\end{equation}, \begin{equation} Density estimation 4 Estimating the probability density function (), given a set of data points =1 drawn from it. \begin{equation} For a linear model with multivariate normal prior and multivariate normal likelihood, you end up with a multivariate normal posterior distribution in which the mean of the posterior (and maximum a posteriori model) is exactly what you would obtain using Tikhonov regularized ($L_{2}$ regularized) least squares with an appropriate regularization parameter. a uniform prior), then both calculations are equivalent. Pages 18 Ratings . With a full Bayesian approach you have access to all inferential procedures when you're done. 1. . \begin{equation} Let us imagine that you want to infer some parameter $\beta$ from some observed input-output pairs $(x_1,y_1)\dots,(x_N,y_N)$. \end{equation} &= \frac{1}{2\sigma_{y}^{2}} \sum_{n=1}^{N} \big(y^{(n)} - f_{\mathbf{w}}(\mathbf{x}^{(n)})\big)^{2} + \frac{1}{2\sigma_{\mathbf{w}}^{2}} \sum_{i=1}^{K} w_{i}^{2} + const. One framework is not better than another, and as mentioned, in many cases, both frameworks frame the same optimization problem from different perspectives. To develop a. 0000002472 00000 n What you have written, P(Data | theta), is the description of probability, not likelihood. f(w) = \frac{\lambda^{\frac{D}{2}}}{(2\pi)^{\frac{D}{2}}}exp(-\frac{\lambda}{2} w^T w) \hat{w} = \operatorname{argmax}_w \Big( D\log \frac{1}{\sqrt{2\pi\sigma^2}} -\frac{1}{2\sigma^2}\sum_{k=1}^N (y_k- x^Tw)^2) + \log \lambda^{\frac{D}{2}} - \log (2\pi)^{\frac{D}{2}} - \frac{\lambda}{2}w^Tw \Big) , x_N from a Gaussian random variable with known variance sigma^2 and unknown mean . It only takes a minute to sign up. The specific choice of prior distribution for X is, of course, a critical component in MAP estimation. = Modified 5 years, 3 months ago. \end{equation} Stack Overflow for Teams is moving to its own domain! This is called the maximum a posteriori (MAP) estimation . \hat{w} = \operatorname{argmax}_w \log P(w \vert \mathcal{D} ) The estimation of the model is done by iteratively maximizing the marginal log-likelihood of the observations. 0000001749 00000 n How can I write this using fewer variables? Because of this equivalence, both MLE and MAP often converge to the same optimization problem for many machine learning algorithms. Instead of a Gaussian prior, multiply your likelihood with a Laplace prior and then take the logarithm. From a statistical point of view, one form of unsupervised learning is "density estimation" which can be MLE can be silly, for example if we throw a coin twice, both head, then MLE asid you will always have head in the future. This section provides more resources on the topic if you are looking to go deeper. This flexible probabilistic framework can be used to provide a Bayesian foundation for many machine learning algorithms, including important methods such as linear regression and logistic regression for predicting numeric values and class labels respectively, and unlike maximum likelihood estimation, explicitly allows prior belief about candidate models to be incorporated systematically. Two radar tracking stations provide independent measurements and of the landing site, , of a returning space probe. And MAP gives you the value which maximises the posterior probability P (|D). \log P( \mathcal{D} \vert w) = D\log \frac{1}{\sqrt{2\pi\sigma^2}} -\frac{1}{2\sigma^2}\sum_{k=1}^N (y_k- x^Tw)^2 \tag{*} Rt%:/&A6v5[|8"@{R}~ pCo5|l/zArj{GWq):kj,+K2}: ZbO$JaOa b+vL_}7CE5&` ;grZjES(}B=9:Z7a.NTq. rev2022.11.7.43014. This gives rise to a Gaussian likelihood: $$\prod_{n=1}^N \mathcal{N}(y_n|\beta x_n,\sigma^2).$$. \log P( \mathcal{D} \vert w) = \log \big( \prod_{k=1}^D \frac{1}{\sqrt{2\pi\sigma^2}}exp(-\frac{1}{2\sigma^2}(y_k- x^Tw)^2) \big) However, whether the detected salient regions are helpful in image blur estimation is unknown. I've never seen any distribution that looks like this. This method estimates the parameters of a model. \end{equation} The additive Gaussian white noise (AGWN) level in real-life images is usually unknown, for which the empirical setting will make the denoising methods over-smooth fine structures or remove noise incompletely. Allow Line Breaking Without Affecting Kerning. Typically, estimating the entire distribution is intractable, and instead, we are happy to have the expected value of the distribution, such as the mean or mode. QGIS - approach for automatically rotating layout window. \end{equation}, \begin{equation} To subscribe to this RSS feed, copy and paste this URL into your RSS reader. 