Markov Models | Markov Chains | Markov Property | Applications | Part 1

Summary

The video script delves into the concept of the Markov process, a stochastic model where future events are dependent solely on the present state, without the need to understand the history of how the present state was reached. It introduces the Markov property and explains it with examples such as weather prediction, customer behavior, and baby activities. The script also covers the transition from first-order to higher-order Markov models and illustrates the concept with a weather example, including state transition diagrams and matrices. The explanation is aimed at helping viewers understand the foundational principles of Markov chains and their applications in various fields.

Takeaways

• 📚 A mock-up model or Markov process is a stochastic model where the future state depends only on the present state, without considering the sequence of events that led to the present state.
• 🔄 The Markov property is characterized by the future state (X3) being dependent solely on the current state (X2), not on any preceding states (X1).
• 🌡️ The script uses weather as an example of a Markov process, illustrating how the probability of tomorrow's weather depends only on today's weather state.
• 📈 The concept of a state transition diagram is introduced, showing the probabilities of transitioning from one state to another.
• 📊 A state transition matrix is explained as a way to represent the Markov process numerically, with rows representing current states and columns representing future states.
• 🔢 The importance of the initial state distribution is highlighted, which provides the probabilities of starting in each state at time zero.
• 🔑 The script explains the first-order Markov model, which only considers the immediate previous state to predict the next state.
• 🔄 The second-order Markov model is introduced, which takes into account the current state and the state two steps back to predict the next state.
• 🎲 The script mentions that higher-order Markov models can be constructed by considering more preceding states, up to an nth-order model.
• 📝 The video script is intended to be followed by a practical session where numerical questions on the Markov model will be solved, continuing the learning process.
• 📢 The speaker encourages viewers to subscribe to the channel for more content on the topic, emphasizing the value of community engagement in learning.

Q & A

What is a Markov process?

A Markov process is a stochastic model describing a sequence of possible events where the probability of each event depends only on the state attained in the previous event.

What are the three important elements in a Markov process?

The three important elements in a Markov process are the system, the event, and the state. The future state depends only on the current state.

Can you explain the Markov property?

The Markov property states that the probability of transitioning to a future state depends only on the present state and not on the sequence of events that preceded it.

What is a Markov chain?

A Markov chain is a sequence of random variables where the probability of each event depends only on the state of the previous event, following the Markov property.

What is the significance of the transition matrix in a Markov chain?

The transition matrix in a Markov chain represents the probabilities of moving from one state to another and is used to predict future states based on the current state.

What is the difference between a first-order and a second-order Markov model?

A first-order Markov model depends only on the immediate previous state, while a second-order Markov model considers both the immediate previous state and the state before that.

Can you provide an example of a Markov chain in everyday life?

An example of a Markov chain is weather prediction, where the likelihood of rain tomorrow depends only on today's weather, not the weather from several days ago.

What is the initial state distribution in the context of a Markov chain?

The initial state distribution in a Markov chain is the probability distribution of the system's state at the beginning of the process.

How is the state transition diagram related to the transition matrix?

The state transition diagram visually represents the possible transitions between states with their associated probabilities, which can be organized into a transition matrix for mathematical analysis.

Why is it important to understand the Markov property when analyzing a system?

Understanding the Markov property is important because it simplifies the analysis of a system by allowing us to focus only on the current state and its immediate transitions, without considering the entire history of the system.

How can the Markov model be applied in artificial intelligence and natural language processing?

The Markov model can be applied in artificial intelligence and natural language processing to predict sequences of events or words based on the current state, such as predicting the next word in a sentence given the previous words.

Outlines

00:00

🔍 Introduction to Markov Models

This paragraph introduces the concept of a mock-up model, also known as a Markov model, which is a stochastic process where the probability of future events depends only on the current state, not on the sequence of events that preceded it. The explanation includes the definition of a Markov process, its three key components: the system, the event, and the state, and emphasizes the Markov property where the future state (X3) depends solely on the current state (X2). Examples are given to illustrate the concept, such as weather prediction and customer behavior in purchasing mobile phones.

05:04

📚 Understanding Markov Chains and Transitions

The paragraph delves deeper into the Markov model, explaining the concept of Markov chains and how they are defined by the Markov property. It discusses the discrete time process and the conditional probability involved in transitioning from one state to another. The explanation includes the representation of these transitions in a state transition diagram and how to interpret the probabilities of moving from one state to another, using examples like a baby's behavior and a tourist's travel plans.

10:05

🌡️ Weather Prediction Using Markov Models

This paragraph uses the example of weather prediction to illustrate how Markov models can be applied. It explains the concept of first-order and second-order Markov models, where the first-order model depends only on the immediate previous state, while the second-order model considers the two preceding states. The paragraph provides a hypothetical scenario with probabilities of weather transitions from sunny to rainy or cloudy, and vice versa, demonstrating how these probabilities can be used to predict future weather states.

