Questions tagged [parameterization]

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

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Fitting variable-dependent normal distribution to data

Given a sample, one can usually find the best fitting normal distribution by matching the mean and variance. What's the correct way to fit a normal distribution to data when the parameters aren't ...
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Does it make sense to worry about stability of parameters?

I'm working on a problem where I'm using grid search on logistic regression and I'm checking two parameters, penalty and C. I ...
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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., $\beta=log(\sigma)$...
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Which data do you use when finding parameters for your model after cross-validation?

Let's say I have a dataset (X, Y) for which I want to find the best fitting polynomial model (say degree = 1 through 10) using k-fold CV. Let's say after doing k-fold CV on degrees 1 through 10, I ...
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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 ...
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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 ...
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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{...
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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 ...
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How to estimate 3 variables using maximum likelihood models using a Ricker model?

I'm trying to fit a non-linear function to discrete population growth data. The data I fit is to a modified Ricker model of the form: ...
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How does max pooling reduce the number of parameters to be learnt in a CNN if we already have parameter sharing?

I've been a bit confused about max pooling for a little while but I've finally understood why it provides more generalisation since you're essentially getting rid of less useful information and ...
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how to fit a set of parametrized data to a set of parametrized distribution?

I have a time series $d_i(a)$ which depends on the parameter $a$. On the other hand, I have a sequence of normal distributions $\mathcal{N}(0,Q_i(a))$, where the variance $Q_i$ depends on time and ...
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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 ...
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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 ...
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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{\...
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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 ...
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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 ...
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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 ...
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estimating a population-average model with known mean and standard deviation

I have a model with some differential equations describing the effect of a drug. There are 100 rat samples, we only know the mean value and its standard deviation for measured drug response. Now I ...
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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 ...
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Does reparameterization change the likelihood function of a distribution?

Given a usual distribution, does reparameterization change the likelihood function? Say for example you have a Poisson distribution $(Y_i \sim \text{Poisson}(\lambda))$, but now you want to ...
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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 ...
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Floating Normalization of Experimental Data Sets while Fitting Multiple Models

I have $N$ data sets of unequal cardinality, and I am told we do not treat each data set with a normalization of $1$. Instead we let the the normalization float and fit it as though it were any other ...
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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(\...
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Likelihood in Bayesian inference: p(x|theta, I) = p(x| I)?

In page 164 of the book “Probability theory: the logic of science” the author says that: $$ p(D|\theta I) = \prod_{i=1}^{n} p(x_i|\theta I) = \theta^r(1-\theta)^{n-r} $$ $ \theta $, in this equation, ...
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Can I still use parametrical test if Shapiro-wilk test saying that my data is not normally distributed?

My data are mostly 0 and 1 so therefore I can't pass normality test. Can I still use parametric tests like repeated measures anova to test for significance? If the answer is yes, how can I justify ...
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How do I calculate the probability according to a geometric distribution given the value of X, its mean and its variance? In R

I want to predict a time series of intermittent demand items. For this, I want to use a geometric distribution for the demand sizes. How do I get the probability that X = k - according to a geometric ...
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Parameter estimation of a model with exponential almon lag structure

Suppose I have the following model: $$y_t = \beta_0\sum_{i=0}^p w(\delta;i)x_{t-i}$$ Where $\displaystyle w(\delta;i)=\frac{\exp(\delta_1 i+ \delta_2 i^2)}{\sum_{i=0}^p \exp(\delta_1 i+ \delta_2 i^2)}$...
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How to perform Chow Test on balanced panel data with multiple breakpoints simultaneously?

I'm currently working with a balanced panel data dataset. I have 27 individuals over a 7 year period. My objective is to perform a Chow Test to determine if ANY of the estimated parameters change from ...
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Standard error in parametrization

I calibrated the parameters of the GARCH model. Now I would like to calibrate the standard errors in MATLAB of the parameters but I don't know how to do it, can someone explain me how?
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Amortized inference in convolutional variational autoencoders

VAEs are an efficient way of performing variational inference at scale. I read that VAEs employ the strategy of amortized variational inference. They approximate the intractable posteriors p(zjx) by ...
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With some shared parameters, and some not, what are my degrees of freedom?

I have a data matrix $X$ of size $m \times (n+1)$ where there are $n$ dependent variables and one independent variable $t$. I also have a collection of $n$ nonlinear functions $f_1, \cdots, f_j, \...
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When does parameterization and LSMEANS vs ESTIMATE in SAS PROC GENMOD make a difference in exponentiated Beta, confidence interval, and p value?

