A show than any random element from the space

A
probabilistic model or method is based on the theory of probability or the fact
that randomness has a vital role in predicting future events. The opposite
is deterministic, that is the opposite of randomness. This explains how something
can be predicted accurately, without the added complication of randomness.

Probabilistic
methods deal random variables and probability
distributions into the model of a
phenomenon or event. While a deterministic model produces a single possible
outcome for an event, a probabilistic method produces a probability distribution as a solution
as shown in Figure 1. These models take into account the
fact that we can rarely know everything about a situation. There’s nearly
always an element of randomness to take into account. For example, life
insurance is based on the fact we know with certainty that we will die,
but we don’t know when. These models can be part deterministic and partially
random or completely random.

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.

Fig. 1. A normal distribution curve (Also called bell curve)

The
probabilistic method, first introduced by Paul Erdos, is a
way to prove the existence of a structure with certain properties in combinatorics. The idea is that you create a probability
space, and choosing elements at random
show than any random element from the space has both a positive probability and
the properties sought after. The method is commonly used in a various areas, such
as quantum mechanics, statistical physics, and theoretical computer science.

Probabilistic
Graphical Model have two types

1) Bayesian Networks

2) Markov Random Field (MRF)

I am going to describe
one probabilistic model Markov Random Field (MRF).

1-
Markov Random Field

A Markov Random
Field (MRF) is a graphical model which has joint probability distribution. It contains
an undirected graph G= (N, E) in which E shows links or edges and N shows
random variables. Suppose

is
the set of random variables and S is a set of nodes. Both sets S and

associated with each other. Then, the
edges E encode conditional
independence relationships using the following rule: there are three disjoint
subsets of vertices A, B, and C,

is
conditionally independent of

given

if
there is no way from any vertices in A to any node in B that doesn’t pass
through a node of C. Then it will be like

.
The subsets are dependent if such a path does exists. The neighbor set

of a node n is can be defined
as the set of all vertices which are linked to n via edges in the graph:

}

Given its neighbor set, a node n is
independent of all other vertices in the graph. Therefore, for the conditional
probability of

This
is called the Markov property, and this is the reason why this model gets this
name markov random field. Figure 2 shows the concept of markov random field.

Fig.2. Given the grey nodes, the black node is conditionally independent
of all other nodes

When
the joint probability density of
the random variables is strictly positive, it is also referred to as a Gibbs random field, because, according
to the Hammersley Clifford theorem,
it can then be represented by a Gibbs measure for
an appropriate (locally defined) energy function. The prototypical Markov
random field is the Ising model;
indeed, the Markov random field was introduced as the general setting for the
Ising model. In the domain of artificial
intelligence, a Markov random field is used to model various low to
mid-level tasks in image processing and computer vision.

2-
Why Markov Models?

Markov model is
very simple and efficient statistical model in which we don’t suppose that all
variables are independent; we suppose that there are significant dependencies,
but also conditional independencies which we can take advantage of Markov
models show up in a lot of ways in computer vision. For example:

·
The
shape of contours can be modeled as a Markov chain. In this method, the future
course of a contour which passes from point x might rely on the tangent of the
contour at x, but not on the shape of the contour prior to reaching x. This is
very simple, but is implicitly used in various methods to contour segmentation
and perceptual grouping.

·
In
modeling texture, a patch of an image can be represented by output of a large
number of filters. If we suppose that few of these filter outputs are dependent
on each other, but that there is conditional independence among various filter
outputs, then we can obtain a tractable model for use in texture synthesis.

·
In
modeling texture, we suppose that the appearance of a pixel in a textured image
patch will reply on some of the neighboring pixels, however, it will be
independent of the rest of the image.

·
In
action recognition, we usually model the movements which consist of an action
as a Markov chain. For example, that if I want to predict about what will
happen when somebody is in the middle of leaving work and going for his car,
that this will depend on their current state i.e., what Is position of the car
and person right now, and what is their current trajectory but not on old
states (once you are standing in front of the car, how you will get in doesn’t
really reply on how you left the building and reached in front of the car).

3-
Markov Chains

A Markov chain is a stochastic
model describing a sequence of possible events in which the probability of each
event depends only on the state attained in the previous event. Diffusion of a
single particle can be thought of as a Markov chain. We can also use Markov
chains to model contours, and they are used, implicitly or explicitly, in many
contour-based segmentation technique. The most important benefit of 1D Markov
models is that they lend themselves to dynamic programming solutions.

In a Markov chain,
there are sequence of random variables, which describes the state of a system,

.  According to Markov conditional independence
property, each state only depends on the old state i.e.

P (

Markov chains have
few significant properties. By solving eigenvalue problem, their steady state
can be found. If a Markov chain involves transitions among a discrete set of
states, it’s very helpful to describe these transitions from state to state
using matrices and vectors.

4-
Difference between
Markov Chains and Markov Random Field

·
A (discrete
time) Markov chain:

A Markov chain is a sequence of
random variables

,

that satisfy

Pr (

·
A Markov Random
Field:

A group  of random variables x1,x2,…,xix1,x2,…,xi that satisfy

Pr (

After comparing the above two, we
can see that for a discrete time Markov chain, we can express the conditional probability

Pr(xi|x1,…,xi?1)=Pr(xi|xi?1)=Pr(xi|Neighbor(xi))Pr(xi|x1,…,xi?1)=Pr(xi|xi?1)=Pr(xi|Neigh-bor(xi))

Pr(

As in the sequence

,

obviously is

is

only neighbor. Therefore, the Markov
field is a simple extension of the Markov chain, also showed by the name
transition: chain ? field.

