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Questions tagged [likelihood]

Given a random variable $X$ which arise from a parameterized distribution $F(X;θ)$, the likelihood is defined as the probability of observed data as a function of $θ: \text{L}(θ)=\text{P}(θ;X=x)$

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Maximum Likelihood Estimator (MLE) for $2 \theta^2 x^{-3}$

I'm having a bit of trouble solving this. $$ f(x_i; \theta) = 2 \theta^2 x_i^{-3}, 0 \le \theta \le x_i \lt \infty $$ I start by finding $f(\textbf{x}; \theta)$: $$ f(\textbf{x}; \theta) = \prod{f(...
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Measure of accuracy for a Bayesian model

I am reading Statistical Rethinking (Section 6.2.1.2). The topic of this section is measuring accuracy for a Bayesian model, i.e. accuracy of the model of predicting correctly an outcome. The ...
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what does non-nested copulas mean?

I read from "Pair-copula constructions of multiple dependence", the following statement: "The likelihood of the Clayton copula is lower than that of the Student copula (39.72 vs. 47.81). However, ...
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Likelihood function vs probability distribution function [duplicate]

I've been reading about Bayesian statistics and data analysis, and constantly see that $\text{posterior} \propto \text{prior} \ \times \text{likelihood}$. I'm familiar with fundamental probability and ...
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Bayesian- How to determine the distribution for likelihood function [on hold]

I have a question about likelihood model. Given I have a set of data, how do i find out what type of distribution is suitable for my likelihood function? (eg. Poisson, exponential etc.)
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Likelihood raised to a power; how to set the power?

Suppose ${\bf{\theta}} = (\theta_1 , \ldots, \theta_d)$ and you have a posterior as below: $$\pi(\theta | D ) \propto L(\theta |D ) \pi(\theta)$$ Suppose we are in active learning setting and need ...
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Likelihood function for linear regression

For linear regression, the likelihood function can be found with: However if your data points are multi-dimensional such that x,...
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How to interprete likelihood equal to 1

I'm trying to interprete the Example 1 from the wikipedia page: the likelihood function of a coin flip with a single parameter p expressing how likely a head will come up. The likelihood is defined ...
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Deriving likelihood function of binomial distribution, confusion over exponents

This question focuses on a specific aspect of this one: How to derive the likelihood function for binomial distribution for parameter estimation? In my own derivation, I start with: $$f(x\mid p) = ...
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Is the Quadratic Approximation of Log-Likelihood Equivalent to the Normal Approximation of the MLE?

Let $X_1, X_2, ..., X_n \sim \text{IID N}(\theta, \sigma^2)$ with $\sigma^2$ known, and let $\hat{\theta}$ be the MLE of the mean. (1) How can I show that in this case, the following is true? $$\...
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Likelihood function when only $\max_{1\le i\le N}X_i$ is observed and $N$ is parameter

Let $X_1,X_2,\ldots,X_N$ be i.i.d random variables having $\text{Exp}(1)$ distribution where $N$ is unknown. Suppose only $T=\max\{X_1,X_2,\ldots,X_N\}$ is observed. I have to derive a most ...
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Likelihood-Based Confidence Intervals for ratio of risk ratios

Ratio of Risk Ratio (RRR), along with Relative Excess Risk Due to Interaction (RERI), has been used to quantify the joint effects of 2 exposures in epidemiology. Quoting Joshua N. Pritikin CIs ...
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Computing the Hessian of maximum log likelihood function

I am trying to find the Hessian matrix for the maximum log likelihood function given training data {(xi, yi)} for i=1:N with yi ∈ {+1, −1} for each i = 1, . . . , N for the function: When I try to ...
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HMMs: How to Interpret The Average Likelihood Of My Data

I have recently trained an HMM using R's depmixS4 package, and am evaluating its performance via the average likelihood of my data. The equation is provided below: However, I noticed the average ...
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Unbiased estimator for Theta of a Normal Distribution

If $X_1,\ldots,X_n\sim \operatorname{iid} \operatorname N(\theta, \sigma^2)$, then verify that $\bar{X}_n$ is unbiased estimator for $\theta$ and that Cramer Rao bound is met? I am facing difficulty ...
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Why is the deviance defined with a factor 2 (or likelihood ratio squared)?