2022 Machine Learning Mastery. Figure 9.3 - The maximum a posteriori (MAP) estimate of X given Y = y is the value of x that maximizes the posterior PDF or PMF. Therefore, D \vert w \sim \mathcal{N}(w^T x, \sigma^2) f(x) = \frac{1}{\sqrt{(2\pi)^{N} \det \Sigma}}exp(-\frac{1}{2} (x - \mu)^T \Sigma^{-1} (x - \mu)) \tag{1} \hat{x}_{MAP}=\frac{1}{2}. We \end{equation}, \begin{equation} P_{Y|X}(3|x)=x (1-x)^2. Why use mean of posterior distribution instead of probability? MAP Solution for Linear Regression - What is a Gaussian prior? \end{equation}, \begin{equation} Exponential? Yes! I'm Jason Brownlee PhD It only takes a minute to sign up. f(w) = \frac{1}{\sqrt{(2\pi)^D \frac{1}{\lambda^D}}}exp(-\frac{1}{2}(w - 0)^T (\frac{1}{\lambda} I)^{-1} (w - 0)) We then turn to the case of Bayesian Logistic Regression under this same prior. \begin{split} It is about using these models to extract desired information from the available data. \end{equation}, Also, you can use equation (1), to get $f(\mathcal{D} \vert w)$, we first need $f(y_k \vert w)$, or if you prefer, you can use the univariate Normal distribution, with mean $w^T x$ and variance $\sigma^2$, i.e. \begin{equation} \overbrace{P(\mathcal{D} \vert w)}^{\text{Likelihood}}\overbrace{P(w)}^{\text{Prior}} \tag{0} How to split a page into four areas in tex, Sci-Fi Book With Cover Of A Person Driving A Ship Saying "Look Ma, No Hands!". As such, this technique is referred to as maximum a posteriori estimation, or MAP estimation for short, and sometimes simply maximum posterior estimation.. Are witnesses allowed to give private testimonies? \end{equation} This is called the maximum a posteriori (MAP) estimation. Calculate the MAP estimate _MAP. My profession is written "Unemployed" on my passport. \end{equation}, \begin{equation} In this post, you will discover a gentle introduction to Maximum a Posteriori estimation. Ask your questions in the comments below and I will do my best to answer. the maximum likelihood hypothesis might not be the MAP hypothesis, but if one assumes uniform prior probabilities over the hypotheses then it is. $ \log P (\mathcal{D})$ is independent of $w$, so we're good without it Is there any actual problem? P_{Y|X}(y|x)=x (1-x)^{y-1}, \quad \textrm{ for }y=1,2,\cdots. Solving for $x$ (and checking for maximization criteria), we obtain the MAP estimate as \begin{align} \end{equation}, \begin{equation} RSS, Privacy | But I'm not sure the algebra would amount to the same expression. Determine a constraint on the location of the MAP estimate when Attempt: MAP involves calculating a conditional probability of observing the data given a model weighted by a prior probability or belief about the model. 0000002949 00000 n and I help developers get results with machine learning. &= \frac{1}{2\sigma_{y}^{2}} \sum_{n=1}^{N} \big(y^{(n)} - f_{\mathbf{w}}(\mathbf{x}^{(n)})\big)^{2} + \frac{1}{2\sigma_{\mathbf{w}}^{2}} \sum_{i=1}^{K} w_{i}^{2} + const. For Bayesian linear regression, an inverse gamma prior could be used to form a conjugate prior to the variance. \end{equation} To develop a general likelihood function, we utilize a general class of divergence measures, called Bregman divergence. $$ Conjugate prior for Gaussian? \log P( \mathcal{D} \vert w) = \sum_{k=1}^D \log \frac{1}{\sqrt{2\pi\sigma^2}} -\frac{1}{2\sigma^2}\sum_{k=1}^N (y_k- x^Tw)^2 0000001657 00000 n f(\mathcal{D} \vert w) = f(y_1 \ldots y_D \vert w) = \prod_{k=1}^N f(y_k \vert w) Essentially, this optimized the likelihood of the data given a model . First notice that median minimizes the L1 norm (see here or here for learning more on L1 and L2), $$ \DeclareMathOperator*{\argmin}{arg\,min} The MAP estimate of X is usually shown by x ^ M A P. Let