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. 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. 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. 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. 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. 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. 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. 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. MHB Max Likelihood: True/False? w/Proofs

Can someone help please? are these statements true or false? With some proofs.
10. MHB Maximum Likelihood: True/False + Proves | Help

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

15. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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...

49. 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. 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...