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

    I am fitting a mass spectrum using pdf(M)=Ns×S(M)+Nb×B(M; a, b) to determine the yield with the extended maximum likelihood fit, where Ns and Nb are the number of signal and background events, S(M) is the function for the signal, B(M;a, b) is the function for the background with parameters a and...
  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

    Hey! :giggle: For $n \in \mathbb{N}$ we consider the discrete statistical product model $(X ,(P_{\theta})_{\theta \in\Theta})$ with $X = \mathbb{N}^n$, $\Theta = (0, 1)$ and $p_{\theta}(x_i) = \theta(1 -\theta)^{x_i−1}$ for all $x_i \in \mathbb{N}, \theta \in \Theta$. So $n$ independent...
  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

    MHB Binomial Distribution: Likelihood Ratio Test for Equality of Several Proportions

    $\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: A survey of voter sentiment was conducted in four midcity political wards to compare the fraction of voters favoring candidate $A.$ Random samples of $200$ voters were polled in each of the...
  8. Ackbach

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

    $\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: Let $S_1^2$ and $S_2^2$ denote, respectively, the variances of independent random samples of sizes $n$ and $m$ selected from normal distributions with means $\mu_1$ and $\mu_2$ and common...
  9. F

    MHB Max Likelihood: True/False? w/Proofs

    Can someone help please? are these statements true or false? With some proofs.
  10. F

    MHB Maximum Likelihood: True/False + Proves | Help

    Is this statement true or false with some proofs? Please help
  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

    Hello community! I am facing a conceptual problem with the correlation matrix between maximum likelihood estimators. I estimate two parameters (their names are SigmaBin0 and qqzz_norm_0) from a multidimensional likelihood function, actually the number of parameters are larger than the two I am...
  13. F

    I Issue about the percentage of falling of height of Likelihood

    I have currently an issue about the height at which the projection of 1 sigma edges in 2D contour should intersect the associated Likelihood. Here a figure to illustrate my issue : At bottom left is represented the joint distribution (shaded blue = contours at 2 sigma (95% C.L) and classic...
  14. B

    Python Minimization likelihood function with parameters

    Hallo at all! I'm learning statistic in python and I have a problem to show you. I have this parametric function: $$P(S|t, \gamma, \beta)=\langle s(t) \rangle \left( \frac{\gamma-\beta}{\gamma\langle s(t) \rangle -\beta}\right)^2\left( 1- \frac{\gamma-\beta}{\gamma\langle s(t) \rangle...
  15. C

    MHB Log likelihood and Maximum likelihood

    I'm not sure how to get this first derivative (mainly where does the 4 come from?) I know x̄ is the sample mean (which I think is 1/2?) Can someone suggest where to start with finding the log-likelihood? I know the mass function of a binomial distribution is: Thanks!
  16. W

    I Distributivity/Inheritance of Max Likelihood Estimators

    Hi, IIRC, Maximum Likelihood Estimators ( MLEs) satisfy an " Inheritance" property , so that if ##m_1,m_2,..,m_n## are MLEs for ##M_1,M_2,...,M_n## respectfully and f is a Random Variable of the ##M_i##, then the MLE for f is given by ##f(m_1,m_2,...,m_n)##. Is this correct? If so, is there a "...
  17. C

    I Akaike Information Criterion Vs Likelihood Ratio Test

    Hello, I want to understand the difference between both goodness-of-fit tests, I would be glad if you could help me: Akaike Information criterion is defined as: ## AIC_i = - 2log( L_i ) + 2K_i ## Where ##L_i## is the likelihood function defined for distribution model ##i## . ##K_i## is the...
  18. W

    A How Does Maximum Likelihood Estimate Factor Loadings in PCA Path Models?

    Hi, I am looking into a text on PCA obtained through path diagrams ( a diagram rep of the relationship between factors and the dependent and independent variables) and correlation matrices . There is a "reverse" exercise in which we are given a correlation matrix there is mention of the use of...
  19. K

    I Is the Likelihood Function a Multivariate Gaussian Near a Minimum?

