Once you click that button, you will be asked, if you want to use a regular single/multiple criteria analysis or a Cost-Effectiveness analysis. Note that is your sample consists of only zeros and one that the proportion is the sample mean. 6b` 3 l> As the parameter for the Bernoulli distribution has range 0 1 (as represents a probability) then it is seen that 0 1 L ( ) d = 1 2. import numpy as np import matplotlib.pyplot as plt They are Investment A and Investment B. 6.2 A description of credibilities: The beta distribution To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Thanks for contributing an answer to Stack Overflow! %%EOF You may have noticed that the likelihood function for the sample of Bernoulli random variables depends only on their sum, which we can write as Y = i X i. We say that has a Bernoulli distribution with parameter if its probability mass function is. In this tutorial, you will learn basically what is a Bernoulli utility function, and how to use a Bernoulli Utility function in a decision tree. I got the following output: Bernoulli. 100, and ask yourself, what is your utility value for that wealth? Bernoulli data, Beta and mixture of Beta priors Priors, likelihood, prior/posterior predictives Hedibert Freitas Lopes 6/1/2021. Why are there contradicting price diagrams for the same ETF? For example, in a binomial distribution, you know the number of successes and fails and would like to . likelihood can still be used if the joint distribution (pmf or pdf) of Y1,. Translations are not retained in our system. The Bernoulli distribution is implemented in the Wolfram Language as BernoulliDistribution [ p ]. def maximum_likelihood (param, pmf): i = symbols ('i', positive=true) n = symbols ('n', positive=true) likelihood_function = product (pmf, (i, 1, n)) # calculate partial derivative for parameter (p for bernoulli) deriv = diff (likelihood_function, param) equation_to_solve = eq (deriv,0) # equate with 0 # solve above equation and return (A.2) A sensible way to estimate the parameter given the data y is to maxi-mize the likelihood (or equivalently the log-likelihood) function, choosing the In the Bernoulli case with a conditional logit model, perfect fit of the model breaks down the maximum likelihood method because 0 or 1 cannot be attained by i = logit1(x i ) = exp(x i ) 1 +exp(x i ). When you have 2 equations with 2 variables, using linear algebra, you can solve the value for those variables, right? and if T(x)is aminimal su cient statistic, then ^ is a function of Form theNeyman-Fisher Factorizaton Theorem Yes. After this module you should be able to recognize and be functional in these key concepts. Since Y has a binomial distribution with n trials and success probability , we can write its log likelihood function as L ( ) = log ( n y) y ( 1 ) n y Bernoulli Distribution and Binomial Trials 15:34 Rewrite the computations of loglik and lik as functions, vectorize them, then optimize. Neither Project Euclid nor the owners and publishers of the content make, and they explicitly disclaim, any express or implied representations or warranties of any kind, including, without limitation, representations and warranties as to the functionality of the translation feature or the accuracy or completeness of the translations. Unfortunately, the plot of the likelihood function prints out just a straight line instead of a curve which is not right. In a steady flow of a fluid, the pressure of the fluid decreases when the velocity of the fluid increases. I have a dataset containing the results of 10 fair coin tosses for 5 different students. 9.998e-05 seems quite small for something that is necessarily positive. As you have checked the box "I want to use a utility function", you will be presented with a utility function editor. SpiceLogic Inc. All Rights Reserved. Distributions, Jan 30, 2003 - 1 - Hence, we might consider the marginal likelihood function based on a function T or the conditional likelihood given a function S. hb```f``+@9n>``>#UY"C2_&a;U8BADv0)& BPsj v03Xw8UUl@)#`UkAn. 504), Mobile app infrastructure being decommissioned, Function to clear the console in R and RStudio, Vectorizing a Large 2D Dataframe for optimx L-BFGS-B efficiency, maximum likelihood in double poisson distribution, Problems plotting log-likelihood-function with ggplot2. He proposes that marginal utility is inversely proportional to wealth. Changes in the log-likelihood function are referred to as "log-likelihood units". y Bernoulli() has probability mass function Pr(y = k)=f(k)= . (i.e. Since the likelihood function looks a Beta distribution and the prior is a Beta distribution, it follows that the posterior of \(\theta\) . If ^(x)is a maximum likelihood estimate for , then g( ^ x))is a maximum likelihood estimate for ). The probability mass function f of this distribution, over possible outcomes k, is given by : If you are not familiar with how to create the decision tree in our decision tree software, please visit the getting started page. p ( 0) = P ( X = 0) = 1 p, p ( 1) = P ( X = 1) = p. The cumulative distribution function (cdf) of X is given by. In particular, the case in which T and S may be taken to be functions of the maximum likelihood estimators is considered and approximations to the resulting likelihood functions are presented. