# Questions tagged [maximum-likelihood]

a method of estimating parameters of a statistical model by choosing the parameter value that optimizes the probability of observing the given sample.

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### How does maximum likelihood estimation from the Kalman filter work?

My understanding is Step 1: You would run through the Kalman filter equations with initial parameter values. Step 2: After you run through the Kalman filter equations, you will have innovations ...
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### Sufficiency of $S=\min\{X_1, ..., X_n\}$ when $X \thicksim Pa(\lambda,\theta)$

Warning The question is the sequel to this question and it was divided into three parts. The first is this, while the second part is found here. Exercise Let $X \thicksim Pa(\lambda, \theta$) with ...
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### Why likelihood is product of probabilities? [duplicate]

I understand following when we say probability we mean probability that a random variable $X$ will have certain value $x_i$ given the parameter $\theta$ that defines underlying probability ...
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### Use of weights in choosing power parameter in Tweedie distribution

I'm looking at the implementation of the tweedie.profile function from the R package tweedie. I have a few questions. When ...
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### Observed Fisher information for the binomial: How is $I(\hat{\theta}) = \frac{n}{\hat{\theta}(1 - \hat{\theta})}$ calculated?

I am currently studying the textbook In All Likelihood by Yudi Pawitan. Example 2.10 of chapter 2.5 Maximum and curvature of likelihood says the following: Example 2.10: Based on $x$ from the ...
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### Justification of the fixed variational distribution in diffusion models

Diffusion models can be regarded as latent variable models (Ho et al., 2020; Section 2), with the latents being an hierarchical chain of random variables $z_T → \dots → z_t → z_{t-1} → \dots → z_1$ (...
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### Maximum likelihood estimator - conditioning input on model parameters

Assume we have a dataset $D=\{(x_i, y_i)\}$ drawn from the joint distribution $(x_i, y_i) \sim P (x=x_i, y=y_i)$. We want to make predictions based on the dataset and for that we use a parametric ...
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### Seemingly contradiction: probability density function and maximum likelihood calculations for continuous random variable

Claim 1: For continuous random variable, $P(X=x)=0$, where $x$ is a particular number. Claim 2: When we use maximum likelihood estimation, we plug-in mean, standard deviation and data point $x$ into ...
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### Is this a correct way of doing maximum likelihood estimation of parameters of a variance analysis model?

I would like to ask you if my way of doing maximum likelihood estimation of the parameters of a variance analysis model with one factor with K categories knowing that the models verifies the ...
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### Why do we minimise a cost function instead of maximising an equivalent? [duplicate]

I don't really understand why we minimise a cost function for gradient descent. Why don't we try to have something like a gradient 'climb', where we maximise some function? Is it due to convention, or ...
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### Smallest threshold for hypothesis test with asymptotic level alpha

Consider a distribution with parameter $\lambda$ that has density $$f_\lambda(x)=\frac{x^4}{24\lambda^5}e^{\frac{-x}{\lambda}},x>0$$ Let $X_1,...,X_n$ be $n$ independent random variables drawn from ...
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### MLE in a logistic regression model

Assuming that the design matrix is of full rank, in non-degenerate cases of the logistic regression model, does the maximum likelihood estimator always exists and is always unique? It would be really ...
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### Can we find some examples for inference distribution that MoM would be better than MLE? [duplicate]

For sample size large enough, we know that the Maximum likelihood estimator (MLE) is asymptotic efficient. So when we have two classical methods (MLE and method of moment estimators) for inference of ...
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### MLE for censored multivariate normal data?

Suppose I have $q$ censored covariates, say $X_1, \ldots, X_q$ which are all left-censored with censoring values $\mathbf{d} = d_1, \ldots, d_q$. If I assume that the $X_i$ are from some multivariate ...
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### Does my solution to an incorrect MAP problem mean anything?

I was doing maximum a posteriori estimation of the variance $v\equiv \sigma^2$ using samples $\{x_n\}$ from a normally distributed random variable $X \sim \mathcal{N}(0,1)$, where the variance has an ...
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### Help me express a tractable complete data log likelihood of this problem with hidden variables

Imagine the following scenario: a couple of friends(total nr of people=7) will all play 5 games. At each game the player will ask an horiscope if he should use one out of two bernoulli distributions. ...
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### Proving that MLE is equal to LSE

Assuming that $\epsilon \sim N(0, \sigma^2)$ I want to prove that the maximum likelihood estimator is equal to the least squares estimator. Now, I think I'm very close to making the proof, I just need ...
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### Do nice properties of MLE still hold in Classical Linear Regression Model?

For simplicity, let's assume that we have the true DGP $$y_i= \beta_0 + \beta_1x_i + \epsilon_i$$ where $\epsilon_i \sim N(0, \sigma^2)$ $(i = 1,2,...,n)$ Assume that these following usual assumptions ...
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### Goodness-of-fit and confidence intervals

I am having trouble grasping the subtleties of determining confidence intervals on parameters from degrading the goodness-of-fit (ie shifting $\chi^2$ or $2\ln {\cal L}$ by 1). For instance, around ...
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