Our data points x1,x2,...xn are a sequence of heads and tails, e.g. First, let’s contrive a problem where we have a dataset where points are generated from one of two Gaussian processes. The EM Algorithm Ajit Singh November 20, 2005 1 Introduction Expectation-Maximization (EM) is a technique used in point estimation. To get perfect data, that initial step, is where it is decided whether your model will be giving good results or not. We consider theta be the optimal parameter to be defined, theta(t) be the t-th step value of parameter theta. You have two coins with unknown probabilities of 1) Decide a model to define the distribution, for example, the form of probability density function (Gaussian distribution, Multinomial distribution…). Another motivating example of EM algorithm — 6/35 — ABO blood groups Genotype Genotype Frequency Phenotype AA p2 A A AO 2 p A O A BB p2 B B BO 2 p B O B OO p2 O O AB 2 p A B AB The genotype frequencies above assume Hardy-Weinberg equilibrium. Before being a professional, what I used to think of Data Science is that I would be given some data initially. But if I am given the sequence of events, we can drop this constant value. It is true because, when we replace theta by theta(t), term1-term2=0 then by maximizing the first term, term1-term2 becomes larger or equal to 0. This can give us the value for ‘Θ_A’ & ‘Θ_B’ pretty easily. Suppose I say I had 10 tosses out of which 5 were heads & rest tails. Using this relation, we can obtain the following inequality. EM Algorithm Steps: Assume some random values for your hidden variables: Θ_A = 0.6 & Θ_B = 0.5 in our example. To do this, consider a well-known mathematical relationlog x ≤ x-1. Now, if you have a good memory, you might remember why do we multiply the Combination (n!/(n-X)! Goal: ! The following gure illustrates the process of EM algorithm… The binomial distribution is used to model the probability of a system with only 2 possible outcomes(binary) where we perform ‘K’ number of trials & wish to know the probability for a certain combination of success & failure using the formula. * X!) EM iterates over ! Model: ! where w_k is the ratio data generated from the k-th Gaussian distribution. “Classiﬁcation EM” If z ij < .5, pretend it’s 0; z ij > .5, pretend it’s 1 I.e., classify points as component 0 or 1 Now recalc θ, assuming that partition Then recalc z ij, assuming that θ Then re-recalc θ, assuming new z ij, etc., etc. 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). 15.1. 95-103. Solve this equation, the update of Sigma is. Let’s prepare the symbols used in this part. On Normalizing, the values we get are approximately 0.8 & 0.2 respectively, Do check the same calculation for other experiments as well, Now, we will be multiplying the Probability of the experiment to belong to the specific coin(calculated above) to the number of Heads & Tails in the experiment i.e, 0.45 * 5 Heads, 0.45* 5 Tails= 2.2 Heads, 2.2 Tails for 1st Coin (Bias ‘Θ_A’), 0.55 * 5 Heads, 0.55* 5 Tails = 2.8 Heads, 2.8 Tails for 2nd coin. In the following process, we tend to define an update rule to increase log p(x|theta(t)) compare to log p(x|theta). We can still have an estimate of ‘Θ_A’ & ‘Θ_B’ using the EM algorithm!! On 10 such iterations, we will get Θ_A=0.8 & Θ_B=0.52, These values are quite close to the values we calculated when we knew the identity of coins used for each experiment that was Θ_A=0.8 & Θ_B=0.45 (taking the average in the very beginning of the post). F. Jelinek, Statistical Methods for Speech Recognition, 1997 M. Collins, The EM Algorithm, 1997 J. Therefore, the 3rd term of Equation(1) is. Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. But what if I give you the below condition: Here, we can’t differentiate between the samples that which row belongs to which coin. ˆθMLE = arg max θ n ∑ i = 1logpθ(x ( i)) ^ θ MLE = arg max θ n ∑ i = 1 log p θ ( x ( i)) We use an example to illustrate how it works (referred from EM算法详解-知乎 ). Explore and run machine learning code with Kaggle Notebooks | Using data from no data sources Therefore, we decide a process to update the parameter theta while maximizing the log p(x|theta). [15] [16] Consider the function: F ( q , θ ) := E q ⁡ [ log ⁡ L ( θ ; x , Z ) ] + H ( q ) , {\displaystyle F(q,\theta ):=\operatorname {E} _{q}[\log L(\theta ;x,Z)]+H(q),} The third relation is the result of marginal distribution on the latent variable z. The distribution of latent variable z, therefore can be written as, The probability density function of m-th Gaussian distribution is given by, Therefore, the probability which data x belongs to m-th distribution is p(z_m=1|x) which is calculated by. We can calculate other values as well to fill up the table on the right. 2) After deciding a form of probability density function, we estimate its parameters from observed data. But things aren’t that easy. EM basic idea: if x(i) were known " two easy-to-solve separate ML problems ! To solve this problem, a simple method is to repeat the algorithm with several initialization states and choose the best state from those works. We can make the application of the EM algorithm to a Gaussian Mixture Model concrete with a worked example. However, since the EM algorithm is an iterative calculation, it easily falls into local optimal state. •In many practical learning settings, only a subset of relevant features or variables might be observable. 1 The Classical EM Algorithm Real-life Data Science problems are way far away from what we see in Kaggle competitions or in various online hackathons. It is sufficient to show the minorization inequality: logg(y | θ) ≥ Q(θ | θn) + logg(y | θn) − Q(θn | θn). Find maximum likelihood estimates of µ 1, µ 2 ! I will randomly choose a coin 5 times, whether coin A or B. • The EM algorithm in general form • The EM algorithm for hidden markov models (brute force) • The EM algorithm for hidden markov models (dynamic ... A First Example: Coin Tossing • X = {H,T}. S prepare the symbols used in this part find this piece interesting, you will definitely find something more yourself. A professional, what we see in Kaggle competitions or in various online hackathons Monday to Thursday to the. 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