# 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 $θ$: $\operatorname{L}(θ | x)=\operatorname{P}(X=x \mid θ)$

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### Why does the score test work for values longer in the tail that have a small log-likelihood derivative?

The score test says that we take the derivative of the log-likelihood at $H_0$ and divide it by the fisher information at $H_0$. $U(\theta )={\frac {\partial \log L(\theta \mid x)}{\partial \theta }}.$...
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### Which parameters optimise the weighted cross-entropy loss for a pre-specified categorical distribution?

Question: Given a categorical distribution $C_q$ with parameters $q_1, \ldots, q_K$ with $K > 2$, $\sum_k q_k = 1$, which (new) categorical distribution $C_p$ with parameters $p_1, \ldots, p_K$ ...
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### Likelihood determination for a step-like pdf

Suppose that random numbers x are generated on the computer using the following procedure: Generate two numbers $x_1$, $x_2$ from a uniform distribution $\mathcal{U}$([0,1]) If $x_1$ > f, take x =...
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### Why does higher dimensional data has higher likelihood?

I am reading about generative models. I came across an example a few times but I cannot come up with an explanation for it. Imagine data is generated according to $p_\text{data}(x)$. It is often said ...
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### Predictive Diagnostic, Comparison of simulated data with observed data

The question is quite abstract, so I display it with only the essential information. Suppose that we have three models $B_{1}, B_{2}$ and $F_{3}$. The $B_{1}, B_{2}$ are Bayesian models and the $F_{3}$...
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### Relation between sample standard deviation from data and maximum likelihood estimates

This is my data:- c(3164, 3362, 4435, 3542, 3578, 4529) I estimated its sample mean and standard deviation via mean & ...
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### Overlapping circular bearing distributions on a plane

I have some directional hydrophones capable of recognizing transient signals/sound and estimating the circular probability density function of the bearing, or direction, that the sound came from. I ...
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### How to calculate the likelihood for a normal distribution N(theta, 1) if we only know the maximum of a sample?

Assuming iid samples x ~ N(theta, 1), we have a sample of 5 observations with maximum value = 3. How to calculate the likelihood?
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### comparing 2 likelihood values

Are likelihood values (density values) comparable across different types of distributions? For example, if you have a data point that has a likelihood value of .05 under a normal distribution and .025 ...
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### What are the undefined constants and functions in Stern's 2011 paper?

I'm reading the 2011 paper on ranking called Moderated Paired Comparisons by Steven E. Stern and there are no definitions given for some of the constants and functions in equation 1. As you can see, ...
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### BIC to test good fitting of data to a model

I want to use the Bayesian Information Criterion in order to measure how well a gaussian and 0 order polynomial fit (using python), the one with the lowest BIC should then be the 'best fit' ? My ...
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### Likelihood ratio test vs p value for Poisson regression

I have a Poisson regression model, from its summary table, I could see the p-value for a certain variable, e.g. gender. Since the p-value is testing the hypothesis whether the coefficient of gender ...
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### Probability of next flip being heads given I have seen h heads and t tails

I am currently attempting to understand "Question 2" at this link but having many difficulties. The problem is as follows: A coin has a chance of landing heads with an unknown probability ...
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### Likelihood function of VAR-MGARCH-BEKK model?

I am doing my dissertation on the spillover effect between countries' markets and looking to use VAR-MGARCH model to do it. For example how would a change/shock of US market index affect Thailand ...
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### Conditional likelihood with missing values

I want to estimate a logistic regression model on a panel data (subject-time) with subject-fixed effects. $$\log(p_{it}/(1-p_{it})) = \alpha_{i} + \beta x_{it} + \epsilon_{it}.$$ To do so, I want to ...
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### Trying to understand log-likelihood estimation for exponential smoothing models in R forecast function ets()

I am doing work on AIC comparisons. For this purpose, I am trying to understand how log-likelihood is calculated for exponential smoothing models (ETS models) in different R packages. In particular, <...
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### Binomial vs product of binomial in likelihood for Bayesian inference

I am working through McElreath's book on statistical rethinking. One of the problems is the following: Using grid approximation, compute the posterior distribution for the probability of a birth being ...
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### MLE for parametric binomial model

I have a model in which $p_i=f(\theta,Z_i)$, where $Z_i$ are iid latent variables distributed with CDF $F_\theta$, and $d_i\sim B(n_i,p_i)$, where $B$ is the binomial distribution. The likelihood ...
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### Why "likelihood is proportional to the probability of the data given the hypothesis" [duplicate]