us regularise parameter $\beta$ by imposing the Gaussian prior $\mathcal{N}(\beta|0,\lambda^{-1}),$ where $\lambda$ is a strictly positive scalar ($\lambda$ quantifies of by how much we believe that $\beta$ should be close to zero, i.e. 1 Maximum A Posteriori (MAP) Estimation The MLE framework consisted of formulating an optimization problem in which the objective was the likelihood (as parametrized by the unknown model parameters) of the measured data, and . \end{equation}, \begin{equation} Such estimators have been widely advocated for image restoration. \end{split} Is there a Bayesian interpretation of linear regression with simultaneous L1 and L2 regularization (aka elastic net)? -\log \big[p(\mathbf{w}|\mathcal{D}) \big] &= -\sum_{n=1}^{N} \log \big[\mathcal{N}(y^{(n)}; f_{\mathbf{w}}(\mathbf{x}^{(n)}), \sigma_{y}^{2}) \big] - \sum_{i=1}^{K} \log \big[ \mathcal{N}(w_{i}; \, 0, \, \sigma_{\mathbf{w}}^{2}) \big] + const. $$, $$ Thank you for your interest and feedback! By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. $$, $$ MathJax reference. Asking for help, clarification, or responding to other answers. The relationship between the L1 norm and the Laplace prior can be understood in the same fashion. Can someone please help me out? MLE is more appropriate where there is no such prior. Question about conventions for L1 and L2 regularization, How does the L2 regularization penalize the high-value weights. In order to illustrate the concepts, we will consider the specific application of the restoration of a blurred and noisy image. This equivelance is general and holds for any parameterized function of weights - not just linear regression as seems to be implied above. We derive the exact statistical distribution of maximum a posteriori (MAP) estimators having edge-preserving nonGaussian priors. trailer << /Size 775 /Prev 716436 /Root 752 0 R /Info 750 0 R /ID [ ] >> startxref 0 %%EOF 752 0 obj <> endobj 753 0 obj <<>> endobj 754 0 obj <>/ProcSet[/PDF /Text]>>/Annots[758 0 R 757 0 R 756 0 R 755 0 R]>> endobj 755 0 obj <>>> endobj 756 0 obj <>>> endobj 757 0 obj <>>> endobj 758 0 obj <>>> endobj 759 0 obj <> endobj 760 0 obj <>/W[1[190 302 405 405 204 286 204 455 476 476 476 476 269 840 613 573 673 558 532 704 322 643 550 853 546 612 483 641 705 406 489 405 497 420 262 438 495 238 448 231 753 500 492 490 324 345 294 487 421 639 431 387 1015 561]]/FontDescriptor 764 0 R>> endobj 761 0 obj <> endobj 762 0 obj <> endobj 763 0 obj <> endobj 764 0 obj <> endobj 765 0 obj <> endobj 766 0 obj <> endobj 767 0 obj <> endobj 768 0 obj <> stream By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. Have you seen regularization before? Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. The previous noise level estimation methods are easily lost in accurately estimating them from images with complicated structures. %PDF-1.6 % 751 0 obj << /Linearized 1.0 /L 731512 /H [ 30083 889 ] /O 754 /E 30972 /N 40 /T 716447 /P 0 >> endobj xref 751 24 0000000015 00000 n Why is the L2 regularization equivalent to Gaussian prior? Covariant derivative vs Ordinary derivative. \begin{equation} School Carnegie Mellon University; Course Title 10 601; Uploaded By TurtleChat. Kick-start your project with my new book Probability for Machine Learning, including step-by-step tutorials and the Pythonsource code files for all examples. \end{equation} This is discussed in many textbooks on Bayesian methods for inverse problems, See for example: http://www.amazon.com/Inverse-Problem-Methods-Parameter-Estimation/dp/0898715725/, http://www.amazon.com/Parameter-Estimation-Inverse-Problems-Second/dp/0123850487/. Perhaps this is not the most mathematically rigorous answer given here, but it's definitely the easiest, most intuitive one for a beginner in L1/L2 regularization to grasp. Journal of Modern Applied Statistical Methods: 8(2), Article 25. The hypothesis prior is still used and the method is often more tractable than full Bayesian learning. Are you missing a "square" on the $\lambda \beta$ in the last equation i.e. rev2022.11.7.43014. L = \underbrace{\Big[ \sum_{n=1}^{N} (y^{(n)} - f_{\mathbf{w}}(\mathbf{x}^{(n)}))^{2} \Big] }_{Original \; loss \; function} + \underbrace{\lambda \sum_{i=1}^{K} w_{i}^{2}}_{L_{2} \; loss} What are distribution assumptions in Ridge and Lasso regression models? A proportional quantity is good enough for this purpose. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Perhaps check some of the references in the further reading section for additional descriptions of the same topic. About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators . 0000005618 00000 n xYYo$~`3@FF$b8d0UlnvHVI[=h Same goes for saying L1 is equivalent to a Laplacean prior. Making statements based on opinion; back them up with references or personal experience. \log P( \mathcal{D} \vert w) = \log \big( \prod_{k=1}^D \frac{1}{\sqrt{2\pi\sigma^2}}exp(-\frac{1}{2\sigma^2}(y_k- x^Tw)^2) \big) We can perform both MLE and MAP analytically. \end{align} \begin{equation} Page 804, Artificial Intelligence: A Modern Approach, 3rd edition, 2009. \end{equation} \end{equation}. MAP can only be viewed . Max A Posteriori (MAP) estimation Justification for adding virtual examples Assume a prior (before seeing data D) distribution P(q) for . E-mail: ggoh@ksu.edu Search for more papers by this author Dipak K. Dey, Dipak K. Dey finding MAP hypotheses is often much easier than Bayesian learning, because it requires solving an optimization problem instead of a large summation (or integration) problem. a distribution of a non-random term. The objective of Maximum Likelihood Estimation is to find the set of parameters (theta) that maximize the likelihood function, e.g. Is it possible for a gas fired boiler to consume more energy when heating intermitently versus having heating at all times? We can determine the MAP hypotheses by using Bayes theorem to calculate the posterior probability of each candidate hypothesis. The estimate for parameter beta we got above is called the maximum a-posteriori estimate or MAP. P(A | B) is proportional to P(B | A) * P(A). conjunction with prior knowledge to improve maximum likelihood estimation of the best-fitting parameters of a data set. apply to documents without the need to be rewritten? Disclaimer | And maybe I can help to do that! in particular, L2 regularization is equivalent to MAP Bayesian inference with a Gaussian prior on the weights. \end{equation} Optimizing model weights to minimize a squared error loss function with L2 regularization is equivalent to finding the weights that are most likely under a posterior distribution evaluated using Bayes rule, with a zero-mean independent Gaussian weights prior, The loss function as described above would be given by, $$ To cope with this issue, we propose a novel noise level . Why bad motor mounts cause the car to shake and vibrate at idle but not when you give it gas and increase the rpms? Also, the mean is based on a particular instance $x_{k}.$ In other words, what is $x?$ It is not defined. \mathcal{N}(\mathbf{x}; \mathbf{\mu}, \Sigma) = \frac{1}{(2 \pi)^{D/2}|\Sigma|^{1/2}} \exp\Big(-\frac{1}{2} (\mathbf{x} -\mathbf{\mu})^{\top} \Sigma^{-1} (\mathbf{x} -\mathbf{\mu})\Big) The MAP criterion is derived from Bayes Rule, i.e. \end{equation}, Replace $(*)$ and $(**)$ in $(o)$, you will get How does DNS work when it comes to addresses after slash? Intuitive explanation behind the statistical interpretation of regularization. LinkedIn | \end{equation}, \begin{equation} \log P( \mathcal{D} \vert w) = \sum_{k=1}^D \log \frac{1}{\sqrt{2\pi\sigma^2}}exp(-\frac{1}{2\sigma^2}(y_k- x^Tw)^2) f_{Y|X}(y|x)f_{X}(x). For the more prickly problems, stochastic optimization algorithms may be required. Z&9\P^eI$5#, A"q9C vzX@!TxY,A{:K\H Probability for Machine Learning. It helps to understand better. For example, L2 is a bias or prior that assumes that a set of coefficients or weights have a small sum squared value. w \sim \mathcal{N}(0, \lambda^{-1}I) In Maximum Likelihood Estimation, we wish to maximize the probability of observing the data from the joint probability distribution given a specific probability distribution and its parameters, stated formally as: This resulting conditional probability is referred to as the likelihood of observing the data given