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. I had copied&pasted from your code without realizing that the likelihood had. 504), Mobile app infrastructure being decommissioned. the data y, is called the likelihood function. Making statements based on opinion; back them up with references or personal experience. Click the button "Set up Criteria". The log-likelihood plot works well but the likelihood function plot doesn't show anything. let's call this constant "a". Consider likelihood inference about a scalar function of a parameter . You have requested a machine translation of selected content from our databases. Anytime, you click the Utility value link shown on each node, the Payoff editor will show up. Say, that excitement value is "10". If total wealth is expressed as W, and the utility function isU(W), then. %PDF-1.5 % Fluid consist of liquid or gas 2. bernoulli_likelihood (p_to_try = 0.0001) ## [1] 9.998e-05 Wow, although I know that this isn't the probability of the observed data , I do know that it's proportional to the data. names(max_loglik) only prints the maximum and the objective values. Discover bayes opimization, naive bayes, maximum likelihood, distributions, cross entropy, and much more in my new book , with 28 step-by-step tutorials and full Python source code. So, here, we used the differential operator . In our expression, we introduced a proportional constant "a". hmm, maybe someone will be somewhat excited. (i.e. But starting in the "Calculating Likelihood Function for theta" section, it seems like we only have 1 feature, and the notation for the X_j changes from . Here I've used an anonymous function that returns the likelihood of our current data given a value of p; I've also specified that the values of p must lie in the interval [0, 1] and asked optimize to maximize the result, rather than minimize, which is the default behavior. Plots of Likelihood and log-likelihood function of Bernoulli random variable, Going from engineer to entrepreneur takes more than just good code (Ep. Same as an exponential utility function or any other utility function, you can use a Bernoulli utility function to associate a payoff in your decision tree. How can I plot this in R ? In logistic regression, we pass the weighted sum of inputs through an activation function that can map values in between 0 and 1. What do you call an episode that is not closely related to the main plot? Again, remember that it is differences in l that matter, not the actual values. This functionality is provided solely for your convenience and is in no way intended to replace human translation. You get a number. Here are the likelihood functions for random samples from some of our favorite distributions: 1. Likelihood of is a measure of how well the given data supports that particular value of . Then click the "Proceed" button. Objectives; Binomial model. The discrete data and the statistic y (a count or summation) are known. Likelihood Function. L A 3. Who is "Mar" ("The Master") in the Bavli? Then, you will be asked if you want to add another criterion. Why is there a fake knife on the rack at the end of Knives Out (2019)? The cumulative distribution function of a Bernoulli random variable X when evaluated at x is defined as the probability that X will take a value lesser than or equal to x. You can determine the value of "a" and "b" like this. You will have access to both the presentation and article (if available). You can fine-tune this value "a" to match someone's net wealth and model his/her utility function. Since ln(0) is the number that we get by solving the equation: There is no value of x that satisfies this equation. Why does sending via a UdpClient cause subsequent receiving to fail? When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. The idea is that, whenever you are running an experiment which might lead either to a success or to a failure, you can associate with your success (labeled with 1) a . A Bernoulli random variable X with success probability p has probability mass function f(x)=px(1p)1x x =0,1 for 0 <p <1. Now, say, you have 100,000$ in your pocket. A negative value tells you the curve is bending downwards. Assuming from your post you already have the first derivative of the log-likelihood function d ln f d p = i x i p n i x i 1 p giving p ^ = i x i n Second deriative The parameters of the model can be estimated by maximizing a likelihood function that predicts the mean of a Bernoulli distribution for each example. So, you will get "a" and "b" accordingly. For example, the likelihood ratio L ( 0 )/L ( 1) is an indicator of whether the observation x=3 favours = 0 over = 1 . Let its support be Let . endstream endobj 77 0 obj <> endobj 78 0 obj <> endobj 79 0 obj <>stream A random variable X has a Bernoulli distribution with parameter p, where 0 p 1, if it has only two possible values, typically denoted 0 and 1. 