I am running a modified Poisson regression (Poisson with robust standard errors) to estimate prevalence ratios (ie relative risk). I am a bit confused about whether PARAM= choice and LSMEANS vs ...
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How does the recursively parametrizing a nonlinear feature extractor of a generalized linear model allow us to model increasingly complex functions?

In Murhpy's Machine Learning: an introduction, he says the following: "Given a GLM: $$f(\mathbf{x};\theta)=W\mathbf{x}+\mathbf{b}$$, We can increase its flexibility by performing a feature ...
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2 votes
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How do Bayesians deal with the fact that the treatment of fixed parameters as random variables can lead to inappropriate subjective probabilities?

Let's assume the tossing of an unfair coin is modeled by a random variable X taking the values head and tail. You know that the objective probability of the coin showing up head is either $p=0.4$ or $...
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Parameter simplification of ARIMA model

I am constructing an ARIMA model on a cryptocurrency price time series. Using the autocorrelation and partial autocorrelation plots I came to the parameters of (p,d,q)=(3,1,2). The resulting RMSE was ...
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Effect of scaling data on ARMA coefficients [duplicate]

For numerical stability, I thought it might be a good idea to scale my data before feeding them into an ARMA GARCH model. I have gone through a few older posts and understand the affect scaling ...
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Bayesian parameter optimization of a Voight matrix

I have constructed a finite element model of a musical instrument. The physical properties of the wood were very difficult to obtain and as they are anisotropic they need all 3 dimensions. I am ...
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What is exactly the structure included in a parameter space?

According to Wikipedia, a space in mathematics is: a set (sometimes called a universe) with some added structure. In statistical literature, I usually find references to a parameter space in the ...
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ARMA GARCH fitting

I've made a few posts regarding a manual ARMA GARCH implementation and I have made some great progress. However, I am still shy of a working program as I am obtaining some rather large forecasts. I've ...
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Issues Manually Implementing ARMA GARCH

I have been working on manually implementing an ARMA GARCH (1,1) model but have been running into a few issues, namely a very large forecasted variance. I am hoping by outlining my process someone can ...
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Fitting ARMA GARCH

I am interested in fitting an ARMA GARCH model by hand (that is without the use of a package such as rugarch), but am unclear on how the parameters are estimated. I have read that one should use MLE, ...
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How to fit ARMA-GARCH parameters for any distributions

To better understand the ARMA-GARCH model I am working on implementing it while avoiding as many packages as I can. For data I am working on returns and for simplicity I am starting with ARMA (1,1) ...
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Change of metric for probability density vs for probability

When one changes the variable in a probability density function, one must account for the jacobian to ensure the elementary probability is constant (eg Derivation of change of variables of a ...
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What are parametric conditions?

My dissertation supervisor asked me to explain further the following question for a GARCH model: "what are the alpha's and the beta's and what are their parametric conditions, and what do the ...
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Fisher Information invariant by a specific reparameterization of the Exponential Distribution

The exponential distribution can be parameterized in two common ways: $$ f(x) = \lambda \exp(-\lambda x) $$ where $E[X] = \frac{1}{\lambda}$ $\text{Var}[X] = \frac{1}{\lambda^2}$, or as $$ f(x) = \...
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Modeling exponential growth with mixed effects

I'd like to create an exponential growth/decay model along the lines of y ~ -a * exp(-b * x) in R. I have been able to do this using nls(), but my dataset has repeated measures, so I need a mixed ...
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Why hyper-parameters of a model should be shared?

Question Can I apply different hyper-parameters for different training sets? I can see the point of using the shared parameters but I cannot see the point of using shared hyper-parameters. The ...
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1 answer
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Is the distinction between parametric and non-parametric statistics always clear-cut?

Is the distinction between parametric and non-parametric statistics always clear-cut or do examples of a statistic exists which cannot clearly assigned to one of these categories?
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Identification of correlated errors with multinominal probit

Consider the multinational probit model where we observe $Y_i \in \{1, \dots, K + l\}$ with $$ \begin{align*} Y_i = l \Leftrightarrow Z_l&\geq \max(Z_1,\dots Z_{K +1}\} \qquad l \in \{1, \dots, ...
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Why do people say linear SVM is a parametrized model and kernel SVM is not?

I am confused by what people mean when they say that linear SVM is a parametrized model and kernel SVM is not. Aren't both methods trying to compute a decision boundary using some optimization program?...
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