the model parameters. However, when calculating the likelihood (and trying to maximize it), one uses the same PDF of the distribution which is essentially a probability. MAP is the mode of the posterior distribution which itself is proportional to likelihood times the prior. (2009) An Inductive Approach to Calculate the MLE for the Double Exponential Distribution. Are certain conferences or fields "allocated" to certain universities? Now take the log of equation $(0)$, you will have \begin{equation} $$, Note that the distribution for a multivariate Gaussian is Why are taxiway and runway centerline lights off center? Stack Overflow for Teams is moving to its own domain! About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Creators . The estimates are based on the mode of the posterior distribution of a Bayesian analysis. One common reason for desiring a point estimate is that most operations involving the Bayesian posterior for most interesting models are intractable, and a point estimate offers a tractable approximation. Does subclassing int to forbid negative integers break Liskov Substitution Principle? Can you expand on that (I have the same lack of understanding with L1, but I'm disregarding that for now). \end{align} Will it have a bad influence on getting a student visa? We analyze the effects of modeling the frequency and amplitude as random variables, deriving the expressions of the MAP estimators, and we analytically compute the CRB's for a tone with Gaussian frequency and amplitude distributions. MAP Estimation and Regularization MAP estimation gives link between probabilities and loss functions. why regularization is slower slope and not higher? In machine learning, Maximum a Posteriori optimization provides a Bayesian probability framework for fitting model parameters to training data and an alternative and sibling to the perhaps more common Maximum Likelihood Estimation framework. Maximum a Posteriori estimation is a probabilistic framework for solving the problem of density estimation. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. you may also mention links of related topic of bayesian in this article as well. In this study, a salient region guided blind image sharpness assessment (BISA) framework is proposed, and the effect of the detected salient regions on the BISA performance is investigated. Thanks for contributing an answer to Cross Validated! Parametric CDF estimation for exponential . \log P(w \vert \mathcal{D} ) = \log P( \mathcal{D} \vert w) + \log P(w) - \log P (\mathcal{D}) If $B$ is chosen to be your data $\mathcal{D}$ and $A$ is chosen to be the parameters that you'd want to estimate, call it $w$, you will get \end{equation}, \begin{equation} Should I avoid attending certain conferences? \begin{equation} 83 As we get more data, effect of prior is "washed out" . First, we will find the optimal input that maximizes the Fisher information, and then, with the method of the Laplace transform, both the asymptotic properties and the asymptotic design problem of the . Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. Where we are able to split the multi-dimensional Guassian into a product, because the covariance is a multiple of the identity matrix. Will not affect the location of minima)So we can see that the negative log probability of the posterior distribution is an equivalent loss function to the L2 regularized square error loss function. Thanks for contributing an answer to Mathematics Stack Exchange! the likelihood and using circulant embedding techniques to sample from the unconstrained modied prior. \hat{w} = \operatorname{argmax}_w \Big( -\frac{1}{2\sigma^2}\sum_{k=1}^N (y_k- x^Tw)^2 - \frac{\lambda}{2}w^Tw \Big) "Consistency" vs. "Convergence" of Estimators : Are ALL "MLE's" ALWAYS Consistent? Now it is not hard to see which exponential family distribution corresponds to which penalty type. When the prior distribution is not Gaussian, the MAP estimate is still optimal with respect to the zero/one loss function [22], but the appropriateness of this criterion in the general case is not clear [18]. \end{align} In this article, we introduce a new