1. You can check the Marginal Utility function, Absolute Risk Aversion, and Relative Risk Aversion from the radio buttons as you can see at the bottom of the panel. . Our Decision Analysis Software (Decision Tree Software or Rational Will) can calculate that parameter based on the Minimum and Maximum possible values in the decision context, which is collected from the user. Space - falling faster than light? The lowest payoff will result in the lowest utility value which can be 0, -1, or -100, depending on the preferences as well. This is called the extended likelihood function. Enter Minimum = 100$ and maximum = 20,000$. Then, create a decision tree like this. The log-likelihood plot works well but the likelihood function plot doesn't show anything. So, for a value of x, your Bernoulli Utility Function will give a value that is undefined. Lets us look at a small example of a Bernoulli trial. Can a black pudding corrode a leather tunic? Answer: To obtain the most likely estimate of the Bernoulli parameter p given your sample data. These are the most fundamental core concepts in mathematical biostatistics and statistics. I need to test multiple lights that turn on individually using a single switch. Maximum likelihood estimation of the log-normal distribution using R, Plotting density graph to show likelihood, Coin tossing simulation unexpected probabilities, How to make an R function that loops over two lists, Compute the posterior probability given a Bernoulli distributed likelihood, Problems plotting log-likelihood-function with ggplot2. How do planetarium apps and software calculate positions? Bernoulli suggests a form for the utility function in terms of a differential equation. Not sure about everyone, but most rational people will feel the "0" temptation to get that 1$ when the 100,000$ he or she has in his/her pocket. jun 2004. This website uses cookies to ensure you get the best experience on our website. 503), Fighting to balance identity and anonymity on the web(3) (Ep. Actually, your data follow a Binomial distribution (in theory) since each student tossed 10 coins, not 1. We define heads as "Success" and tails as "Failure, though reversing this definition will make no difference. @Edward so you are saying that i should use the likelihood function for the binomial distribution instad f bernoulli ? 76 0 obj <> endobj The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so n would be 1 for such a binomial distribution). The second derivative tells you how the first derivative (gradient) is changing. The quantile is defined as the smallest value x such that F (x) p, where F is the distribution . p (x) is computed using Loader's algorithm, see the reference below. We can solve this differential equation to find the function "U(W)". The likelihood function (often simply called the likelihood) is the joint probability of the observed data viewed as a function of the parameters of the chosen statistical model.. To emphasize that the likelihood is a function of the parameters, the sample is taken as observed, and the likelihood function is often written as ().Equivalently, the likelihood may be written () to emphasize that . Parameters The Bernoulli distribution uses the following parameter. This occurs at a maximum. (3) Also, the optimize function doesn't work. You can also see a green vertical line that indicates where your utility stands in the plot based on the currently set payoff. I need help with this one. In a likelihood function, the data/outcome is known and the model parameters have to be found. It can also be used as an approximation to the binomial distribution when the success probability of a trial is very small, but the number of trials is very large. Suppose that Y1,. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide. The likelihood function is not a probability function; but it is a positive . It is also a special case of the two-point distribution, for which the possible outcomes need not be 0 and 1. Connect and share knowledge within a single location that is structured and easy to search. I need help with this one. We will show how to calculate this constant in this tutorial. The process of finding that best fit is called maximum likelihood estimation (MLE). Definition Let be a discrete random variable. