A popular clustering algorithm is known as K-means, which will follow an iterative approach to update the parameters of each clusters.

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Probabilistic
Learning and
Reinforcement
Learning

 Here, μ1 and μ2 are the centroids of each cluster and are
parameters that identify each of these.
 A popular clustering algorithm is known as K-means, which will
follow an iterative approach to update the parameters of each
clusters.
 More specifically, what it will do is to compute the means (or
centroids) of each cluster, and then calculate their distance to
each of the data points.
 The latter are then labeled as part of the cluster that is identified
by their closest centroid.
 This process is repeated until some convergence criterion is met

 One important characteristic of K-means is that it is a hard
clustering method, which means that it will associate each
point to one and only one cluster.
 A limitation to this approach is that there is no uncertainty
measure or probability that tells us how much a data point
is associated with a specific cluster.
 This is exactly what Gaussian Mixture Models, or simply
GMMs, attempt to do.

 A Gaussian Mixture is a function that is comprised of
several Gaussians, each identified by k {1,…,
∈ K},
where K is the number of clusters of the dataset.
 Each Gaussian k in the mixture is comprised of the
following parameters:
 A mean μ that defines its centre.
 A covariance Σ that defines its width. This would be
equivalent to the dimensions of an ellipsoid in a multivariate
scenario.
 A mixing probability π that defines how big or small the
Gaussian function will be.

A popular clustering algorithm is known as K-means, which will follow an iterative approach to update the parameters of each clusters.

 There are three Gaussian functions, hence K = 3.
 Each Gaussian explains the data contained in each of the
three clusters available.
 The mixing coefficients are themselves probabilities and
must meet this condition:

 How do we determine the optimal values for these
parameters?
 To achieve this we must ensure that each Gaussian fits the
data points belonging to each cluster.
 This is exactly what maximum likelihood does.
 In general, the Gaussian density function is given by:

 Where x represents our data points, D is the number of
dimensions of each data point.
 μ and Σ are the mean and covariance, respectively.
 If we have a dataset comprised of N = 1000 three-
dimensional points (D = 3), then x will be a 1000 × 3
matrix.
 μ will be a 1 × 3 vector, and Σ will be a 3 × 3 matrix.

 we will also find it useful to take the log of this
equation, which is given by:
 If we differentiate this equation with respect to the
mean and covariance and then equate it to zero, then
we will be able to find the optimal values for these
parameters, and the solutions will correspond to the
Maximum Likelihood Estimates (MLE).

Expectation
-
maximizatio
n algorithm
 Finding the parameters would prove to be very hard
 Fortunately, there is an iterative method we can use to
achieve this purpose.
 It is called the Expectation — Maximization, or simply EM
algorithm.
 It is widely used for optimization problems where the
objective function has complexities such as the one just
encountered for the GMM case.
 Let the parameters of our model be

 Step 1: Initialise θ accordingly. For instance, we can use
the results obtained by a previous K-Means run as a good
starting point for our algorithm.
 Step 2 (Expectation step): Evaluate
 Step 3 (Maximization step): Find the revised parameters
θ* using:

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1. Probabilistic Learning and Reinforcement Learning

2. Gaussian Mixture Models

3.  Here, μ1 and μ2 are the centroids of each cluster and are parameters that identify each of these.  A popular clustering algorithm is known as K-means, which will follow an iterative approach to update the parameters of each clusters.  More specifically, what it will do is to compute the means (or centroids) of each cluster, and then calculate their distance to each of the data points.  The latter are then labeled as part of the cluster that is identified by their closest centroid.  This process is repeated until some convergence criterion is met

4.  One important characteristic of K-means is that it is a hard clustering method, which means that it will associate each point to one and only one cluster.  A limitation to this approach is that there is no uncertainty measure or probability that tells us how much a data point is associated with a specific cluster.  This is exactly what Gaussian Mixture Models, or simply GMMs, attempt to do.

5.  A Gaussian Mixture is a function that is comprised of several Gaussians, each identified by k {1,…, ∈ K}, where K is the number of clusters of the dataset.  Each Gaussian k in the mixture is comprised of the following parameters:  A mean μ that defines its centre.  A covariance Σ that defines its width. This would be equivalent to the dimensions of an ellipsoid in a multivariate scenario.  A mixing probability π that defines how big or small the Gaussian function will be.

7.  There are three Gaussian functions, hence K = 3.  Each Gaussian explains the data contained in each of the three clusters available.  The mixing coefficients are themselves probabilities and must meet this condition:

8.  How do we determine the optimal values for these parameters?  To achieve this we must ensure that each Gaussian fits the data points belonging to each cluster.  This is exactly what maximum likelihood does.  In general, the Gaussian density function is given by:

9.  Where x represents our data points, D is the number of dimensions of each data point.  μ and Σ are the mean and covariance, respectively.  If we have a dataset comprised of N = 1000 three- dimensional points (D = 3), then x will be a 1000 × 3 matrix.  μ will be a 1 × 3 vector, and Σ will be a 3 × 3 matrix.

10.  we will also find it useful to take the log of this equation, which is given by:  If we differentiate this equation with respect to the mean and covariance and then equate it to zero, then we will be able to find the optimal values for these parameters, and the solutions will correspond to the Maximum Likelihood Estimates (MLE).

11. Expectation - maximizatio n algorithm  Finding the parameters would prove to be very hard  Fortunately, there is an iterative method we can use to achieve this purpose.  It is called the Expectation — Maximization, or simply EM algorithm.  It is widely used for optimization problems where the objective function has complexities such as the one just encountered for the GMM case.  Let the parameters of our model be

12.  Step 1: Initialise θ accordingly. For instance, we can use the results obtained by a previous K-Means run as a good starting point for our algorithm.  Step 2 (Expectation step): Evaluate  Step 3 (Maximization step): Find the revised parameters θ* using:

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