Markov Chains and Markov Processes

Here the concept of conditional probability discussed in previous days  blogs (July 14 - July 30, 2017) is extended to Markov chain and Markov Processes. 

Let,

P(Rainy Day) = 0.3

P(Dry Day) = 0.7

We know that if it rains today then there are high chances that it might rain tomorrow that is the probability that it will rain tomorrow is high.  So the conditional probability that it will rain tomorrow given that it rains today is higher than the unconditional probability that it will rain tomorrow.

Notation wise

P(Rain Tomorrow|Rain Today)>P(Rain Tomorrow)

Conditional Probability>Unconditional Probability

P(Rains tomorrow)=P(Dry Today)P(Rains Tomorrow|Dry Today)+P(Rains Today)P(Rains Tomorrow|Rain Today)

Markov Processes are systems where outcome of the current state is highly dependent on the outcome of immediately preceding state.  Examples can be weather systems and state of well being of an individual. Here outcome of state today is heavily dependent on the  outcome of immediately preceding state. So probability that it rained one month ago has less influence on the probability that it rained today than the probability that it rained yesterday. State of well being of an individual can also be explained in the following manner. So these examples explain special case of conditional probability namely markov chains. This will be discussed in detail in the next blog.

Expectation and Conditional Expectation

The discussion on probability and conditional probability (previous day's blog) is continued to expectation and conditional expectation.

Let Z be a random variable denoting money spent by the state on the cardiac care of a citizen below 40 years.

E(Z) is the average money spent by the state per citizen below 40 years on its cardiac care.

E(Z)=P(Y1)E(Z|Y1)+P(Y2)E(Z|Y2)

=(Probability of a Heart attack before 40)(Average money spent by the state on caradiac care of a citizen with an incidence of heart attack before 40 years)+(Probability of no heart attack before 40 years)(Average money spent by the state on cardiac care of a citizen with no incidence of heart attack before 40)

so,

Expectation=Probability(condition1)* Conditional expectation+Probability(condition2)* Conditional expectation

Probability and Conditional Probability

The discussion is being continued. 

So,

Y1 stands for  incidence of a heart attack before 40.

Y2 stands for no incidence of a heart attack before 40.

X stands for the event of Blood pressure and Blood sugar beyond normal limits and cholesterol within normal limits.

Expenses on Cardiac Care

Probability and Conditional Probability

Probability and Conditional Probability

This is in continuation of the discussion on Probability and Conditional Probability of the Previous Blog.
Now we are using a general notation. Then the equation get's reduced to the following form.

Probability and Conditional Probability

Visualization of Probability & Conditional Probability

The tree diagram given in the image below helps visualize the main difference between conditional and unconditional probability. Here we continue with the example of the previous blog where conditions leading to a heart attack before 40 years are minutely analyzed.  

Visualization of conditional probability

Probability and Conditional Probability

Understanding conditional probability through Venn Diagrams

Conditional probability is normally computed under some additional conditions and "unconditional probability" is the probability where no additional conditions are provided. This is illustrated with example mentioned in the previous blog. Here the probability of having a heart attack before 40 years of age is minutely analyzed by a Venn Diagram. The venn diagram  below shows a rectangle representing all people ( in an area) below 40 years. The circles H, S, C and N  are those who have suffered a heart attack. The other region represents those without  a heart attack. The shaded region in the diagram below gives the  probability that a patient with  high level blood pressure and sugar but the level of cholesterol is within limits has a heart attack before 40 years of age. This is the ratio of number of elements in the shaded region divided by the total number of elements in H, S, C and N.

Understanding Venn Diagrams

Incidence of Heart Attack Below 40 years of Age

We are interested in finding the probability of a heart attack below 40 years of age. Venn diagrams help visualize the scenario of people having heart attack before age of 40 including those with high blood pressure denoted by H, high sugar denoted by S and high cholesterol denoted by C or a combination of these. N denotes those individuals who have stroke before 40 and have blood pressure, sugar and cholesterol within normal limits. The image below shows the venn diagram of those cases where the blood pressure and sugar are beyond normal limits but cholesterol is within the limits.  This way all the possible causes resulting to a heart attack before 40 years can be minutely analyzed and it's probability can be calculated. This analysis is based on collected data and can aid in correct diagnosis.

This venn diagram shows probability of a stroke with high levels of blood pressure and blood sugar