Markov Models | Markov Chains | Markov Property | Applications | Part 1
Summary
TLDRThe 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
🔍 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.
📚 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.
🌡️ 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.
📊 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.
🎓 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.
Mindmap
Keywords
💡Mock-up Model
💡Markov Process
💡Stochastic Process
💡State
💡Markov Property
💡State Space
💡Transition Matrix
💡State Transition Diagram
💡First-Order Markov Model
💡Second-Order Markov Model
💡Conditional Probability
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
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