors into a product two simpler distributions. Independence is a simplifying modeling assumption. #### Example:

    

    So we turn it into two more tables where we can work with two variables instead of three.
    and make order from 2 power n to 2n .

• ### Conditional Independence Let us first give an example:

  Suppose we have three variables in tooth decay : 1.cavity
  2.toothache
  3.catch
  • If I have a cavity, the probability that the probe catches in it.

  doesn't depend on whether I have a toothache:   
  
  

  • P(+catch | +toothache, +cavity) = p(+catch | +cavity)

    • The same independence holds if I don’t have a cavity:

    • P(+catch | +toothache, -cavity) = p(+catch| -cavity)

    Now to reduce the dependencies, imagine that the person's tooth has a cavity. Two variables of the person have a toothache and the catch of a bad tooth is defined as two separate variables.
    The catch and toothache dependence is due to the cavity. If we know that the cavity has occurred, it is as if we have eliminated the catch and toothache dependence.

    Equivalent statements:

  1. P(Toothache | Catch , Cavity) = P(Toothache | Cavity)

  2. P(Toothache, Catch | Cavity) = P(Toothache | Cavity) P(Catch | Cavity)

     One can be derived from the other easily. #### Example :

 Here are three variables:  
1. smoke
2. Fire
3. alarms  

Are these three variables conditional or not? Can a variable be found here that causes two other things? Here, unlike the previous example, when the breakdown caused two other reasons, there is a chain that connects this fire and alarm to the existence of a smoke variable, and if we remove smoke from the environment in some way, the other two variables, fire and alarm, have nothing to do with They do not have each other

In general, all world events can be divided into common and causal chains

Introduction to Bayes’ Networks¶

• ### Definition • Two problems with using full joint distribution tables as our probabilistic models:

  1. Unless there are only a few variables, the joint is WAY too big to represent explicitly
  2. Hard to learn (estimate) anything empirically about more than a few variables at a time

  Bayes' nets: a technique for describing complex joint distributions (models) using simple, local distributions (conditional probabilities)

  • More properly called graphical models
  • We describe how variables locally interact
  • Local interactions chain together to give global, indirect interactions
    • we’ll be vague about how these interactions are specified

• ### Graphical Model Notation
  • Nodes: variables (with domains)
    • Can be assigned (observed) or unassigned (unobserved)
  • Arcs: interactions
    • Similar to CSP constraints
    • Indicate
      direct influence between variables
    • Formally: encode conditional independence (more later)
    • For now: imagine that arrows mean direct causation (in general, they don’t!)

The question that was first created for us for insurance now we show with Bayes Net :