Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [invariance]

a property of remaining unchanged regardless of changes in the conditions of measurement

In particular, statistical models that are closed under a class of transformations are called invariant. Often this invariance has important implications for inferential problems associated with the model. The principle of invariance asserts that whenever a problem is invariant under a group of transformations, then the solution to the problem should also be invariant. Applications of this principle occur in both estimation and hypothesis-testing problems. For example, maximum-likelihood estimators and likelihood-ratio tests are invariant solutions to inference problems when the model is invariant.

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Example for a prior, that unlike Jeffreys, leads to a posterior that is not invariant

I am reposting an "answer" to a question that I had given some two weeks ago here: Why is the Jeffreys prior useful? It really was a question (and I did not have the right to post comments at the time, either), though, so I hope it is OK to do…
Christoph Hanck
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Is the invariance property of the ML estimator nonsensical from a Bayesian perspective?

Casella and Berger state the invariance property of the ML estimator as follows: However, it seems to me that they define the "likelihood" of $\eta$ in a completely ad hoc and nonsensical way: If I apply basic rules of probability theory to…
user56834
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Is there any difference between estimating $\sigma^2$ and $\sigma$ in a simulation study?

In a simulation study, is there any difference between $\bullet$ estimating the variance $\sigma^2$, $1000$ times and taking its average, and $\bullet$ estimating the standard deviation $\sigma$, $1000$ times and taking its average? Can I do…
user81411
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Ways of implementing Translation invariance

Is there any literature about the different ways translation invariance can be achieved when classifying images with Convolutional Neural Networks? Aside from using the structure of CNN, did anyone attempt something different? Pre-processing, for…
user_1177868
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Meaning of Invariance of Maximum Likelihood Estimator

In Casella-Berger, the invariance of MLE is defined as: Assuming that $\hat{\theta}$ is MLE of $\theta$, then for any function $\tau$, $\tau(\hat{\theta})$ is MLE of $\tau(\theta)$. In the case of a one-to-one transformation, everything is…
SWIM S.
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Significance of parameterisation invariance of Jeffreys prior

I often hear it said that the Jeffreys prior is well-motivated because it is invariant under reparametrization. The proof of this is quite straight-forward (I know the proof on e.g., wiki). I'm a bit confused about what the proof really means,…
innisfree
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What is Exact definition of Invariance principle

Books wrote a lot about invariance definition and method to obtain invariance estimators, tests, and etc. However I couldn't find exactly the definition of principle of invariance. Is there any good source of clear definition of this principle?
SirSaleh
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Find the invariant measure $\pi=(\pi_{1},\pi_{2},\pi_{3})$ for a Markov Chain with transition matrix given

Let $(X_{n})_{n\in\mathbb{N}_{0}}$ be a Markov Chain with state space $M=\left\{x_{1},x_{2},x_{3}\right\}$ and transtition matrix $$ \Pi=\left(\begin{array}{ccc}p_{1} & p_{2} & 1-p_{1}-p_{2}\\ q_{1} & q_{2} & 1-q_{1}-q_{2}\\ r_{1} & r_{2} &…
Diego Fonseca
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Neural network to read short strings - translational invariance in CNNs

I have a series of short strings that each describe some item (one item per string). The people who write these strings can get pretty creative when it comes to spelling. For each string, I also have the label of the true object the strings refers…
Jivan
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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 $$ f(x) = \frac{1}{\beta} \exp(-\frac{1}{\beta} x) $$ where…
max
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Doubt in the Invariance Property of Consistent Estimators

Let $X_1,X_2,X_3,..,X_n,X_{n+1}$ be random samples from $N(\mu,1)$. Let us define $\bar {X}_n = \frac{\sum X_i}{n}$ and $T = \frac{1}{2}(\bar {X}_n + X_{n+1})$. It is required to test whether $T$ is consistent and unbiased for $\mu$. We can clearly…
userNoOne
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Finding a distribution with a particular invariance property: F(x/b) - F(x/a) independent of x

Suppose $F$ is a cdf for some random variable on some support, and that $a,b$ are constants with $a<1
bhalperin
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Are HPD intervals invariant to reparameterization?

Suppose I have a parameter (or just any random variable) $\theta$, which we can assume to be absolutely continuous. If I compute a Highest Posterior Probability $I$ for $\theta$ and another one (let's call it $I'$) for $g(\theta)$, will it be true…
PedroSebe
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What determines the functional form of maximum entropy constraints?

I'm familiar with the maximum entropy (ME) principle in statistical mechanics, where, for example, the Boltzmann distribution $p(\epsilon_i|\beta)$ is identified as the ME distribution constrained by normalizability and a given average energy…
marnix
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