What is Likelihood: Definition and 139 Discussions

In statistics, the likelihood function (often simply called the likelihood) measures the goodness of fit of a statistical model to a sample of data for given values of the unknown parameters. It is formed from the joint probability distribution of the sample, but viewed and used as a function of the parameters only, thus treating the random variables as fixed at the observed values.The likelihood function describes a hypersurface whose peak, if it exists, represents the combination of model parameter values that maximize the probability of drawing the sample obtained. The procedure for obtaining these arguments of the maximum of the likelihood function is known as maximum likelihood estimation, which for computational convenience is usually done using the natural logarithm of the likelihood, known as the log-likelihood function. Additionally, the shape and curvature of the likelihood surface represent information about the stability of the estimates, which is why the likelihood function is often plotted as part of a statistical analysis.The case for using likelihood was first made by R. A. Fisher, who believed it to be a self-contained framework for statistical modelling and inference. Later, Barnard and Birnbaum led a school of thought that advocated the likelihood principle, postulating that all relevant information for inference is contained in the likelihood function. But in both frequentist and Bayesian statistics, the likelihood function plays a fundamental role.

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  1. S

    Likelihood function of the gamma distribution

    There is a random sample of size n from a gamma distribution, with known r. Please help me formulate the likelihood function of the gamma distribution. I understand that the density function is the following: f\left(y;r,\lambda\right)=\frac{\lambda}{\Gamma\left(r\right)}\left(\lambda...
  2. T

    Obtain maximum likelihood estimates

    Problem : It is assumed that the arrival of the number of calls X per hour follow a poisson distribution with parameter \lambda, A random sample X1 = x1 , X2 = x2,...xn is taken .Obtain the maximum likelihood estimates of the average arrival rate. Please can you provide me with some...
  3. T

    Maximum Likelihood Estimator of Bin(m,p)

    Homework Statement Let \displaystyle X_1 ,..., X_n \stackrel {\text{i.i.d.}}{\sim} \text{Bin} (m,p) where m is known. Find the MLE \hat{p}_n of p and hence show that \hat{p}_n is unbiased. The Attempt at a Solution Can anyone check my attempt please? \displaystyle L(m,p) =...
  4. S

    Maximum likelihood estimator of mean difference

    Homework Statement A sample of size n_{1} is to be drawn from a normal population with mean \mu_{1} and variance \sigma^{2}_{1}. A second sample of size n_{2} is to be drawn from a normal population with mean \mu_{2} and variance \sigma^{2}_{2}. What is the maximum likelihood estimator of...
  5. B

    Likelihood of making a career out of clean energy?

    So, here's the situation: I'm a second-year engineering student at a not-exactly-prestigious school (Wright State University for the morbidly curious), and my current plan is to carry my education all the way through to a Master of Science in Renewable and Clean Energy. Do you really think...
  6. Y

    Maximum Likelihood Estimator for a function

    Homework Statement Consider the following density function: f(x) = ABB/xB+1; A<= x, zero elsewhere, where A > 0 and B> 0 Homework Equations The Attempt at a Solution f(x1,...,xn)= ABnBn(x1...xn)B+1 ln f(x1,...,xn)= Bn ln A + n ln B + (B+1)ln((x1...xn) After differentiating...
  7. H

    Likelihood Function Homework: Data, Parameter & Solution

    Homework Statement A series of independent measurements of the length of a rod delivers the following result (units of cm): {57.0, 49.3, 60.6, 74.5, 58.6, 62.4} (1) Assume that the measurements are Gaussian distributed, with...
  8. A

    How do I normalize the likelihood distribution?

    Hey everyone, i am new to using statistics and have come across a problem. I am trying to regenerate the result of a paper in which a theoretical parameter is constrained. I have calculated the chi squared and plotted the likelihood function (exp[-chisquared])to see what value of the...
  9. K

    Correct Form of Likelihood Function for Data w/ Upper/Lower Bound

    So, I have this problem I am tackling where I am doing a Bayesian scan of a multi-dimensional model. Most of the quantities predicted by the model have likelihood functions which are normal distributions (as functions of the possible data values), however there are some pieces of experimental...
  10. E

    Likelihood of light being absorbed/re-emitted in 50 light years

    What's the likelihood of light striking some interstellar gas and being absorbed, then re-emitted, and the re-emitted light actually being the light we see rather than the original light from the source? Also, what about the likelihood of reemission after passing through the heliosphere...
  11. L

    What does the log likelihood say?

