Contents

- 1Basic Rules of Probability
- 1.1Probability summation rule
- 1.2Probability multiplication rule
- 2Event sets used in examples section
- 2.1Event Set 1
- 2.2Event Set 2
- 3Bayes' Theorem
- 4Causal Graph
- 5D-Separation
- 5.1Type 1 (causal chain)
- 5.1.1Active Triplets :
- 5.1.2Inactive Triplets
- 5.2Type 2 (common cause)
- 5.2.1Active Triplets (Conditional Independence)
- 5.2.2Inactive Triplets
- 5.3Type 3 (common effect OR v-structure)
- 5.3.1Active Triplets
- 5.3.2Inactive Triplets (Absolute Independence)
- 5.4Type 4 (common effect on descendant OR v-structure with a bottom tail)
- 5.4.1Active Triplets
- 5.4.2
- 6Examples
- 6.1Cancer example
- 6.1.1Given
- 6.1.2Directly Inferred
- 6.1.3Computations for Cancer Example
- 6.2Two test Cancer example
- 6.2.1Given
- 6.2.2Computations for two test cancer example
- 6.3Example for Converged Conditionals on a Single node
- 6.3.1Given
- 6.3.2Computations for converged conditionals on a single node

## Basic Rules of Probability

represents probability of occurrence of events

Probability of any event can be calculated as the ratio of the number of chances favorable for event to the total number of chances:

*i.e.*;

where, is the total number of chances & is the number of chances favorable for event

Probability of any random event lies between &

*i.e.*;

Event is called *practically sure* if its probability is not exactly but very close to

*i.e.*;

Event is called *practically impossible* if its probability is not exactly but very close to

*i.e.*;

### Probability summation rule

The Probability that one of two (or more) mutually exclusive events occurs is equal to the sum of the Probabilities of these events

*i.e.*;

if events are ** mutually exclusive** and

**, the sum of their Probabilities is equal to**

*exhaustive**i.e.*;

From the above statement it follows that Probability of an event and its opposite(non occurrence) *i.e.*; , is equal to

**i.e.**;

### Probability multiplication rule

The Probability of the combination of two events (simultaneous occurrence) is equal to the Probability of one of them multiplied by the probability of the other provided that the first event has occured.

*i.e.*;

where is called ** conditional probability** of an event calculated for the condition that event has occurred.

## Event sets used in examples section

### Event Set 1

: Event of the occurrence of a specific type of Cancer

: Probability of occurrence of event C.

: Event that a test is positive for C. (Sometimes we will represent it with just a )

OR

### Event Set 2

: Event that the weather is Sunny.

: Event that the person gets a raise.

: Event that the person is happy.

## Bayes' Theorem

where:-

: is Posterior

: is Likelihood

: is Prior

: Marginal Likelihood

Bayes' Theorem gives us a framework to modify our beliefs in light of new evidences. Bayesian statistics gives us a solid mathematical means of incorporating our prior beliefs and evidence, to produce new posterior beliefs.

(NOTE: I will expand upon this in some future blog post).

## Causal Graph

** Causal Graphs** (also called

**) are directed acyclic graphs which are used to encode assumptions about the Probabilistic data represented by variables.**

*Causal Bayesian Networks*For example in the above graph, *joint Probability* represented by *Bayes' Network* is given by:

This example requires 10 values to represent the network. These are:

## D-Separation

D-separation is a criterion for deciding, from a given *causal graph*, whether a variable (OR set of variables) is independent of another variable (OR set of variables) , given a third variable (OR set of variables) . This has to do with *path* or *reachability*.

So when we think in terms of types of D-Separation we need to further break it down in terms of *Active* or *Inactive triplets. *

(NOTE: In listed types, I have highlighted node(s) in gray color to show involvement.)

### Type 1 (causal chain)

#### Active Triplets :

Given that is involved, & are independent.

*i.e.*;

#### Inactive Triplets

Given that is not involved, & are dependent

*i.e.*;

### Type 2 (common cause)

#### Active Triplets (Conditional Independence)

Given that is involved, & become independent (Conditional independence) .

*i.e.*;

Remember,* Conditional Independence* does not guarantee absolute independence

*i.e.*;

#### Inactive Triplets

Given that is not involved, & become dependent.

*i.e.*;

### Type 3 (common effect OR v-structure)

#### Active Triplets

Given that is involved, & become dependent (loose their absolute independence).

*i.e.*;

Also used for probabilistic reasoning called *explain away effect*.

#### Inactive Triplets (Absolute Independence)

Given that is not involved, & are independent(*Absolute Independence*).

*i.e.*;

### Type 4 (common effect on descendant OR v-structure with a bottom tail)

Type 4 is derived directly from Type 3.

#### Active Triplets

Given that is involved, and become dependent (loose their absolute independence).

*i.e.*;

## Examples

### Cancer example

#### Given

#### Directly Inferred

#### Computations for Cancer Example

**resolution:** (*Bayes' Rule*)

### Two test Cancer example

#### Given

#### Computations for two test cancer example

**resolution:** and (Since C is the root of both and , they become independent)

**resolution:** and (Since C is the root of both and , they become independent)

**resolution:** (since both and are conditionally independent in presence of )

### Example for Converged Conditionals on a Single node

#### Given

#### Computations for converged conditionals on a single node

resolution: (*Absolute Independence*)

resolution: (Since and are conditionally independent in absence of (*absolute independence*) )

[…] My notes on simple Causal Probability – moebiuscurve http://moebiuscurve.com/my-notes-on-simple-causal-probability/ […]

In you last example "Example for Converged Conditionals on a Single node", the given probability for "P(H|S,!R)=0.7" should instead read "P(H|!S,!R)=0.1"

Also it will be great if you could explain how in the two test cancer example, P(+1,+2|C) expands to P(+1|+2,C)P(+2|C)

Great article nonetheless.