In Etz, there is "likelihood is proportional to the probability of the data given the hypothesis" and "L(H) = K × P(D|H)", are there more detailed explanations to understand it? ...
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### Log likelihood decreases with posterior obtained after fitting GP

Possibly related to Log posterior probability in MCMC is decreasing but I do not have a MCMC process and the details there are not sufficient for me to understand fully (I'm a mathematician with basic ...
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### Implementing a 2-PL Dichotomous IRT Module in Python from scratch

I am trying to implement a 2-PL dichotomous IRT Model for my dataset from scratch in Python. Here is my code so far: ...
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### Likelihood function is a product of PDFs [duplicate]

I am learning about the likelihood function given iid random variables $X_i$ and realizations $x_i$: $\mathcal{L}(\theta | x) = \prod_{i=1}^n \mathbb{P}(X_i = x_i)$. One thing I am confused about is ...
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### Conditional Maximum Likelihood Estimation with Subsample

Suppose that we have an $i.i.d$ sample, $\{Y_i,X_i\}_{i=1}^N$, and a correctly specified conditional density of $Y$ given $X$, $f(Y|X; \theta)$, where $\theta$ is the parameters of the density. Then, ...
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### Log-likelihood skew-t

I am trying to write down the log-likelihood for the multivariate skewed student-t distribution, but I don't really get how to define it exactly. Could someone please tell me the definition as in how ...
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### If Likelihood is not a PDF then why is the PDF of Multivariate Normal equivalent to the likelihood of I.I.D. Normals?

I am understanding why likelihoods are not PDFs using links such as What is the reason that a likelihood function is not a pdf. However I am getting more confused. For instance, the likelihood of I.I....
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### Why is it that we can talk about the probability of the data given the parameter in bayesian inference though the data is considered to be fixed?

Basically, the title of the question is all there is. quoting from bishop's pattern recognition and machine learning: In both the Bayesian and frequentist paradigms, the likelihood functions $p(D/w)$ ...
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### How to write the likelihood for a multivariate gaussian linear model

I have a lasso-like bayesian graphical model where we try to estimate precision matrices between two conditions (0 and 1), $\Sigma_0^{-1}$ and $\Sigma_1^{-1}$, respectively. The model can be ...
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### Confusion with the "lower bound"-term in diffusion models

I am trying to understand the maths of diffusion models following this video explanation on youtube and this blog post. Here is what how I understood it so far: The overall goal is, that we want to ...
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### Odds Ratio - how many times more likely?

Problem: I need to figure out how many times are women more likely to continue attending a class after a certain period of time. Data: Solution: I used odds ratio (see picture). Is it correct to ...
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### Using a Normal Distribution to motivate the use of a Parameter Penalty in Negative Log-Likelihood

This is a question arose from an argument I had about a statistical explanation. We are comparing two models using their negative log-likelihoods. Given that we are unable to split the data into a ... 70 views

### Likelihood and log-likelihood for Weibull distribution

Question Suppose there are two groups on treatments and an individual follows a Weibull distribution with the following probability density function. \begin{align} f(x; \alpha, \lambda) = \alpha \...
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### Calculate log-likelihood of logistic regression

I am trying, without success, to calculate the log-likelihood of the most basic logistic regression model - a constant probability model (i.e. only $\beta_0 \ne 0$). For the simplest model with 1 ...
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### Model, Likelihood & ABC

I'm struggling to understand what likelihood free means in ABC, since ABC is using a model as simulator to produce $y_{simulated}$. However, to me is not clear the difference between model/simulator ...
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### Why is the likelihood the product of the conditional PDF and the PDF of the parameter

Please help me understand the following: Suppose a tester recorded the quantity $Y=X_1+\cdots X_n$ where $X_i$ has a Poisson distribution with mean $\theta$. Now, the tester lost all samples $X_i$ ...
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### Function proportional to the log likelihood for the Gaussian distribution

The following question is crossposted from MathStackExchange upon recommendation from the MSE community and a lack of responses on my post over there. Consider the following problem from a course on ...
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### Loss function for estimating the conditional variance by fitting $y_i^2$
I'm trying to detect anomolies in a dataset $i \in \{1,2,...,N\}$ where a random variable $y_i$ is expected to be drawn from a normal distribution with mean $\mu_i=0$ and variance $\sigma_i^2 (X_i)$ ...