    Hello! I am reading Data Reduction and Error Analysis by Bevington, 3rd Edition and in Chapter 8.1, Variation of ##\chi^2## Near a Minimum he states that for enough data the likelihood function becomes a Gaussian function of each parameter, with the mean being the value that minimizes the...
  20. F

    I Likelihood, posterior, prior interpretation and credibility/confidence

    I try to understand the following article : testing general relativity from curvature and energy contents at cosmological scale I don't understand the title of figure 1 : where it is indicated the prior values for ##\omega_{b}, \omega_{\text{cdm}}, \text{h}, ...## : what do authors mean by...
  21. S

    MHB Maximum Likelihood Estimation Formula Simplification and Solution for p?

    Hey guys ! My mother language is not English by the way. Sorry for spelling and gramme. :) I'm curious to see if you can help me with my problem.I have already tried for almost a week and did not get to a solution. I also know, that the Maximum likelihood estimation is part of statistics and...
  22. F

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

    I am curently working on Forecast in cosmology and I didn't grasp very well different details. Forecast allows, wiht Fisher's formalism, to compute constraints on cosmological parameters. I have 2 issues of understanding : 1) Here below a table containing all errors estimated on these...
  23. astroman707

    Programs What is the likelihood of a physics major getting an engineering job?

    I’ve been told that physics majors are looked at favorably when applying for entry-level engineering jobs. Is this true? How qualified are they when compared to engineering majors? Are applied physics majors looked at any differently? Do employers care about an undergraduate thesis?
  24. R

    MHB Likelihood of things happening on with regularity depending on initial probability

    Hello All, Firstly, what a wonderful place for people to help one and other. I am glad that I have stumbled across : ) I have posted this query in advanced because to me there it is advanced but if it is basic to others then please excuse me. I can easily work out the probability of...
  25. F

    I Expectation Maximization (EM) : find all parameters

    I am tackling a technique to determine the parameters of a Moffat Point Spread Function (PSF) defined by: ## \text {PSF} (r, c) = \bigg (1 + \dfrac {r ^ 2 + c ^ 2} {\alpha ^ 2} \bigg) ^ {- \beta} ## with the parameter "(r, c) =" line, column "(not necessarily integers). The observation of a...
  26. F

    I Maximum Likelihood to find the original data or estimation directly

    I make confusions in the using of Maximum Likelihood to find (approximately) the original signal knowing the data observed and the using of Maximum Likelihood to find estimations of parameters of the PSF 1) Find (up to some point) the original signal : I start from this general definition (in...
  27. M

    MHB Finding the Maximum Likelihood Estimator for a Density Function

    Hey! :o We have the density function $f_x(x)=\frac{2c^2}{x^3}, x\geq 0, c\geq 0$. I want to calculate the maximum Likelihood estimator for $c$. We have the Likelihood Function $$L(c)=\prod_{i=1}^nf_{X_i}(x_i;c)=\prod_{i=1}^n\frac{2c^2}{x_i^3}$$ The logarithm of the Likelihood function is...
  28. FallenApple

    I Understanding Maximum Likelihood Estimation: Unpacking the Basics

    I'm getting a bit lost on some of the basics. So a Likelihood function determines the plausibility of parameters given the observed random variables. This is fine and all, but something seems a bit off. The observed random variables themselves must be generated from a probability distribution...
  29. B

    I How can I convert chi-square into a matrix and graph likelihood contours?

    Hello, I'm new to the world of "numerical" astronomy/cosmology and I've been studying maximum likelihood estimation for model tests. The (analytical) theory is somewhat easy to understand, but I've been struggling with the numerical aspect of the computations. Suppose you have some model with a...
  30. D

    I Likelihood of the maximum of a parabola

    I have a quadratic regression model ##y = ax^2 + bx + c + \text{noise}##. I also have a prior distribution ##p(a,b,c) = p(a)p(b)p(c)##. What I need to calculate is the likelihood of the data given solely the extremum of the parabola (in my case a maximum) ##x_{max} = M = -\frac{b}{2a}##. What I...
  31. M

    MHB Maximum of the Likelihood estimation

    Hey! :o I am looking at the Likelihood function. I have understood how we define that function, but having find the maximum of the Likelihood estimation, what is the meaning of that? What information do we have, when we have found the $\theta$ so that the Likelihood function L is maximized...
  32. Avatrin

    I Why is the maximum likelihood estimation accurate?