Furthermore, a Markov chain is illustrated
by a directed graph while a Markov random field showed by undirected. However,
every directed graph as an undirected version, and the undirected graph version
of a Markov chain looks the same as the directed version except without the
arrow directions 15.

5-
Extension of
Markov Chain in two Dimension of 2-D MRF

Markov chain has
extension into 2 dimensions of 2-dD Markov Random field.

·
Lattice

Rather than a
subset of the integer line, R ? Z = {0, ±1, ±2, .
. .}, consider some deterministic lattice structure or grid structure (for
instance R ?

= = {(x, y) : x, y ?
Z}). All coordinate pair is basically the location of a site i ?
S ? Z. This is shown in Figure 3.

Fig.3.
Various types of Lattice

The square lattice
has value |?i| = 4, the triangular lattice has value |?i| = 6, and the
honeycomb lattice has value |?i| = 3.

6-
Properties of
Markov Random Field

There are three
properties of Markov Random Field

·
Pairwise Markov
Property:
Two nodes that are not directly connected can be made independent given all
other node.

·
Local Markov
Property:
A set of the nodes (variables) can be made independent form the rest of the
node variables given its immediate neighbors.

·
Global Markov
Property:
A vertex set A is independent of the vertex set B (both B and A are disjoint)
given set C if all chains among A and B intersect C.

7-
Types of Markov
Random Field

·
Pairwise Markov
Random Field: Associated
energy is factorized into a sum of potential functions defined on cliques of
order strictly less than three.

·
Higher Order
Models:
Potential of factor term depends on more than two random variable. Difficult to
train and get inference form.

·
Conditional Random
Field:
In conditional random field observed random variable are not included in
graphical model. Only unobserved variables are latent variable.

8-
Inference Methods
in MRF

There are various
methods for inference process in MRF. Inference process has an important role
in MRD for classification purpose. Some of these inference methods are
described as follow:

·
Graph Cut
Algorithm: This
algorithm utilizes max flow problem to draw useful inference from Markov Random
field. Graph Cut algorithm rapidly compute a local minimum, in the sense that no
“permitted move” will produce a labeling with lower energy.

·
Belief Propagation
Algorithm:
1) Relies upon message passing of various vertices in graph. 2) Exact Method
for Tree models. 3) Improvement in complexity with the help of dynamic
programming.

·
Loopy Belief Propagation: This is approximate
version of belief propagation. This method cannot converge for all the cases.  We can guarantee convergence for certain cases
using factor graph.

·
Junction Tree
Algorithm:
1) This is an exact inference technique for arbitrary graphical models. 2)
Possibility of this algorithm is only if graph is triangulated. 3) Due to
exponential complexity, it is not practically easy to implement.

·
Dual Method: Dual method focuses
at reformulating inference issue as linear integer programming problem. We can
extend most of the above algorithms in case of higher order graphical models.

9-
Training of Markov
Random Field

MRF follows two
types of training procedures:

·

1- Climb up in
steepest direction

2- Not guaranteed
to reach global maximum.

·
Max-margin
Learning:

This algorithm focuses
at adjusting the weight vector such that energy of desired ground truth
solution is less than energy of any other solution by a fixed amount.

10-
MRF
Applications to Vision

In
computer vision, mostly problem are related to noise, and so exact solutions
are most often impossible. Moreover, the latent variables of interest also have
the Markov property. For example, the pixel values in an image mostly rely on
those in the immediate vicinity, and have only weak correlations with those
further away. Hence, computer vision issues can be solve easily by using MRF optimization
method. Below are some examples of vision problems to which MRFs have been used:

Segmentation
Image reconstruction
Edge detection
Image restoration

11-  Some Other Applications
of Markov Random Field

Markov
random fields (MRF) have many application in a various fields, ranging
from computer graphics to
machine
learning and computer vision.

·
MRFs are used in image processing to produce textures as they
can be used to create stochastic and flexible image models. In image modelling,
the aim is to search a suitable intensity distribution of a specific image,
where suitability relies on the type of task.

·
MRFs are flexible enough to be used for texture and image
synthesis, image restoration and compression, surface reconstruction, image
segmentation,  texture synthesis, super-resolution, image
registration,
stereo
matching and information
retrieval.

·
They can be used to solve various computer vision problems
which can be posed as energy minimization problems or problems where different
regions have to be differentiate using a discriminating features set, within a
Markov random field framework, to predict the class of the region.

Conclusion

Graphical
models analyze those probability distributions whose conditional dependencies
appears in specific graph structures. Markov random field is an undirected
graphical model along with specific factorization properties. 2-Dimensional
MRFs have been largely used in image processing problems. In this report, I
explained two very important concepts Markov Random Field (MRF) and Markov
Chains. The basic difference between them is extension of probability means MRF
is simply an extension of Markov Chains. Both of them have deep rotes in
various applications such as image processing, classification, texture and
image synthesis and many more. Integration of Markov chains with MRF produce
more accurate and efficient results.

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