Deviance is defined as I see the motivation in why we would define the deviance as a difference of logLikelihoods or just the log(Likelihood Ratio), but why the factor 2? Why square the ratio? Does ...
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Calculation of the likelihood in the Boxcox transformation

I have an outcome which may be transformed in cases of egregious departures from normality to its Box-Cox optimal normal transformation in an unconditional model. How is the likelihood calculated for ...
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Infill likelihood for a continuously observed continuous-time process

Consider a continuous-time stochastic process $y(t)$ having the following linear (Gaussian) state-space representation for $t \geq 0$ $$ \left\{ \begin{array}{c c l} \text{d}{\boldsymbol{\...
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An example where the likelihood principle *really* matters?

Is there an example where two different defensible tests with proportional likelihoods would lead one to markedly different (and equally defensible) inferences, for instance, where the p-values are ...
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From likelihood P(Data|H) to P(H|Data)

If there are four possible hypotheses and I calculate the likelihood of the data given each of these hypotheses, can I calculate the probability of one of the hypotheses as ...
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Is it ever convenient to maximize different functions of the likelihood than the logarithm?

We all know that it's often much more convenient to maximize the log-likelihood rather than the likelihood to get a parameter estimate, since it amounts to the same thing by the fact that the ...
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The likelihood function: Why is it no pdf? [duplicate]

I know that there have already been a lot of questions about why the likelihood is no probability density function and I ve read most of the answers. However, to me the point is still not clear yet ...
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Likelihood of one datapoint given $k$ models

Introduction I'm currently facing a problem where I'm constraining a set of (physical) parameters $\theta_k$ with $k\in [1,2,...,K]$ via several independent datasets. One of those datasets, however, ...
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Writing Likelihood of Poisson in R

Here is my attempt to make the likelihood function for Poisson distribution for data x and parameter theta in R: ...
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Likelihood of a linear model in matrix form

I have difficulty finding the likelihood of the data represented in the matrix form. The mapping between target variable $\mathbf{T}$ and observed variable $\mathbf{X}$ is given as $f:\mathbf{X}\...
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Joint probability distribution of correlated data points

I have a query with respect to joint distributions. Here, each output data point in $\mathbf{y}$ is conditionally independent given the inputs $\mathbf{x}$ and the mapping $f:\mathbf{x}\rightarrow \...
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Bayesian inference on multiple datasets

I have a data-set of synthetic observations that contain an x, y and z component. The model parameters used to make it is known, including the noise level added. I am using a probabilistic ...
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1answer
38 views

Likelihood when points aren't i.i.d

If we assume that we have a set of N data points given as $\textbf{X}$ and corresponding targets vectors $\textbf{T}$, where both represents matrices in this case. For an i.i.d we could write the ...
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If values for relative likelihood function is given then how to find likelihood interval

$$R(\theta) = (\frac{\theta}{2})^{48}e^{48-24\theta}$$ You are told that R(1.5) = 0.164;R(2.5) = 0.276, and R(3) = 0.011. Give a value of $$\theta$$that is: (i) Inside a 10% likelihood interval: ...
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What is the probability/likelihood of a sample being drawn from a probability distribution over binary values

Suppose we have a known discrete probability distribution $X$ over $\{0,1\}^k$. Given a sequence of binary values $e = (e_1, ..., e_n)\text{, where } e_i\in \{0,1\}^k$, what is the probability (or the ...
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Marginal Log Likelihood of Count Data to Simplify MLE

My question concerns the calculation of a marginal likelihood given some priors for my underlying exponential mixture distribution. Background My data is the (orderd) set of integers $\{N_\ell\}$. ...
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What Is Meant by “Maximising” Posterior Probability?