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. my values are [8,8,4,5,6] and probability is 0.5 (p = 0.5) since it is a fair coin toss. D is a np matrix with elements i/ r, the derivatives of () with respect to the parameters. The properties and construction of marginal and conditional likelihood functions based on parameter-dependent functions are studied. 10 I don't understand the use of diodes in this diagram. Almost everything we cover below will apply to this case, as well. Kick-start your project with my new book Probability for Machine Learning, including step-by-step tutorials and the Python source code files for all examples. Maximum Likelihood Estimation for the Bernoulli Distribution For reproducibility, we can include a random_state argument assigned to a number. These constants "a" and "b" are called scaling parameters. 135 2008 Jon Wakefield, Stat/Biostat 571 Often we work with the natural logarithm of the likelihood function, the so-called log-likelihood function: logL(;y) = Xn i=1 logf i(y i;). Will Nondetection prevent an Alarm spell from triggering? Thomas A. Severini. The R syntax for the cumulative distribution function of the Bernoulli distribution is similar as in Example 1. Then you will be presented with the following screen. Not the answer you're looking for? Discuss. Let's say that we have 100 samples from a Bernoulli distribution: In [1]: import torch import numpy as np from torch.autograd import Variable sample = np. 0.9). Click "No". Consider the below image: What's the proper way to extend wiring into a replacement panelboard? How much excited will you feel? L () = 3 e - / 6. "Likelihood functions based on parameter-dependent functions." .,Yn and independent and Yi B(p). The parameters of the model can be estimated by maximizing a likelihood function that predicts the mean of a Bernoulli distribution for each example. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. The likelihood is a function of the parameter, considering x as given data. The function will take the probability of success (p) as a shape parameter The size parameter decides the number of times the trials are to be repeated. The Bernoulli distribution is associated with the notion of a Bernoulli trial . Find centralized, trusted content and collaborate around the technologies you use most. Say, you have 50$ in your pocket, how much more excited will you feel if you just get 1$ (so your total wealth will be 51$)? bernoulli_likelihood <- function(theta, data) { # `theta` = success probability parameter ranging from 0 to 1 # `data` = the vector of data (i.e., a series of 0s and 1s) n <- length(data) z <- sum(data) return(theta^z * (1 - theta)^(n - sum(data))) } This will come in handy in just a bit. In the "Bernoulli Naive Bayes" section and in the previous article (Naive Bayes Classifier) I was under the impression that we have P features (the X_j) for each of the N observations. Edwards in his 1972 monograph, Likelihood) takes the likelihood function as the fundamental basis for the theory of inference. A planet you can take off from, but never land back. In particular, the case in which T and S may be taken to be functions of the maximum likelihood estimators is considered and approximations to the resulting likelihood functions are presented. The word "quasi" refers to the fact that the score may or not correspond to a probability function. I have a dataset containing the results of 10 fair coin tosses for 5 different students. What is Bernoulli distribution? 3.3 Properties of the Maximum Likelihood Estimator Removing repeating rows and columns from 2d array. If you would like to express the inverse relationship function you obtain the logistic transformation $$ \pi = \frac{1}{1 + \exp \left( - \theta \right)} $$ Regarding you two questions, and as far as I understand the issues: The logistic function rises from the Bernoulli distribution. Like any proportional expression, we can introduce a constant and express with an equal sign. By solving the differential equation by integrating both sides, we get. Estimation of parameter of Bernoulli distribution using maximum likelihood approach f(y_i ~| x_i; \beta, \sigma^2) & = & \frac{1}{\sqrt{2 \pi \sigma^2}} ~ \exp \left\{ Figure 8.1 illustrates finding the maximum likelihood estimate as the maximizing value of for the likelihood function. Thanks for the help. Can an adult sue someone who violated them as a child? I am trying to obtain plots for both the likelihood and the log-likelihood. Python code for plotting bernoulli distribution in case of a loaded coin-from scipy.stats import bernoulli 87 0 obj <>/Filter/FlateDecode/ID[<0EDBC39EEC9E9B59A8D736F2E4BE322F><204D4D443219184C9688C1D712B4681F>]/Index[76 23]/Info 75 0 R/Length 69/Prev 188514/Root 77 0 R/Size 99/Type/XRef/W[1 2 1]>>stream A discrete random variable is one that has a finite or countable number of possible valuesthe number of heads you get when tossing three coins at once, or the number of students in a class. The Bernoulli distribution is the discrete probability distribution of a random variable which takes a binary, boolean output: 1 with probability p, and 0 with probability (1-p). Argument y is the data. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Did Great Valley Products demonstrate full motion video on an Amiga streaming from a SCSI hard disk in 1990? Contents 1 Properties 2 Mean 3 Variance 4 Skewness Select the "Bernoulli Utility Function" button. The maximizer of L T (zT;), T,is known as the quasi-maximum likelihood estimator (QMLE) of . Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. For example the difference in the support for q = 0.3 and q = 0.35 is l (0.3)-l (0.35) = 0.5630377 log-likelihood units. This module covers Conditional Probability, Bayes' Rule, Likelihood, Distributions, and Asymptotics. When viewed as a function of y and X with a fixed , it is just the probability density function. If, in the model with held fixed, T is ancillary, then a marginal likelihood may be based on the distribution of T, which depends only on ; alternatively, if a statistic S is sufficient when is fixed, then a conditional likelihood function may be based on the conditional distribution of the data given S. The statistics T and S are generally required to be the same for each value of . In simple words , . Examining the output of optimize, we can see that the likelihood of the data set was maximized very near 0.7, the . Why does sending via a UdpClient cause subsequent receiving to fail? Below, I have written the SAS code to . Note that, by the above definition, any indicator function is a Bernoulli random variable. If your payoff's Minimum and Maximum value is such a range where 0 can be a possible number, then our software will show an error. Why don't math grad schools in the U.S. use entrance exams? Select "Numerical Type". Is there a way to extract the variance/standard error of mu_hat? We can now use "a" and "b" to scale the utility function to fit any specific scenario. Bernoulli Distribution SAS Code Example. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. The covariance matrix of U() is also the negative expected value of U / , and is i =D T V1 D/ 2. Bernoulli distribution (with parameter ) - X takes two values, 0 and 1, with probabilities p and 1p - Frequency function of X p(x) = x(1)1x for x 2 f0;1g 0 otherwise - Often: X = 1 if event A has occured 0 otherwise Example: A = blood pressure above 140/90 mm HG. This new function is the logit function. The shorthand X Bernoulli(p)is used to indicate that the random variable X has the Bernoulli distribution with parameter p, where 0 <p <1. In these cases, the overall likelihood function is the product of the probability of finding a given value of n (given by equation (4.47)) and the usual likelihood function for the n values of x. So, you can understand that the more wealth is gained, the temptation to get an extra 1$ gets diminished. Now use algebra to solve for : = (1/n) xi . Connect and share knowledge within a single location that is structured and easy to search. So the combined likelihood function is. Index of the Bernoulli number or polynomial, specified as a nonnegative integer, symbolic nonnegative integer, variable, expression, function, vector, or matrix. Is this meat that I was told was brisket in Barcelona the same as U.S. brisket? Then, set another value to W, i.e. The likelihoodist approach (advocated by A.W.F. If an element of x is not 0 or 1, the result of dbern is zero, without a warning. i = l o g i t 1 ( x i ) = exp ( x i ) 1 + exp ( x i ). My profession is written "Unemployed" on my passport. https://doi.org/10.3150/bj/1089206405, Business Office 905 W. Main Street Suite 18B Durham, NC 27701 USA. Within the payoff editor, click the Utils link to open the utility function chart. Then, we can apply the pbern function to this vector: y_pbern <- pbern ( x_pbern, prob = 0.7) # Apply pbern function. Consequences resulting from Yitang Zhang's latest claimed results on Landau-Siegel zeros. I don't understand the use of diodes in this diagram. Sci-Fi Book With Cover Of A Person Driving A Ship Saying "Look Ma, No Hands!". For instance suppose our sample is 0, 1, 1, 0, 1 Now computer the sample mean \bar{x} = \frac{0+1+1+0+. Here is the Marginal Utility Function for the above-generated function. First available in Project Euclid: 7 July 2004, Digital Object Identifier: 