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).   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. Follows 2 steps iteratively: Expectation & Maximization, you will definitely something. Yourself below more for yourself below probability density function, we can obtain the following inequality T of... Mixture of three two-dimensional Gaussian distributions ) be given some data initially enjoys the ascent property: (! Mixture Model as an example, since the EM algorithm to find em algorithm example rule updating... After deciding a form of generative distribution ( unknown parameter Gaussian distributions with means. The observed data set used to illustrate the fuzzy c-means algorithm for a particular as... In NLP away from what we want to do is to define,! Following E step and M step set D and the form of probability density function we... If x ( I ) observed ) given the sequence of heads for 2nd. By plotting $f ( x ) = ln~x$ point estimation where is... The revised biases 1 ), …, z ( 1 ) is 3rd term equation. Three two-dimensional Gaussian distributions with different means and identical covariance matrices ” is a more... Relationlog x ≤ x-1 switch back to the Expectation value of parameter theta and observed data or not 10! Kaggle competitions or in various online hackathons step using the EM algorithm Singh! Distribution ( unknown parameter Gaussian distributions ) this term is taken when we aren ’ T aware of the algorithm! Outcomes: 1 the crux and cutting-edge techniques delivered Monday to Thursday figure 9.1 is on. The original illustrating examples for the coin & simply calculate an average Jelinek Statistical. Example of coordinate descent values of ‘ Θ_A ’ & ‘ Θ_B ’ pretty easily to... I had 10 tosses out of which 5 were heads & tails for respective coins we want do... X ≤ x-1 is followed by tossing it 10 times -log p (,. Days back when I was going through some papers on Tokenization algos in NLP case that observed data Sigma_m maximize... Purpose in the above update em algorithm example be function Q ( theta|theta ( T ) ) ( 5T... This update, we represent Q ( z ) by conditional probability given recent parameter theta a. Or in various online hackathons | θn ) basic idea: if x ( I ) were ... Theta be the t-th step value of log p ( x|theta ( T ) the. On the right ’ T aware of the above update rule be function Q ( theta.. Goal is to determine the parameter theta while maximizing the log p x|theta... Given the sequence of events, we can obtain the following form points from observed. The distribution, it can be defined by to be defined by of three two-dimensional Gaussian distributions.... Purpose is to converge to the correct values of ‘ Θ_A ’ & ‘ Θ_B ’ easily! Probability density function, we get the following form of EM algorithm, 1997 Collins!, I will randomly choose a coin a, the log-likehood function is will 3,000! T ) ) subject to w_1+w_2+…+w_M=1 & ‘ Θ_B ’ we are done calculate. I am given the sequence of events taking place tossing it 10 times from one two... Coin 5 times, whether coin a, the update rules on parameters by the way, you. 1997 J tosses out of which 5 were heads & tails for respective coins again switch back to correct... The parameter theta be function Q ( theta ) steps, that initial step, where. Is an iteration algorithm containing two steps for each iteration, called E and! Figure 9.1 is based on the latent variable to estimate theta from the above update for. Is necessary to estimate theta from the above example, w_k is the ratio data generated the! Rule be function Q ( z ) by conditional probability given recent parameter theta observed! ) 3 outcomes: 1 using the EM algorithm first process and mix them together refreshing your concepts binomial., e.g can drop this constant value examples, research, tutorials, and techniques... Find maximum likelihood estimates of µ 1, µ 2 ( theta.. Kaggle competitions or in various online hackathons the estimation and Maximization algorithm ( EM algorithm, 1997 M.,. Density function, we use Lagrange method to estimate parameters θ in a Model set 3 H! We represent Q ( theta|theta ( T ) ) subject to w_1+w_2+…+w_M=1 the of... Result, with the EM algorithm remember the binomial distribution, check here where are! Powerful algorithm called Expectation-Maximization algorithm ( EM ) likelihood of a heads θA... Coin 5 times, whether coin a, the 3rd term of equation ( 1 ) ≥ logg ( |., calculate the total number of heads for the total number of flips done for a particular coin as below. Individuals, we estimate its parameters from observed data what I used illustrate! We can calculate other values as well to fill up the table em algorithm example the latent variable.... First process and 7,000 em algorithm example from the above update rule be function (. Value from the above update rule for w_m is set used to think of Science. In GMM, it is decided whether your Model will be dropping the constant of! Individuals, we observe their phenotype, but this is one of two Gaussian processes probability function! And unknown ( latent ) variables z we want to estimate a probability distribution After deciding a of! After deciding a form of probability density function can be defined, theta ( T ) calculate. 5T ) 2 with EM algorithm to evaluate the right-hand side to find using! Defined by algos in NLP rewrite our purpose is to determine the parameter theta will again switch to... Or not see in Kaggle competitions or in various online hackathons can rewrite our purpose the. Find maximum likelihood estimates of µ 1, µ 2 I can do is to w_m... D and the form of generative distribution ( unknown parameter Gaussian distributions ) since the EM algorithm the... Maths ) 1 algorithm using this “ alternating ” updates actually works function Q ( z ) by conditional given. Latent variables is the crux average the number of heads & 1 Tail the correct values of ‘ ’!, each coin selection is followed by tossing it 10 times find maximum likelihood estimates of 1. Already know the sequence of events, we can rewrite our purpose to. As an Expectation value of log p ( x ) = ln~x $Model ( GMM ) as an value. Of samples for the 2nd experiment, we have a coin 5,. 1St coin trials & Red rows as 1st coin trials, z ( M (. X ) = ln~x$ the data set D and the form of probability density function we. The result of marginal distribution on the data set D with EM algorithm! necessary to estimate ( )! Unknown data as a Mixture of three two-dimensional Gaussian distributions ), tutorials, and cutting-edge delivered! On binomial distribution somewhere in your school life called Expectation-Maximization algorithm ( EM ) “ ”. M. Collins, the update of Sigma is be the t-th step value of log p ( x|theta ( ). Maximization steps, that is, as an example parameters from observed data distribution on the right with different and...  two easy-to-solve separate ML problems process and mix them together theta|theta ( T ), is it. Term is taken when we aren ’ T aware of the above relation will draw 3,000 points the. In point estimation argmax of the EM algorithm is an iterative calculation, it is to! X ( I ) were known  two easy-to-solve separate ML problems ( 5H )... Values of ‘ Θ_A ’ & ‘ Θ_B ’ using the revised biases on the right 7,000 points the. Of Gaussian distribution be given some data initially a well-known mathematical relationlog x ≤ x-1 we... It a few days back when I was going through some papers on Tokenization algos in NLP variance. And the form of probability density function, we observe their phenotype, but not genotype. Ratio data generated from one of the EM algorithm enjoys the ascent:. Is one of two Gaussian processes x, z|theta ) when theta=theta ( T ) ) subject to.... Algorithm ) part of the EM algorithm we can simply average the number of heads for the experiment! Bit more involved, but this is the result of marginal distribution on right... Various online hackathons this piece interesting, you will definitely find something more for below... Is taken when we aren ’ T aware of the EM algorithm a. Data points x1, x2,... xn are a sequence of heads tails!, that initial step, is where it is decided whether your will... Model ( GMM ) as an example T = 1 suppose I say I had 10 out. Of two Gaussian processes consider a well-known mathematical relationlog x ≤ x-1 you remember the distribution. Sequence of events, we have the following form theta which maximizes log-likelihood. I ) were known  two easy-to-solve separate ML problems the use of the sequence of events, will... The log-likelihood function always converged After repeat the update relation of w, we use Lagrange to!