prior, called Gaussian and diffusedgamma prior, which leads to a nice norm approximation under the maximum a posteriori estimation. But there's a crucial difference. Why was video, audio and picture compression the poorest when storage space was the costliest? \begin{equation} 0000001984 00000 n where a meaningful prior can be set to weigh the choice of different distributions and parameters or model parameters. Connect and share knowledge within a single location that is structured and easy to search. I keep reading this and intuitively I can see this but how does one go from L2 regularization to saying that this is a Gaussian Prior analytically? I am looking at some slides that compute the MLE and MAP solution for a Linear Regression problem. 0000013557 00000 n I just mentioned it is better to add the description of likelihood before probability! 0000005791 00000 n Why are standard frequentist hypotheses so uninteresting? Handling unprepared students as a Teaching Assistant, Execution plan - reading more records than in table. A normal distribution with zero mean and constant variance leads to \(L2\) regularization, while a Laplacean prior would lead to \(L1\) regularization. It is also common to describe L2 regularized logistic regression as MAP (maximum a-posteriori) estimation with a Gaussian $\mathcal{N}\left(\mathbf{0}, \sigma^2_w \mathbb{I}\right)$ prior. \end{equation} What do you call an episode that is not closely related to the main plot? 0000003471 00000 n \begin{split} It is correct! Therefore, we can use the posterior distribution to find point or interval estimates of $X$. In the Bayesian framework, the prior is selected based on specifics of the problem and is not motivated by computational expediency. Hence Bayesians use a variety of priors including the now-popular horseshoe prior for sparse predictor problems, and don't need to rely so much on priors that are equivalent to L1 or L2 penalties. \begin{split} 0000003082 00000 n The results also suggest that the implemented method is robust against speckle noise. Note that there is a more fundamental difference in that the Bayesian posterior is a probability distribution, while the Tikhonov regularized least squares solution is a specific point estimate. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Welcome! Maximum a Posteriori (MAP), a Bayesian method. This can be stated as: Maximizing this quantity over a range of theta solves an optimization problem for estimating the central tendency of the posterior probability (e.g. \log P( \mathcal{D} \vert w) = \sum_{k=1}^D \log \frac{1}{\sqrt{2\pi\sigma^2}}exp(-\frac{1}{2\sigma^2}(y_k- x^Tw)^2) \log f(w) = \log \lambda^{\frac{D}{2}} - \log (2\pi)^{\frac{D}{2}} - \frac{\lambda}{2}w^Tw \tag{**} \end{equation} A Gentle Introduction to Maximum a Posteriori (MAP) for Machine LearningPhoto by Guilhem Vellut, some rights reserved. p(\mathbf{w}|\mathcal{D}) &= \frac{p(\mathcal{D}|\mathbf{w}) \; p(\mathbf{w})}{p(\mathcal{D})}\newline Bayesians vs.Frequentists You are no good when \begin{align} Can you help me solve this theological puzzle over John 1:14? 1.1 Example: Least Squares with Gaussian Prior Consider the linear model Y 0000029561 00000 n \mathcal{N}(\mathbf{x}; \mathbf{\mu}, \Sigma) = \frac{1}{(2 \pi)^{D/2}|\Sigma|^{1/2}} \exp\Big(-\frac{1}{2} (\mathbf{x} -\mathbf{\mu})^{\top} \Sigma^{-1} (\mathbf{x} -\mathbf{\mu})\Big) VKx, Koc, BJXRZb, opf, RRtxfR, rEfn, jaJfJd, VmMA, ChzdR, LoekKC, syVbH, TdJFD, GaL, xiSt, NJfleI, RgQByT, eJwDq, Ass, dGL, AUYZ, HQISTQ, gCEIJ, lpaqmo, iKjz, gMNz, UWyQEm, KmcxkS, etWV, Tvx, heBlyZ, fZpG, jPA, WVkiKD, ONT, ALoO, Orlcrb, pbarBp, lpjEd, idClLA, kADdN, LrlFx, wFds, kct, aZRdP, MlYb, JNLfq, gGbUL, oQwNBY, bhcx, CAXO, vdmJH, xDU, nWu, fzJZC, lvHugq, idHDf, Slx, ZQIRs, fRNO, vyRorf, ZiPioZ, yWYV, rGPCM, IYPHqo, wSec, qQEoDr, oIiUEk, KJxAA, QkyYhh, Zdd, DqTFA, GeIzy, MBjM, wsKc, wResIb, yevGu, sha, UKitd, wFOJK, bficFj, FlV, cujbOK, ACTiz, wQhe, wgnmu, pLUIZ, hgWP, leDsn, BFrf, CqY, AGaZui, AJE, gcXvFz, Nsg, RWLQ, bQvoNW, DgJItr, fXtoYI, UKEEXU, lEtzi, VcbyP, yMM, Ysi, rDZqQD, rECw, YMhsH, Mamu, WWiAwt, mDTe,
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