    Hi, here's some information after fitting measurements to a lognormal distribution. What exactely does it mean that the log likelyhood is -67.175?? As of my understanding the log likelihood, is the natural logaritm of the likelihood function, which is the probability that these measurements...
  12. A

    Expectation of an Uniform distribution maximum likelihood estimator

    Hi had this question on my last "Statistical Inference" exam. And I still have some doubts about it. I determined that the maximum likelihood estimator of an Uniform distribution U(0,k) is equal to the maximum value observed in the sample. That is correct. So say my textbooks. After that the...
  13. R

    Find ξ Value with Maximum Likelihood

    http://bbs.mathchina.com/usr1PvjRWKew/4/22/graph_1261406609.jpg I can't understand the equation, may I know is there any good idea to get the value of ξ?
  14. C

    What Defines a Maximum Likelihood Estimator in Ordered Statistics?

    Homework Statement Let Y1<Y2<...<Yn be the order statistics of a random sample from a distribution with pdf f(x; \theta) = 1, \theta - 0.5 < x < \theta + 0.5. Show that every statistic u(X1,X2,...,Xn) such that Y_n - 0.5<u(X_1,X_2,...,X_n)<Y_1 + 0.5 is a mle of theta. In particular (4Y_1 +...
  15. C

    Finding the MLE for a Given Probability Using iid PDF

    Homework Statement Suppose X1...Xn are iid and have PDF f(x; \theta) = \frac{1}{\theta} e^{\frac{-x}{\theta}} \ \ \ 0<x<\infty Find the MLE of P(X<2). Homework Equations The Attempt at a Solution I know the MLE of theta is \overline{X} so would P(X<2) = 1 -...
  16. sylas

    A low likelihood for high climate sensitivity

    I've been aware of this work for a couple of months, and it has at last got through the review processes and has been accepted for publication. Annan, J.D., and Hargreaves, J.C. (2009) On the generation and interpretation of probabilistic estimates of climate sensitivity, to appear in Climatic...
  17. E

    Statistics: Method of Moments/Maximum Likelihood Estimation

    Homework Statement f(x;theta)=Exp(-x+theta) Find parameter estimates for variable 'theta' using maximum likelihood Estimator and Method of Moments. Homework Equations Log(x; theta) = Log(Exp(-x + theta)) -- For MLE Integral from theta to infinity of (x*Exp(-x + theta)) = xbar -- For Method of...
  18. S

    Maximum likelihood estimator of binominal distribution

    L(x_1,...,x_n;p)=\Pi_{i=1}^{n}(\stackrel{n}{x_i}) p^{x_i}(1-p)^{n-x_i} Correct so far? The solution tells me to skip the \Pi: L(x_1,...,x_n;p)=(\stackrel{n}{x}) p^{x}(1-p)^{n-x} This is contradictory to all the examples in my book. Why?
  19. S

    What is the Confidence Interval Formula for Maximum Likelihood Estimation?

    L(x_1,x_2,...,x_n;\theta)=\Pi _{i=1}^n (\frac{\theta}{2})^x (1-\frac{\theta}{2})^{1-x} = (\frac{\theta}{2})^{\Sigma^n_{i=1}x_i}(1-frac{\theta}{2})^{n-\Sigma^n_{i=1}x_i} Correct so far if f(x) = (\frac{\theta}{2})^x (1-\frac{\theta}{2})^{1-x} ? lnL(x_1,x_2,...,x_n;\theta) =...
  20. M

    Exponential distribution likelihood

    In general what do you do when some kind of data is missing from some data that follows exponential distribution. For example, say 3 observations are made by an instrument where x1=5, x2=3, but for x3 the instrument can not give a specific answer because it can't measure past 10. So the only...
  21. M

    Likelihood Function - Exponential Distribution

    Homework Statement X is exponentially distributed. 3 observations are made by an instrument that reports x1=5, x2=3, but x3 is too large for the instrument to measure and it reports only that x3 > 20 . (The largest value the instrument can measure is 10) a)What is the likelihood function...
  22. D

    Determining the Likelihood function

    I was under the impression that the likelihood function was simply the probability density function but viewing the parameter theta as the variable instead of the observations x. Ie p(x|theta) = L(theta|x) However, the likelihood function is no longer a probability function See Example 1...
  23. S

    What is the maximum likelihood estimator for a given density function?

    Homework Statement pdf: f(x)=ax^(a-1) ; 0<x<1, a>0 estimate a by maximum likelihood Homework Equations let L be maximum likelihood L=(a(x[1])^(a-1))(a(x[2])^(a-1))...(a(x[n])^(a-1)) The Attempt at a Solution Im trying to make this into a nicer expression: L=a^n... (now I am...
  24. S

    Maximum Likelihood Fitting normalization (Bevington)

    Hi, I'm posting this in this particular forum because, though this's a statistics question, my application is in high energy. My question is regarding a problem in Bevington's book (Data Reduction and Error Analysis..., Page 193, Ex. 10.1), but I'll give a general description here... Say...
  25. T

