Questions tagged [parameterization]

For questions about how to parameterize some statistical model, or comparisons between different ways to parameterize.

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How do I determine a good parameter grid for cross-validation? [closed]

Can anyone help me understand how best to determine parameter grids for e.g. XGboost, RandomForests and CatBoost? Currently I'm using the below based on random examples online, but I'd appreciate any ...
cebuq's user avatar
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What's the difference and relationship between theta, theta star and theta hat?

I understand that $\theta$ is the true distribution parameter (great explanation here). I also know that $\hat\theta$ is an estimator of the true $\theta$ (so for example, MLE is an example of $\hat\...
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parameterize, estimate & interprete interaction terms between two factors in a Cox Proportional Hazard model

Overview I was trying to fit a cox proportional hazard model to look at interactions between two time-constant covariates, both of which are factors. I parameterized the model in two different but ...
Xuan's user avatar
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Determining the Identifiability of Models

I am completing exercises in the book Mathematical Statistics: Basic Ideas and Selected Topics regarding proving or disproving that a model is identifiable. The problem I am struggling with considers $...
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About Estimating Parameters from Unpaired Datasets

I possess three datasets: $x$, $y$ and $z$. It's hypothesized that a relationship exists between these variables, represented by the equation $z=a*x+b*y$. My goal is to estimate the values of $a$ and $...
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Defining parameters so that they obey multiple constraints

I'd like to define parameters $\beta_i$ for $i=1,\ldots,I$ for a problem so that they automatically obey some constraints. The constraints are: $\sum_{i=1,\ldots,I} w_i \beta_i = c_1$ and $\sum_{i=1,\...
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Is there an exponential family such that its natural parameter mapping is non-invertible or has non-convex range?

On the Wikipedia article for exponential families the density of a distribution on a measure space $(X, \xi)$ from an exponential family is written as $$f_{\theta} \colon X \to \mathbb{R}_{\ge 0}, \...
ViktorStein's user avatar
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Reparameterization of Poisson Distribution

In deep learning, especially generative models, sometimes we need to add some random noise to the input of model. To make the sampling of random noise learnable (or differentiable), we need to ...
Lorin60's user avatar
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Understanding a parameter in a bayesian Poisson model ($\beta$)

I would like to know the meaning or signification of the parameter $\beta$ in this Bayesian model. I have a Poisson model : $ s_{i} \mid \lambda_{i} \sim Poisson(\lambda_{i}t_{i})$ Where $\lambda_i\...
xenuti's user avatar
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Bayesian reparametrization are they equivalent?

Suppose that we are in a Bayesian context, we we have the following matrix $n,$ $K\times K,$ as parameter, and we assume that $$n_{ij}\sim Pois(w*w_{ij})$$ where $w\sim Gamma(N+1,1)$ and $w_{ij}$ is ...
Fiodor1234's user avatar
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How to extract the correct parameters for the lognormal distribution when using the survreg() function in survival analysis?

I am testing simulation of the lognormal distribution against the lung dataset, as an example of right-censored data, from the ...
Village.Idyot's user avatar
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Options for transforming the variance-covariance matrix generated by the survreg() function to the original scale of the Weibull distribution?

I'm working with the survreg() function of the R survival package, and I understand that the default scale parameter for the ...
Village.Idyot's user avatar
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What is a good way to automate distribution fitting in python?

I have to do some distribution fitting of 120 data subsets. They take the form of financial transactions amounts and timestamps. Timestamp, BTC, EUR, USD I know from some other analysis that each ...
oldquant's user avatar
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Estimation of Distribution using multiple ECDFs

Every day, I keep track of the processing times for each input to my CPU and create empirical cumulative distribution functions (ECDFs) based on this data. Let's assume I have 100 observations per day ...
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Parameterizing a Gaussian distribution

I am reading this blog post where the author talks about diffusion models. Let's keep diffusion out of the conversation for now. The author showcased that we can parameterize a Gaussian distribution ...
enterML's user avatar
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Parameterization of inverse gamma prior in Bayesian methods

For a prior of $\sigma^2 \sim IG(0.01, 0.01)$, often recommended as an uninformative prior for the variance parameter in MCMC approaches and other Bayesian methods, which parameterization does this ...
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How to detect an unknown number of segments, each to be fitted with an unknown parametric curve/surface equation?

Let's say I have a set of points (possibly noisy) in an N-dimensional space that represent an arbitrary number of curved segments, each segment having an arbitrary type of curve. See the 2D sample ...
Anson Kao's user avatar
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Excercise 6.1 and its solution in Bishop's PRML, Question 1

The problem comes from Exercise 6.1 of "Pattern Recognition and Machine Learning" by Christopher M. Bishop: Consider the dual formulation of the least squares linear regression problem ...
zzzhhh's user avatar
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How do I impose restrictions $ 0\leq \alpha \leq \beta <1$?