10.3150/bj/1089206405, Rights: Copyright 2004 Bernoulli Society for Mathematical Statistics and Probability, Thomas A. Severini "Likelihood functions based on parameter-dependent functions," Bernoulli, Bernoulli 10(3), 421-446, (jun 2004), Registered users receive a variety of benefits including the ability to customize email alerts, create favorite journals list, and save searches. Thus for bernulli distribution L ( ) = k ( 1 ) n k Where k = i X i and ( 0; 1) To maximize L it is easy use logL l o g L = k l o g + ( n k) l o g ( 1 ) Derivating in and setting =0 you get ^ = k n = X n Share Cite Follow In this paper, we consider the case in which either T or S is allowed to depend on . Where to find hikes accessible in November and reachable by public transport from Denver? I am trying to obtain plots for both the likelihood and the log-likelihood. Is this homebrew Nystul's Magic Mask spell balanced? likelihood of p=0.5 is 9.7710 4, whereas the likelihood of p=0.1 is 5.3110 5. In order to avoid such a problem, we recommend setting the Minimum Payoff value greater than 0. Multiply both sides by 2 and the result is: 0 = - n + xi . Select "Maximize" and enter "Revenue" as shown below. Thanks for contributing an answer to Stack Overflow! 503), Fighting to balance identity and anonymity on the web(3) (Ep. Two methods of constructing a likelihood function for are conditioning and marginalizing. Demo: Blow It Up Hold a sheet of paper in front of your mouth and blow; the paper will rise. Find centralized, trusted content and collaborate around the technologies you use most. The Bernoulli distribution is a special case of the binomial distribution with n=1. Name for phenomenon in which attempting to solve a problem locally can seemingly fail because they absorb the problem from elsewhere? Each instance of an event with a Bernoulli distribution is called a Bernoulli trial. Probability Density Function This will count as one of your downloads. scipy.stats.bernoulli () is a Bernoulli discrete random variable. Let's get started. If n is a vector or matrix, bernoulli returns Bernoulli numbers or polynomials for each element of n. Bernoulli's principle 1. This content is available for download via your institution's subscription. but, when the net wealth is very low, the extra 1$ gets much more exciting. What is the Bernoulli Utility Function? When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. When {f t} is specied correctly in its entirety for {z t}, the . Copyright 2007-2020. The formula is given as follows: CDF = F (x, p) = 0 if x < 0 1p if 0 x < 1 1 x 1 { 0 i f x < 0 1 p i f 0 x < 1 1 x 1 Mean and Variance of Bernoulli Distribution We are ready to learn the model using maximum likelihood: In [4]: learning_rate = 0.00002 for t in range . The likelihood is a function of . Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide. I need to test multiple lights that turn on individually using a single switch. . f Bernoulli ( x; ) = x ( 1 ) 1 x, sampling x = 1 the likelihood is then L ( ) = . Stack Overflow for Teams is moving to its own domain! He proposes that marginal utility is inversely proportional to wealth. The . First, we have to create a vector of quantiles: x_pbern <- seq (0, 10, by = 1) # Specify x-values for pbern function. Your use of this feature and the translations is subject to all use restrictions contained in the Terms and Conditions of Use of the Project Euclid website. .,Yn is specied. Using some parameters, you can adjust the utility function in that way. We distinguish the function for the log-likelihood from that of the likelihood using lowercase l instead of capital L. The log likelihood for n coin flips can be expressed in this formula. The above is the code I wrote to try and plot the likelihood of observed data from a Bernoulli distribution as a function of $\theta$ . . Set any value to W, i.e. Investment A can bring 20,000$ in revenue with a probability of 0.2 and 500$ with a probability of 0.8. Click Proceed. You will be asked about the type of criterion. This matrix plays the same role as the Fisher information for likelihood functions. Bernoulli random variables are characterized as follows. Then, you will get 2 equations where the variables are just "a" and "b". The parameter to fit our model should simply be the mean of all of our observations. Let me give you an example. Why should you not leave the inputs of unused gates floating with 74LS series logic? 98 0 obj <>stream Plotting the Likelihood of a Bernoulli Distribution, Confidence interval for Bernoulli sampling, Compute the posterior probability given a Bernoulli distributed likelihood . Did Great Valley Products demonstrate full motion video on an Amiga streaming from a SCSI hard disk in 1990?
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