My textbook says the following: The optimal coding decision (optimal in the sense of having the smallest probability of being wrong) is to find which value of $\mathbf{s}$ is most probable, ...
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If both prior and likelihood are Gaussian what can we say about the posterior? [closed]

If X is a random variable that has Gaussian prior and Gaussian likelihood. What can be inferred about the posterior? As posterior is proporional to prior*likelihood which are Gaussians, the posterior ...
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What is the Cox Partial Likelihood the partial likelihood of?

In linear regression, the likelihood works as follows: Suppose that $Y \mid X, \beta \sim \mathcal{N}(\beta^T X, 1)$. Then the likelihood of $\beta$ given a datum $(x, y)$ is $L(\beta; x, y) = \...
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Notation confusion in likelihood question [closed]

Asking this question based on an equation I found in my homework: "Given a training set, $\mathcal{S}$, find an expression for the likelihood, $p_\mathcal{D}(\mathcal{S}\mid w)=\mathbb{P}[\mathcal{S}...
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ML algorithm to determine the likelihood of one value

I’m a noob in machine learning algorithm. I have two possible outcome: hand and foot. I have a ratio computed based on different characteristics. For instance, the ratio can be 0.3 : 0.7 With this ...
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A confusion about Bayes's theorem

I am reading a paper on the differences between bayesian outlook and frequentist outlook. The exact pic from the paper is: I have read a decent amount about what likelihood is and how it is not a ...
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1answer
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Likelihood Function is Minimal Sufficient

What does it mean to say that "Likelihood Function is Minimal Sufficient"? Is this a general statement, or does it apply to only exponential family of distributions? I think I understand the concept ...
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HMM Weather Example Likelihoods

I recently created an HMM following the Viterbi algorithm for predicting the sequences of weather (rainy, sunny) given a sequence of observations X (walk, shop, clean) but computationally inefficient ...
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Can negative of empirical second derivative of the log likelihood with respect to the parameters not be semi-positive definite?

This is the empirical Fischer Information. Also consider the outer product with itself of the first derivative of the log likelihood with respect to the parameters. This will always be semi-negative ...
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Classification Using The Hidden Markov Model

I am having difficulty understanding certain concepts regarding the classification using HMM. There are numerous post here and in the internet about that, but they never get to detail or they all ...
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How to compare a new measurement to an existing multivariate distribution?

I have a dataset that describes the position and rotation of an object at different points in time using four dimensions. I want to use this sample of observations to get a sense of what positions and ...
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Bayesian Inference: Modeling checkout times at a store

I am currently learning how to use Bayesian inference. I have been making up problems (by defining some population parameters) and then trying to infer those values from samples. I recently made up a ...
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Why in ABC algorithms, when likelihood is intractable, we can draw observations from $p(y|\theta)$? [duplicate]

One of the main reasons for using the Approximate Bayesian Computation(ABC) algorithm is when we have a situation where direct computation of $p(y_{obs}|\theta)$ is numerically intractable. However, ...
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Variance of distribution for maximum likelihood estimator

We're looking at maximum likelihood estimators in my stats course at the moment. As I understand it, the idea is that we have some data. We come up with a model. The model has some parameter $\theta$. ...
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Problems fitting a model to a variogram

I am having problems fitting a variogram model. I tried to change some parameters to estimate or fix them but I am still not achieving any improvement. I remove trend of the data and use logarithms ...
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Rejection Region for Likelihood Ratio Test

I have $((Y_1,x_1),(Y_2,x_2),\ldots,(Y_n,x_n))$ where $Y_i$ is distributed as $N(\theta x_i,1)$. I want to find the rejection region $[0, c]$ associated with $\lambda$ for a test with significance ...
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Find an estimator for $\theta$ when PDF is:

Find an estimator for $\theta$ when PDF is: $$ f(x) =(1-\theta)\mathbb{I}_{[-1/2,0)}(x)+ (1+\theta)\mathbb{I}_{(0,1/2]}(x). $$ I know that one way is to write the likelihood function then do Log-...
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AIC and BIC and number of quantization level

I want to test how many quantization levels (discretizing levels) are the best for the given data(time series) set I have. Therefore I am applying different levels of binning (like discretisize data ...