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. Johny Boy

    A Fisher information from likelihood function for quantum circuits

    In the context of a single phase estimation problem of a quantum photonics experiment. For example consider a 3-photon quantum circuit (such as the Mach-Zehnder which depends on some phase shift operator which encodes a parameter ##\theta##) with a photon counting measurement (two detectors) at...
  2. X

    A Minimum-variance bound for the extended maximum likelihood estimation

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  3. tworitdash

    A Finding Global Minima in Likelihood Functions

    I have a likelihood function that has one global minima, but a lot of local ones too. I attach a figure with the likelihood function in 2D (it has two parameters). I have added a 3D view and a surface view of the likelihood function. I know there are many global optimizers that can be used to...
  4. M

    I Maximum likelihood to fit a parameter of this model

    Hi PF! Given random time series data ##y_i##, we assume the data follows a EWMA (exponential weighted moving average) model: ##\sigma_t^2 = \lambda\sigma_{t-1}^2 + (1-\lambda)y_{t-1}^2## for ##t > 250##, where ##\sigma_t## is the standard deviation, and ##\sigma_{M=250}^2 =...
  5. M

    MHB Finding MLE for $\theta$ of Statistical Product Model

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  6. ORF

    Binned Maximum Likelihood fit in python?

    Hi, I have been using Python for a while now, but so far for Least-squares fits using curve_fit from Scipy. I would like to start using Likelihood method to fit binned and unbinned data. I found some documentation in Scipy of how to implement unbinned likelihood fit, but I have not managed to...
  7. Ackbach

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  8. Ackbach

    MHB Likelihood Ratio Test for Common Variance from Two Normal Distribution Samples

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  9. F

    MHB Max Likelihood: True/False? w/Proofs

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  10. F

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  11. Arman777

    I What Does the Likelihood Function Tell Us?

    I am trying to understand the meaning of the Likelihood function or the value of the Likelihood itself. Let us suppose we are tossing coins and we have a sample data given as $$\hat{x} = \{H,H,T,T,T,T\}$$ For instance in this case the Likelihood Function can be given as $$L(\theta|\hat{x}) =...
  12. A

    A Correlation between parameters in a likelihood fit

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  13. F

    I Issue about the percentage of falling of height of Likelihood

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  14. B

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  15. C

    MHB Log likelihood and Maximum likelihood

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  16. W

    I Distributivity/Inheritance of Max Likelihood Estimators

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  17. C

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  18. W

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  19. K

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  20. F

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  21. S

    MHB Maximum Likelihood Estimation Formula Simplification and Solution for p?

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  22. F

    I What do "marginalized" or "marginalized error" mean? (contours - posterior)

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  23. astroman707

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  24. R

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  25. F

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  26. F

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  27. M

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  28. FallenApple

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  29. B

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  30. D

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  31. M

    MHB Maximum of the Likelihood estimation

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  32. Avatrin

    I Why is the maximum likelihood estimation accurate?

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  33. U

    MHB Showing that likelihood function is sufficient but not minimal sufficient

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  34. F

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  35. Ma Xie Er

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  36. T

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    Homework Statement I look at the distribution ##(Y_1,Y_2,...,Y_n)## where ##Y_i=μ+(1+φ x_i)+ε_i## where ##-1<φ<1## and ##-1<x_i<1## . x's are known numbers. ε's are independent and normally distributed with mean 0 and variance 1. I need to find the the maximum likelihood estimator for μ and...
  37. T

    A Improving intuition on applying the likelihood ratio test

    I am trying to better understand likelihood ratio test and have found a few helpful resources that explicitly solve problems, but was just curious if you have any more to recommend. Links that perhaps work out full problems and also nicely explain the theory. Similar links you have found...
  38. T

    MHB Calculating Likelihood of Proposal Passage in Council Vote of 9 Members

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  39. pluviosilla

    I Why is the Maximum Likelihood Function a product?

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  40. J

    MHB Maximizing Likelihood Estimator of β

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  41. S

    I Likelihood of some points fitting a derived function....

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  42. Alltimegreat1

    B Orbit Likelihood: The Odds of Objects Falling into Stable Orbits Explained

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  43. D

    Maximum Likelihood and Fisher Information

    Homework Statement Let X1, X2,...Xn be a random sample from pdf, f(x|θ) = θx-2 where 0 < θ ≤ x < ∞ Find the MLE of θMy attempt: Likelihood fxn: L(θ|x) = ∏θx-2 = θn∏ θx-2 And to find MLE, I take Log of that function and partial derivative (w.r.t θ, of log L(θ|x) and set that = 0, and get...
  44. D

    Find MLE of θ: Maximizing Likelihood fxn

    Homework Statement Let X1, X2,...Xn be a random sample from pdf, f(x|θ) = θx-2 where 0 < θ ≤ x < ∞ Find the MLE of θMy attempt: Likelihood fxn: L(θ|x) = ∏θx-2 = θn∏ θx-2 And to find MLE, I take Log of that function and partial derivative (w.r.t θ, of log L(θ|x) and set that = 0, and get...
  45. W

    Bayesian estimation via MCMC

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  46. NATURE.M

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  47. D

    Likelihood of confirming past life on Mars

    If there was microbial life on Mars in the distant past, would there be any way to conclusively confirm it? Or would any such attempt be reduced to speculation, like the Martian meteor which "possibly" contained evidence?
  48. Julio1

    MHB Maximum Likelihood Estimators for Uniform Distribution

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  49. K

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  50. Overt

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