15:05

📊 State Transition Diagram and Matrix

The paragraph explains the construction of a state transition diagram and matrix for a Markov chain, using the weather prediction example. It describes how to represent the transition probabilities in a matrix format, where each element represents the probability of moving from one state to another. The importance of the transition matrix in understanding the dynamics of the Markov chain is highlighted, along with the concept of the initial state distribution, which is essential for making predictions.

20:06

🎓 Conclusion and Future Content Tease

The final paragraph wraps up the explanation of Markov processes, chains, state transition diagrams, and matrices. It informs viewers that the next video will continue with practical applications, specifically solving numerical questions on the Markov model. The speaker encourages viewers to subscribe to the channel for more content and provides a link to the next video in the description, emphasizing the value of understanding these concepts for further learning.

Keywords

💡Mock-up Model

A mock-up model is a simplified representation of a system or process, often used in various fields such as probability, artificial intelligence, and natural language processing to visualize and understand complex systems. In the video, the term is used to introduce the concept of a Markov process, which is a type of stochastic process that the video focuses on explaining.

💡Markov Process

A Markov process is a stochastic model where the probability of future states depends only on the current state and not on the sequence of events that preceded it. This property is known as the Markov property. The video script discusses the Markov process in detail, explaining how it is used to predict future events based solely on the present state.

💡Stochastic Process

A stochastic process is a collection of random variables indexed by time or space. It is used to represent systems whose future states are not entirely predictable but can be described in terms of probabilities. The video script uses the stochastic process as a broader category of which the Markov process is a specific type.

💡State

In the context of the video, a state refers to a condition or situation of a system at a particular time. The Markov process is concerned with the transition of states over time, where the future state is dependent on the current state. For example, the video uses weather conditions like sunny, rainy, and cloudy as states in a Markov chain.

💡Markov Property

The Markov property is the defining characteristic of a Markov process, stating that the future state depends only on the present state and is independent of the past states. The script emphasizes this property as the core concept of the Markov model, using it to explain how predictions are made in various scenarios.

💡State Space

The state space is the set of all possible states that a system can be in. In the video, the state space is used to describe the different conditions or events that can occur within a Markov chain, such as different weather conditions in the weather prediction example.

💡Transition Matrix

A transition matrix is a matrix used in Markov chains to describe the probabilities of moving from one state to another. The video script explains how to construct a transition matrix from a state transition diagram, which is essential for understanding the probabilities of state changes in a Markov process.

💡State Transition Diagram

A state transition diagram is a graphical representation of the possible transitions between states in a Markov chain. The video script uses this diagram to illustrate how states can change over time according to the Markov property, providing a visual aid to the concept of state transitions.

💡First-Order Markov Model

A first-order Markov model, as mentioned in the video, is a type of Markov chain where the next state depends only on the immediate previous state. The script explains this concept by contrasting it with higher-order Markov models, which consider additional previous states.

💡Second-Order Markov Model

A second-order Markov model extends the first-order model by considering the state two steps back in addition to the immediate previous state when predicting the next state. The video script introduces this concept to illustrate how the order of a Markov model can affect its predictive capabilities.

💡Conditional Probability

Conditional probability is the probability of an event given that another event has occurred. In the context of the video, conditional probability is used to describe the likelihood of transitioning from one state to another in a Markov chain, given the current state.

Highlights

Introduction to the concept of a mock-up model and its importance in various fields such as probability, artificial intelligence, and natural language processing.

Definition of the Markov process as a stochastic process where future events depend only on the present state, not on how the present state was reached.

Explanation of the three key components of a Markov process: the system, the event, and the state.

Illustration of the Markov property using a pictorial representation of states and transitions over time.

Clarification that in a Markov process, the future state depends solely on the current state, not on any preceding events or states.

Introduction of the concept of state space and the number of distinct states in a Markov process.

Examples of how the Markov model can be applied to real-world scenarios such as weather prediction and customer behavior.

Explanation of the first-order Markov model, which only considers the immediate previous state for predictions.

Introduction to second-order and higher-order Markov models, which take into account more than one preceding state.

Description of a state transition diagram as a visual tool to represent the probabilities of transitioning between states.

Conversion of a state transition diagram into a transition matrix format for mathematical representation.

Importance of the transition matrix in understanding the probabilities of state transitions in a Markov chain.

Explanation of the Markov property in the context of a Markov chain and its significance for predictions.

Definition of a Markov chain as a sequence of random variables that satisfy the Markov property.

Discussion on the practical applications of Markov chains in various fields and the types of problems they can solve.

