jensen awm965

method.Derivation of 2.1 logarithmic likelihood function by EM algorithmSuppose we have a sample set {x (1),..., X (M)}, which contains m independent samples. But each sample I corresponds to the category Z (i) is unknown (equivalent to clustering, contact Kmeans), also known as the implied variable, so we need to estimate the probability model P (x,z) of the parameter θ, but because it contains the implied variable z, it is difficult to use the maximum likelihood solution.That is to say our go

, it is still not given quantitatively, and the derivation process of generalized em is still not given. The next article focuses on these topics.1. Jensen InequalitiesReview some concepts in the optimization theory.Set F is a function that defines the field as a real number, and if for all real numbers x, then f is the convex function.When x is a vector, if it hessian the matrix H-type semi-positive definite (), then f is the convex function. if or h

EM is one of the algorithms that I've always wanted to learn deeply, the first time I heard it was in the HMM section of NLP class, in order to solve the hmm parameter estimation problem, the EM algorithm was used. The words in the MT later are also used in the Zizhong. It is also mentioned in Mitchell's book that EM can be used in Bayesian networks.The following mainly introduces the entire derivation process of EM.1. Jensen InequalitiesReview some c

the denominator, "two normal distribution, respectively, and then add" to do the fractional form of the molecule. M this thing add up to make it equal to 0, ask for the analytic solution of θ, you think of your maths level not too high.What to do? First introduce an inequality, called Jensen Inequality, is said:　　X is a random variable, f (x) is a convex function (the second derivative is large or equal to 0), then there are:　　　　When and only if X is

1. IntroductionThe probability models we discussed before are all only the observed variables (observable variable), that is, these variables can be observed, so the given data can be directly using the method of maximum likelihood estimation or Bayesian estimation, but when the model contains Implicit variables (latent variable) , it is not possible to simply use these estimation methods.as in Gaussian mixture and EM algorithm The Gaussian mixture discussed in this paper is a typical example o

categories of the sample I z to the right of the equality of the joint probability density function and the equation to the left of the random variable x edge probability density, that is, the likelihood function, but you can see there is "and" logarithm, after the derivation form will be very complex (you can imagine the log (F1 (x) + F2 (x ) + F3 (x) + ...) Complex functions, so it is difficult to solve the unknown parameters z and θ. OK, can we make some changes to the (1) formula? We see (2

em algorithm , which is a more well-known CV world algorithm, although very early heard, but really dig into the recent days to see the Stanford Public Lecture notes. The reason that EM and Mog are put together is because we need em to solve the MOG model, so here we introduce the EM algorithm first.Before introducing the EM algorithm, we will first popularize the knowledge of the Jensen inequalities. First, let's give a definition of the

distribution of differential entropy is Gaussian distribution. When we calculate, we do not assume that the differential entropy must be non-negative, therefore, it is not a necessary condition. Differential entropy expression for normal distribution: It can be found that the entropy becomes larger with the increase of variance . Relative entropy (Relative entropy) and mutual information (mutual information) relative entropy (relative entropy), also known as KL divergence (Ku

takes the logarithm, and organizes; 3 The derivative is 0, and the likelihood equation is obtained; 4) to solve the likelihood equation, the obtained parameter is the requestIn fact, the maximum likelihood can think, we assume already know to θ, in the case of known θ, produce y, it is natural, if we see the result produced a lot of Yi, then P (yi|θ) must be relatively large. Now we think in turn, we already know Y,, then the most probable parameter to make the result appear is the parameter we

by an equal function, or there is a "sum of the logarithm" ah, or can not solve, then why do this? Let's take a look at (3) and find that (3) becomes "logarithmic and", so it's easy to take a derivative. We notice that the equal sign becomes an equal, why can it be so changed? This is the place where the Jensen inequality is greatly apparent.Jensen Inequalities:Set F is a function that defines the field as a real number, if for all real numbers x. If

? We see (2), (2) formula is only the numerator denominator multiplied by an equal function, or there is a "sum of the logarithm" ah, or can not solve, then why do this? Let's take a look at (3) and find that (3) becomes "logarithmic and", so it's easy to take a derivative. We notice that the equal sign becomes an equal, why can it be so changed? This is the place where the Jensen inequality is greatly apparent.Jensen Inequalities:Set F is a function

estimator. Before solving, we need to make a stronger assumption that all sample data are independent of each other. Then we can get the logarithmic likelihood estimator as shown in Formula 3. L (θ) =∑ni=1logp (xi∣θ) =∑ni=1log∑kk=1wkϕ (xi∣θk) Formula 3 The optimal value to be estimated is θ^=argmaxθl (θ) for the equation shown in Formula 3, it is difficult to obtain the maximum value by the direct derivation order minus zero. Therefore, using the EM algorithm to solve this problem, the princip

(1−λ) a+λb) ≥ (1−λ) u (a) +λu (b). Further, because gis nondecreasing, r≥ simplies g (r) ≥ g (s). Hence g (U (1−λ) a+λb) ≥ g ((1−λ) U (a) +λu (b)). But now by the concavity of Gwe have g ((1−λ) U (a) +λu (b)) ≥ (1−λ) G (U (A)) +λg (U (b)) N Bsp;= (1−λ) f (a) +λf (b). so fis Concave. Jensen ' s inequality:another characterization of concave and convex functions If we letλ1= ; =λin the earlier definition of a concave functionand replace aby x 1and bnbs

algorithmIn the problem of implicit variables, the parameters in the model can not be obtained directly by the maximum likelihood estimation, and the EM algorithm is an effective method to solve the problem of implicit variable optimization. The EM algorithm is the short name of the desired maximal (expectation maximization) algorithm, and the EM algorithm is an iterative algorithm, which is divided into two main steps in each iterative process: the desired (expectation) step and the maximizati

race, which is actually a hidden intermediate variable. Muk and sigmak^2 are the parameters of various Gaussian distributions. According to our above pattern is to seek the logarithm likelihood probability derivative to obtain the parameter estimation, then first looks the likelihood functionThis embarrassing situation arises, with a plus sign inside the logarithm, the derivation becomes complex and impossible to solve, in fact, the formula does not have an analytic solution. But let's just say

SQL having instance tutorial HavingIn the listen to have added to SQL, because where the keyword cannot use aggregate functions.SQL with syntax SELECT column_name, Aggregate_function (column_name) from table_name WHERE column_name operator Value GROUP by column_name has aggregate_function (column_name) operator value Let's look at an example of having one. _id OrderDate Orderprice Customer 1 2008/11/12

parameter is the request. (4) Summary In most cases we calculate the result based on a known condition, and the maximum likelihood estimate is the one that already knows the result and then seeks to make the result the most probable condition, as an estimate. 2. Jensen Inequalities (1) Definition Set F is a function that defines the field as a real number, if for all real numbers x. If the two derivative of all real x,f (x) is greater than or equal t