    Hi I've been googling maximum likelihood estimation. While I do understand how to compute it, I don't understand why maximizing the likelihood function will give us a good estimate of the actual parameter. In some cases, like the normal distribution, it seems almost obvious. However, in the...
  33. U

    MHB Showing that likelihood function is sufficient but not minimal sufficient

    Let $f(x|\theta)$ be a family of densities where the parameter space is finite, i.e., $\theta \in \Theta = \{\theta_1, \cdots, \theta_p\}$. Now consider the likelihood function statistic, defined to be $T(\mathbf{X}) = (f_{\theta}(\mathbf{X}))_{\theta \in \Theta} = (T_1(\mathbf{X}), \cdots...
  34. F

    I Maximum likelihood w/ histogram, zero probability samples

    I'm trying to replicate a machine learning experiment in a paper.The experiment used several signal generators and "trains" a system to recognize the output from each one. They way this is do is by sampling the output of each generator, and then building histograms from each trial. Later you...
  35. Ma Xie Er

    A Logistic Regression Interpretation

    I was trying to find an easy interpretation of the predicted probabilities of a logistic regression model, when one of my coworkers claimed that the logistic regression model is a likelihood. Now, I know that maximum likelihood estimation is used to estimate the parameters, but I didn't think...
  36. T

    Maximum likelihood of a statistical model

    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

    In a council vote for a proposal, there a nine council members. There is a 60% chance that each council person will vote in favour of the proposal. What is then likelihood that the proposal will pass? So, this means that 5 or greater members must vote in favour. These are independent events...
  39. pluviosilla

    I Why is the Maximum Likelihood Function a product?

    Why is the Maximum Likelihood function a product? Explanations of how the Maximum Likelihood function is constructed usually just mention that events are independent and so the probability of several such events is just the product of the separate probabilities. I get the logic w.r.t...
  40. J

    MHB Maximizing Likelihood Estimator of β

    Assuming α is known, find the maximum likelihood estimator of β f(x;α,β) = , 1 ,,,,,,, .(xα.e-x/β) ,,,,,, ,,,,,,α!βα+1 I know that firstly you must take the likelihood of L(β). But unsure if I have done it correctly. I came out with the answer below...
  41. S

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

    Hello! I have some concentrical spheres with many points inside. And I need to plot the total mass of points in each shell (so between 2 spheres) versus the radius of that shell (defined as (r1+r2)/2, where r1 and r2 are the radius of the 2 spheres forming the shell). I have 10 shells so my plot...
  42. Alltimegreat1

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

    How likely or unlikely is it for a passing object to fall into a stable orbit of another object? For example, if an Earth-sized rogue planet came near the sun, would the odds be something like 10 million to 1 that it would fall into a stable orbit?
  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

    How Can MCMC Methods Aid in Bayesian Parameter Estimation for Complex Models?

    Hi folks. I have the following question.I have a model M containing 20 adjustable parameters k = {k_j}. I also have 40-50 measured temporal profiles e = {e_i} at my disposal. I can use M to predict the experimental values after solving complex systems of differential equations.Consequently, I...
  46. NATURE.M

    Derivative of Log Likelihood Function

    So looking through my notes I can't seem to understand how to get from one step to the next. I have attached a screenshot of the 2 lines I'm very confused about. Thanks. BTW: The equations are for the log likelihood in a mixture of gaussians model EDIT: To elaborate I am particularly...
  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

    Find maximum likelihood estimators of an sample of size $n$ if $X\sim U(0,\theta].$ Hello MHB :)! Can any user help me please :)! I don't how follow...
  49. K

    Calculating log liklihood: Zero value of likelihood function

    Hello, I am analysing hydrology data and curve fitting to check the best probability distribution among 8 candidate distribution. (2 and 3 parameter distributions) The selection is based on the lowest AIC value. While doing my calculation in excel, how is it suggested to treat very low (approx...
  50. Overt

    Likelihood of postdoc position?

    I'm nearing the end of choosing a physics graduate school to attend and the decision is coming down to what research groups I'm interested in. My area of interest is in biological physics but I'm not quite sure whether I want to do computational or experimental work. A major factor in choosing...