Introduction of the concept of initial state distribution and its role in the beginning of a Markov chain.

Emphasis on the importance of understanding the Markov property for analyzing and predicting sequences of events.

Preview of upcoming video content that will solve numerical questions on the Markov model, encouraging viewers to follow for more.

Transcripts

you we are going to talk what is mock-up

model this is very important topic and

being used in probability in artificial

agency next language processing and many

more places so with us understand what

is the mock-up model or what is the

Markov process

so actually Markov process is simple

stochastic process in which distribution

of the future event only depend upon the

present state or the present event

without knowing how this present event

has been arrived and this is the same

where kind of definition has been given

by the Wikipedia and three things are

important when they were a-changin

system you can say the system you can

say the event and you can see the state

whatever you want to call and then the

future state depends upon the current

state these three things are important

so let us understand in a in pictorial

form it so suppose if you have the one

state another state another state and

say okay let me say this is the X 1

state X 2 State or extra state and say

if this is the current state and if this

is the past state and if this is the

future state and assume the time wise if

this is at the T then obviously this

past was the t minus 1 and this is the t

minus 2 so Markov process says this X 3

will depend upon X 2 only this X 3 you

know depend upon the ax 1 very important

so the future state

we depend upon the current state not how

this current estate has been arrived

this current estate arrived by any other

event or state it doesn't make any sense

in the Markov process Markov says this

next of the future event or state will

depend upon the current this is the

current states so it depend upon the

current estate and this is called the

Markov property means T so indeed this

is the t plus 1 this is the current T

and T minus 1 in future is a t plus 1 so

T plus 1 event will depend on T event

not how this T event has been arrived

and this is called the Markov property

and here you see we have the three state

so we can say X 1 X 2 and the X 3 are

the state space or we can say set of

states and in this process we have the n

distinct state so in this case n is

equal to 3 if you want to know what is

the when I call the event or the state

or the system what does it mean let me

explain you with some example suppose in

the case of weather in the way that we

have the three kind of weather sunny

rainy and the cloud this is called the

event state or the system so if two days

the rainy tomorrow

it would be the cloud or it could be

so this system are the randomly changing

system means it could be anything that's

well it is applicable that Markov

property is a Picabo here and if you

want to predict whether tomorrow

crowd would be there or not it will

depend upon just previous event mainly

it with no depend upon the sunny did

neither example if you want to predict

the customer which mobile product he is

going to by his the next mobile phone so

again if you have the three kind of

product we can assume this is the estate

or the event or the or the system so if

person having the self current phone he

is having the same Sun what is the

probability he is going to purchase

iPhone or the Nokia or he might be

continued with the Samsung so he knew

phone might purchase the samson only so

this is the example of the more copper

model next example funny example infant

a baby if baby is eating right now what

is the probability baby would be

sleeping or crying or the pairing so

these are the event if any police is

there visiting city what is the

probability he BPG to the next city that

is depend upon the which city currently

he is there and the game reserved for

any game they are the three

possibilities either trio can lose the

game or draw the game or the winter game

so let me explain in other way in more

detail I am going to explain the same

thing so what happened if suppose I have

the state or the event

and this is my current so we are the or

here say these are the past and this are

the future right so if this is the X and

this is X event at the time of T this

would be the T verse 1 this would be the

T plus 2 this is the t minus 1 and t

minus 2 just for denote purpose

this state is XT minus 2 X t minus 1 X T

XT plus 1 X T plus 2 this ax ax T first

to depend upon this event this depend

upon this event and this is the sequence

of event but this is independent of the

time so what important thing is that

this these are the state and this

process happening of independent of the

time or you can say discrete time okay

this is not the continuous process this

is the discrete time process okay now

according to the Markov property what he

says that X t1 will depend upon the only

de this one it will not depend upon the

other one felt this indicator I am

saying that of you getting confused to

remove this so this we depend upon the T

first one depend upon the t t ax T

depend upon the XT minus 1 so if I say

the probability of XT you depend upon XT

minus 1 and you have already and know

this is the conditional probability

[Music]