I want to restrict values s.t. I get $\theta = (\alpha, \beta) =g(\theta_1, \theta_2)$ with the following restrictions. $ 0\leq \alpha \leq \beta <1 $ I know the correct answer should be $\beta = (...
user773674's user avatar
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Writing exponential family in canonical form

I have the following pdf with support $x>0$: $$f_{\mu}(x)=\frac{1}{\sqrt{2\pi x^3}}\textrm{exp}\left(-\frac{(x-\mu)^2}{2\mu^2x}\right)$$ This belongs to the exponential family, and I write this in ...
pecer10012's user avatar
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What are we modelling when a gamma distribution has non-integer shape parameter

I wish to receive a clear and concise answer as to what is being modeled for a gamma distribution with non-integer shape parameter, and a more detailed derivation of its distribution function for all ...
Cai's user avatar
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Estimating time varying parameters of ODE with the help of solution data

I am trying to extend a parameter estimation of ODE model from constant parameter estimation to time-varying parameter estimation. I have completed the constant parameter estimation (where parameter ...
Formal_that's user avatar
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Weibull distribution parameterization

I have the following Weibull distribution: $f(x;\lambda,\beta) = (\lambda\beta)x^{(\beta-1)}e^{(-\lambda x^b)} $ where $\lambda$ is scale parameter and $\beta$ is shape parameter. I have an ...
forecaster's user avatar
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Is the OLRE term meaningful in the negative binomial model? + Is overdispersion in the NB model an issue?

I'd like to ask three questions regarding the negative binomial (NB) regression / distribution. The NB model with NB2 parameterization ($var(Y_{NB2}) = \mu + \frac{\mu^2}{\theta}$) is sometimes ...
Eva Šragová's user avatar
7 votes
2 answers
914 views

What does "parameterized by" mean?

Sometimes I have seen likelihood written as $L(\mu,\sigma |y)$ and sometimes as $L(y|\mu,\sigma)$. I have been told that in the first case it means that there is a pre-assumed model depicting the ...
Kirsten's user avatar
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Aren't ALL Parameters Eventually "Nuisance Parameters"?

I am an MBA student taking some courses in statistics. We attended a seminar on GLM Models for Count Data in which the presenter was introducing us to the concept of "Nuisance" Parameters. I ...
stats_noob's user avatar
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Differentiation on the conditional variables of a probability

I have been questioning how to calculate the partial derivatives of a conditional probability function with respect to its parameters. Assume $x$ is data and $\theta$ is a parameter(s). If I have a ...
Yutaka Tsuzuki's user avatar
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63 views

Likelihood function-expectation

Given likelihood is a function of parameters, I cannot understand why the expectation of likelihood functions is not calculated with respect to the the parameter space but the sample space, as put ...
Wenxu's user avatar
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How many parameters on a Bayesian network

I'm taking Coursera's course on probabilistic graphical models, and I'm stuck on a question. The discussion forums there are dead, and I can't find any resource to help me, so I hope someone could ...
João Areias's user avatar
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220 views

Interpretation of coefficients in GLM: coefficients associated to continuous covariates interpreted as MD's or OR's

I was having a discussion with someone regarding OR’s estimated trough a logistic regression and then he claims that OR’s for continuous variables can only be estimated trough a logistic regression. ...
Nicolas Molano's user avatar
1 vote
1 answer
256 views

The effect of over-parameterization on local minima

While reading some papers about over-parameterization in deep learning models, I also read that "over-parametrization is a simple method to introduce additional dimensionality and help make the ...
AGM's user avatar
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Is there a sampling method to find multiple local minima for a multidimensional parameter space?

Firstly, I just want to declare that I'm not a statistician and I apologize for any obvious errors. Let's say I have a dataset with x and y values. Now, I have a model with 10 parameters/coefficients ...
Agnibha Banerjee's user avatar
2 votes
1 answer
591 views

Interpretation of drm parameter estimates and p-values for EXD.3 function in 'drc' package in R

I was wondering if someone could help me understand what the parameter estimates and p-values are saying in a three-parameter exponential decay function using the drm function in the 'drc' package in ...
tedwin183's user avatar
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Multinomial likelihood function with data for only 2 of 3 outcomes

Can/should I use a binomial likelihood function if the data were generated from a multinomial process (3 possible outcomes) but data were only collected for two of the possible outcomes? In each trial ...
Alex's user avatar
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1 answer
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Need for reparameterization trick in RL (and others)?

