EM algorithm example from "Introducing Monte Carlo Methods with R" - em_algorithm_example.py Equation (1): Now, we need to evaluate the right-hand side to find a rule in updating parameter theta. Make learning your daily ritual. Let’s prepare the symbols used in this part. Most of the time, there exist some features that are observable for some cases, not available for others (which we take NaN very easily). Proof: \begin{align} f''(x) = \frac{d~}{dx} f'(x) = \frac{d~\frac{1}{x}}{dx} = -\frac{1}{x^2} < 0 \end{align} Therefore, we have $ln~E[x] \geq E[ln~x]$. This result says that as the EM algorithm converges, the estimated parameter converges to the sample mean using the available m samples, which is quite intuitive. θ A. . Then the EM algorithm enjoys the ascent property: logg(y | θn + 1) ≥ logg(y | θn). But if I am given the sequence of events, we can drop this constant value. The EM algorithm can be viewed as two alternating maximization steps, that is, as an example of coordinate descent. It is often used for example, in machine learning and data mining applications, and in Bayesian statistics where it is often used to obtain the mode of the posterior marginal distributions of parameters. For example, in the case of Gaussian distribution, mean and variance are parameters to estimate. Coming back to EM algorithm, what we have done so far is assumed two values for ‘Θ_A’ & ‘Θ_B’, It must be assumed that any experiment/trial (experiment: each row with a sequence of Heads & Tails in the grey box in the image) has been performed using only a specific coin (whether 1st or 2nd but not both). Let’s take a 2-dimension Gaussian Mixture Model as an example. Before being a professional, what I used to think of Data Science is that I would be given some data initially. Our data points x1,x2,...xn are a sequence of heads and tails, e.g. EM iterates over ! As saw in the result(1),(2) differences in M value(number of mixture model) and initializations offer different changes in Log-likelihood convergence and estimate distribution. The probability shown in log-likelihood function p(x,z|theta) can be represented with the probability of latent variable z as the following form. In the above example, w_k is a latent variable. Random variable: x_n (d-dimension vector) Latent variable: z_m Mixture ratio: w_k Mean : mu_k (d-dimension vector) Variance-covariance matrix: Sigma_k (dxd matrix) Randomly initialize mu, Sigma and w. t = 1. I will randomly choose a coin 5 times, whether coin A or B. Example in figure 9.1 is based on the data set used to illustrate the fuzzy c-means algorithm. The missing data can be actual data that is missing, or some ... Before we get to theory, it helps to consider a simple example to see that EM is doing the right thing. We start by focusing on the change of log p(x|theta)-log p(x|theta(t)) when update theta(t). To do this, consider a well-known mathematical relationlog x ≤ x-1. We can translate this relation as an expectation value of log p(x,z|theta) when theta=theta(t). Now, if you have a good memory, you might remember why do we multiply the Combination (n!/(n-X)! 15.1. What I can do is count the number of Heads for the total number of samples for the coin & simply calculate an average. Examples that illustrate the use of the EM algorithm to find clusters using mixture models. Solving this equation for lambda and use the restraint relation, the update rule for w_m is. Therefore, the 3rd term of Equation(1) is. The intuition behind EM algorithm is to rst create a lower bound of log-likelihood l( ) and then push the lower bound to increase l( ). The third relation is the result of marginal distribution on the latent variable z. Given a set of observable variables X and unknown (latent) variables Z we want to estimate parameters θ in a model. Here, we represent q(z) by conditional probability given recent parameter theta and observed data. Therefore, in GMM, it is necessary to estimate the latent variable first. An effective method to estimate parameters in a model with latent variables is the Estimation and Maximization algorithm (EM algorithm). Similarly, If the 1st experiment belonged to 2nd coin with Bias ‘Θ_B’(where Θ_B=0.5 for the 1st step), the probability for such results will be: 0.5⁵x0.5⁵ = 0.0009 (As p(Success)=0.5; p(Failure)=0.5), On normalizing these 2 probabilities, we get. For a random sample of n individuals, we observe their phenotype, but not their genotype. Then I need to clean it up a bit (some regular steps), engineer some features, pick up several models from Sklearn or Keras & train. constant? The EM algorithm is particularly suited for problems in which there is a notion of \missing data". To get perfect data, that initial step, is where it is decided whether your model will be giving good results or not. The first and second term of Equation(1) is non-negative. And if we can determine these missing features, our predictions would be way better rather than substituting them with NaNs or mean or some other means. One considers data in which 197 animals are distributed multinomially into four categories with cell-probabilities (1/2+θ/4,(1− θ)/4,(1−θ)/4,θ/4) for some unknown θ ∈ [0,1]. And next, we use the estimated latent variable to estimate the parameters of each Gaussian distribution. The EM algorithm has many applications throughout statistics. Let the subject of argmax of the above update rule be function Q(theta). Let’s go with a concrete example by plotting $f(x) = ln~x$. Set 1: H T T T H H T H T H(5H 5T) 2. This term is taken when we aren’t aware of the sequence of events taking place. As we already know the sequence of events, I will be dropping the constant part of the equation. 2 EM as Lower Bound Maximization EM can be derived in many different ways, one of the most insightful being in terms of lower bound maximization (Neal and Hinton, 1998; Minka, 1998), as illustrated with the example from Section 1. By the way, Do you remember the binomial distribution somewhere in your school life? The algorithm follows 2 steps iteratively: Expectation & Maximization. The points are one-dimensional, the mean of the first distribution is 20, the mean of the second distribution is 40, and both distributions have a standard deviation of 5. The grey box contains 5 experiments, Look at the first experiment with 5 Heads & 5 Tails (1st row, grey block). But things aren’t that easy. Interactive and scalable dashboards with Vaex and Dash, Introduction to Big Data Technologies 1: Hadoop Core Components, A Detailed Review of Udacity’s Data Analyst Nanodegree — From a Beginner’s Perspective, Routing street networks: Find your way with Python, Evaluation of the Boroughs in London, UK in order to identify the ‘Best Borough to Live’, P(1st coin used for 2nd experiment) = 0.6⁹x0.4¹=0.004, P(2nd coin used for 2nd experiment) = 0.5⁹x0.5 = 0.0009. Now, our goal is to determine the parameter theta which maximizes the log-likelihood function log p(x|theta). Suppose that we have a coin A, the likelihood of a heads is θA. Model: ! This can give us the value for ‘Θ_A’ & ‘Θ_B’ pretty easily. An example: ML estimation vs. EM algorithm qIn the previous example, the ML estimate could be solved in a closed form expression – In this case there was no need for EM algorithm, since the ML estimate is given in a straightforward manner (we just showed that the EM algorithm converges to the peak of the likelihood function) The following gure illustrates the process of EM algorithm… Our goal in this step is to define w_m, mu_m, Sigma_m which maximize Q(theta|theta(t)). The EM algorithm helps us to infer(conclude) those hidden variables using the ones that are observable in the dataset and Hence making our predictions even better. 4 Gaussian MixtureWith Known Mean AndVariance Our next example of the EM algorithm to estimate the mixture weights of a Gaussian mixture with known mean and variance. Suppose bias for 1st coin is ‘Θ_A’ & for 2nd is ‘Θ_B’ where Θ_A & Θ_B lies between 0

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