and and suppose if this state name is

the I and this is J we can also put like

this T IJ means from I to J probability

would be J probability the same thing

but just giving the state name also so

this is the property of the ax T the

same way if you want to know the

property of the xt-1 it will happen X T

plus 1 given that ax T and as I say this

is called the Markov property and this

is called the Markov chain any chain

this is the chain belt any chain that

follow the Markov property that chain is

called the Markov chain

ok I am repeating the same thing if this

XT depend only on the XT minus 1 this XT

person only depend upon the ax T this

property is called the Markov property

for any chain of event that follow the

Markov property we can say that chain is

called the Markov chain let me define

what is the Markov chain in a proper

definition so if this is the sequence of

the distinct random variable then this

chain x1 x2 x3 would call the Markov

chain

EPT satisfy the Markov property and what

is Markov property means if you want to

predict the xt purse 1 state the state

name of at the xt 1 is equal to s will

not depend upon this fool if we just

only depend upon the previous

so XT plus 1 only depend upon the ax T

it will not depend upon the complete one

this is called the Markov property and

if this is followed then this equation

is called the Markov chain okay if you

understand this then and this is called

the first order Markov model why it is

called the first order because this

depend upon on the immediate this up

depend upon on the immediate so let me

here is this part if you understand the

first order Markov let me explain the

second-order Markov model second-order

Markov model says it not only depend

upon the immediate one but immediate per

s1 so if the case of the xt xt in first

order what was that it only depend upon

xt minus 1 but in the second order it

will depend upon x t minus 1 and also

depend upon coma xt minus 2 this is

called the second order so see it is the

second two here so this is called the

second order what was the first order

just for your reference xt t minus 1 so

this is the first order if you want to

say the third order or the n order n

order you can understand so the X 3

would be the 1 2 T this is called the

third order if you understand all those

things let me take one example and I we

explain how the questions come in the

and what we have to solve so this time I

will be take the example of weather so

it will easy to explain and in the

weather I have already explained we have

the three event event sunny rainy and

the cloud so suppose this is the sunny

and this is the rainy and this is a

crowd Claudia fan is the third evil to

vent right and this property has been

given if today is sunny then what is the

property of tomorrow DB the sunny this

is the point

it means 80% it has been given right and

if today sunny what is the probability

for tomorrow is the raining it is the

0.15 so this is the 15% okay and as we

know we have only the three event then

all together three event property must

be equal to one okay so even if this is

not given we can calculate 0.5 plus 0.5

first 1.15 how much I should add so we

become the 1 so obviously it should be

the point 0 5 so if today is sunny

what is the poverty of tomorrow be sunny

0.8 what is the property tomorrow

raining 0.15 what is the property

tomorrow it be the crowd this is the

point 0 5 the same thing if you take for

the rainy if today's rainy what is the

poverty tomorrow is the rainy this is

the 0.6% and what is the property

tomorrow is the crowd so point zero two

percent then we can calculate this part

all safe so it becomes the G 0.38 right

the same way if today's crowd what is

the property of tomorrow is the crowd

0-2 and if it is that what is the

poverty of tomorrow is the sunny 0.75

then you can calculate about what is the

property of tomorrow will be the rainy

so point u5 in the exam it be already

given and this is called the state

transition diagram stilt tell Angie cell

daga here if you see what is the state

space or the set of space s is equal to

as a and C and one more information you

will be have that initial state

distribution what is that initial state

distribution or initial state

probability that would be a pageant by

the PI and very square 20.7 1 to 5 and

0.5 we can say 0.05 idea so this is

called the initial state of initial

state of all the initial probability of

or or the state is has been given so you

would have the three informants in the

exam sun-ho this state transition

diagram and the initial state

distribution so this one information to

information in the three information now

if I have this thing we can convert into

the matrix format so let me convert into

the matter

form is also important so this is the as

I see as I see you understand this this

part is the current state this as sunny

rainy and this one is the current state

this sunny rainy is the next or the

future future state

[Music]

so if I want to convert this transition

diagram into and this is called the

transition matrix so let me give that

name this is called the transition

matrix why it is called the transition

matrix means of more both children

[Music]

because it's transit from one state to

another state so now let me put the data

okay so what is the data here if today

is sunny what is the property tomorrow

be any point eight so you can give the

point eight if two days sunny what is

the property of tomorrow debe de rainey

it is the point one five this point

eight is this point eight right this

point one five this is the this point

one five and this point six point zero

zero five is this one right so that's

why you can draw this diagram right and

this is called the transition matrix

change and each one is called the each

each value is called the transition

probability

if you can define pIJ is called - same I

told you P t1 is equal to J and D is

equal to I means the H behavior and if

you have noticed this this the this is

called the view right so if you add this

one it should cause the 1 right

distribution probability so each would

be equal to have you the 1 and these are

the state right so we have all the state

we have the 3 state so this matrix would

be the TV n 2 3 if you have the n state

then this matrix would be the and into n

matrix so this was the complete so I

explained already what is Markov process

Marko poverty Markov chain I explained

the state transition diagram I explained

the state transition matrix more for

chair for Markov chain what are the

different examples are there so in next

video I am going to solve at least three

questions

Numerical questions on the Markov model

it is highly recommended and it would be

the continuous of this video the next

video

the description I will be given by the

video in video description I believe the

link of that next video and one more

thing

they don't forget to subscribe my

channel and it motivates me a lot to

make many more videos on that thank you

very much