This is a multi-fold question that has a number of closely related questions; that is why I will pose them all here, instead of separate questions. In RL you have a parameterized policy that dictates ...
Schach21's user avatar
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1 answer
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T-distribution parameters with QRM package

I am fitting a t-distribution on some data I have using the fit.st function from the QRM package. The function returns 2 set of ...
Mayeul sgc's user avatar
2 votes
1 answer
271 views

Is the Jacobian term needed if the prior is on the transformation parameter?

Suppose I have a strictly positive parameter $\sigma$ and I need to estimate it using the random walk Metropolis-Hasting algorithm. I know that I can do a parameter transform, i.e., $\beta=log(\sigma)$...
Ding Li's user avatar
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M-estimator: There is no "of something" in the definition

I see that when talking about estimator, we have "of something", where "something" refers to a fixed parameter. For example, we say that the sample mean is an estimator of the ...
TrungDung's user avatar
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3 answers
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Why does the von Mises-Fisher distribution need two parameters?

The von Mises-Fisher distribution has two parameters: the mean $\mu \in \mathbb{R}^p$ and concentration $\kappa \geq 0$, where $\mu$ is constrained to have unit norm. Why not instead define the ...
Rylan Schaeffer's user avatar
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1 answer
68 views

More stable reparametrization of a parameter on $(-1,1)$?

Suppose that a distribution contains a parameter $\theta \in (-1,1)$. I want to reparametrize this model in terms of $\beta = h(\theta) \in (-\infty,\infty)$. I am considering: $$h(\theta) = \mbox{...
Calip's user avatar
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Understanding "In Bayesian inference, the difference between data and a parameter is that one is observed (data) and one isn't (parameter)" [duplicate]

In his statistical rethinking course, Richard Mclreath states "In Bayesian inference, the difference between data and a parameter is that one is observed (data) and one isn't (parameter)" I ...
Mir Henglin's user avatar
2 votes
1 answer
330 views

Anova models: different parametrizations give different results

In class our teacher explained that there are different parametrization which can be used to make the design matrix called CornerPoint parametrization: the first coefficient represents the mean value ...
Tortar's user avatar
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2 votes
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Guide to self-starter estimators (parameter initialization) for "simple" functions

Background I have a collection of functions with trainable parameters that I am implementing as Keras model classes, which enables immediate use of a variety of objective functions, optimizers, and ...
Galen's user avatar
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2 votes
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Bootstrap instances of a statistic for MLE?

Let's say that I have real-valued random variables, $\{X_1. \cdots, X_n \}$, and some statistic $T(X_1, \cdots, X_n)$ for which I hypothesize might have a distribution $f(T(X_1, \cdots, X_n); \vec{\...
Galen's user avatar
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Vanishing partial derivative of least squares w.r.t. Verhulst growth parameter

The Verhulst growth model can be given as $$P(t) = \frac{k}{1+ \left( \frac{k-P_0}{P_0} \right)\exp(-rt)}$$ where $P(t)$ is the population size at time $t$, $k$ is the carrying capacity, $P_0$ is the ...
Galen's user avatar
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1 vote
1 answer
110 views

Function fit to skewed data and non-zero beginning of the function

I would like to find a function that would represent the best fit to represent this type of biological data. More precisely, I would like to estimate expected daily egg production by an insect, based ...
MIH's user avatar
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Appropriate term for "seemingly unrelated regression with shared parameters"?

I have paired values for three variables $z$, $y$, and $t$, and I wish the perform the regressions $z = g(t)$ and $y = f(t)$. I happen to know there is a bias in the variable $t$, thus the true times ...
Galen's user avatar
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218 views

Parameter Optimization in RF and rpart

I am using rpart and random forest in R to predict GPA (regression tree). On what basis do I decide the value of cp, minsplit, and minbucket? And on what basis do I decide the values of mtry and ntree ...
Tannya Kumar's user avatar
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394 views

Understanding difference between Maximum Likelihood and Levenberg Marquardt result

In some of my regression results I noticed a deviation between Maximum Likelihood (via Monte Carlo Markov Chain, initialised by parameter result of Nelder-Mead, median value pictured) result and ...
Fabian Pascher's user avatar
1 vote
0 answers
458 views

Understanding natural parameterization of exponential family

I'm going through section 3.4 on exponential families in Statistical Inference by Casella and Berger. They first cite the following general form of an exponential family: $$f(x|\mathbf{\theta})=h(x)c(\...
Shivashriganesh Mahato's user avatar

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