    Estimation of parameters using maximum likelihood method

    Homework Statement Let's have random value X defined by its density function: f(x; \beta) = \beta^2x \mbox{e}^{-\beta x} where \beta > 0 for x > 0 and f(x) = 0 otherwise. Expected value of X is EX = \frac{2}{\beta} and variance is \mbox{var } X = \frac{2}{\beta^2}. Next...
  26. D

    Maximum likelihood of Poisson distribution

    Homework Statement Suppose X has a Poisson distribution with parameter lambda. Given a random sample of n observations, Find the MLE of lambda, and hat lambda. Find the expected value and variance of hat lambda. Show that hat lambda is a consistent estimator of lambda. Homework...
  27. I

    Maximizing θ with Probability Mass Function and Marbles Data

    in need of help for how to do this question given probability mass function: x 1 2 3 4 p(x) 1/4(θ+2) 1/4(θ) 1/4(1-θ) 1/4(1-θ) Marbles 1=green 2=blue 3=red 4=white For 3839 randomly picked marbles green=1997 blue=32 red=906...
  28. R

    Effect of spreading center on likelihood to subduct

    Let's say that two oceanic plates run into each other in a head-on collision. Pretend that one is 2000 miles from its spreading center, and the other one is 4000 miles from its spreading center. My question to you is "which one of these would subduct and why?" Bill in Miami
  29. T

    Question about a Force Table - Likelihood of Uncertainty? (lab based)

    It's not a homework problem at all.. but it would help me with my understanding in my coursework, so I thought I'd ask here. I did a lab with a force table recently. I got my measured values easily, and understand the vector nature of forces... but I'm wondering how accurate/precise is a...
  30. Simfish

    What Does arg Mean in Maximum Likelihood Estimation?

    http://en.wikipedia.org/wiki/Maximum_likelihood What exactly does the "arg" here mean? It seems to be an unnecessary - the max L(\theta) seems to be sufficient enough. Or am I missing something? \widehat{\theta} = \underset{\theta}{\operatorname{arg\ max}}\ \mathcal{L}(\theta).
  31. A

    Poisson distribution, likelihood ratios

    Homework Statement An independent, identically distributed sample, x = (x1, ... , xn) of size n, is drawn from a Poisson distribution, parameter A. We want to test the null hypothesis H0 : A = A1 against the alternative hypothesis H1 : A = A2 where A1 < A2. Write down the likelihood ratio...
  32. G

    Method of moments/maximum likelihood

    I just needed some help with a few questions. Consider N independent random variables having identical binomial distributions with parameters Θ and n= 3. If n0 of them take on the value 0, n1 of them take on the value 1, n2 of them take on the value 2 and n3 of them take on the value 3, use...
  33. C

    Maximum Likelihood Estimation: Exploring Solutions

    Hi, I'm taking a basic course in statistical methods, and we recently learned of maximum likelihood estimation. We defined the likelihood as a function of some parameter 'a', and found the estimator of 'a' by requiring a maximum likelihood with respect to it. As an example, we took the...
  34. N

    Challenges of the Course and the Likelihood of Success

    As far as I have seen on the web, this course is holy abstract. Anyone willing to discuss the difficulties in it? Are most people likely to fail it? What's the consensus?
  35. B

    Generalized likelihood ratio test

    This is the problem (t for theta): X ~ Expo(t) = t * e ^ (-t * x), x>0, t >0 0 otherwise Test H0: t <= 1 vs. H1: t >1 using the generalized likelihood ratio test where you have a random sample from X {X1, X2, ... , X50} and the sum of all Xi = 35. Use alpha =...
  36. B

    How can I improve my understanding of Maximum Likelihood estimators?

    Can anyoe help with likelihood estimtor problems? :cry:
  37. T

    Estimate Standard Deviation of Mean Average w/ Maximum Likelihood

    How do I estimate the standart deviation for the mean average of an poisson-distribution ? The mean average was estimated with the maximum-likelihood method by graphing the likelihood in dependence of the mean average, then just reading off the value for which the likelihood became maximal. Up...
  38. S

    How Do You Find the Maximum Likelihood Estimator for Theta?

    Maximum likelihood estimator... ok, I'm stil a bit lost...so tell me if this is right: f_y(y;\theta) = \frac{2y}{\theta^2}, for 0 < y < \theta find the MLE estimator for theta. L(\theta) = 2yn\theta^{-2 \sum_1^n y_i . is this even right to begin with? then take the natural...
  39. Ivan Seeking

    ET Visitors: Scientists See High Likelihood

    http://www.space.com/searchforlife/et_betterodds_050114.html The link is good but as you can imagine, very slow. A few tries works best if needed. For some very striking government reports [NSA, CIA, USAF, etc], see also:https://www.physicsforums.com/showthread.php?t=2805
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