Questions tagged [bounds]

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Bounds on Ratio of Likelihood to Marginal?

Bayesian inference tells us that the posterior over parameters $\theta$ given data $X$ is given by: $$p(\theta|X) = \frac{p(X|\theta)}{p(X)} p(\theta)$$ Are there any known bounds on the ratio of the ...
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30 views

Is there an example of a Lipschitz function of a Gaussian vector for which $f(Z)-\mathbb{E}[f(Z)]$ is not sub-Gaussian

Definitions: A random variable $X$ is called sub-Gaussian with parameter $\sigma^2$ if there exists $\sigma \in \mathbb{R}$ such that $$\forall \lambda \in \mathbb{R} \quad \mathbb{E}[e^{\lambda X}]\...
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47 views

Why is the lower bound of the confidence interval of a model's error relatively constant compared to the upper bound? [closed]

I am interested in studying the effect of increasing data samples for a regression model on train error and test error. For this I have used 95% confidence intervals for different values of a sample ...
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29 views

What is the definition and upper bound on the variable "m" in the definition of the multivariate normal Fisher Information?

Multivariate normal distribution [edit] The FIM for a $N$-variate multivariate normal distribution, $X \sim N(\mu(\theta), \Sigma(\theta))$ has a special form. Let the $K$-dimensional vector of ...
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52 views

How to compute a non-trivial lower bound on $P \left[|X| > \frac{|\alpha|}{2} \right]$?

Let $X$ be a random variable such that $E[X] = \alpha$, $\alpha \in \mathbb{R}$ and $E[X^2] = \beta$. The problem is to find a lower bound on the following probability $$ P \left[|X| > \frac{|\...
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1answer
45 views

What is the probability that every variable of combination of random variables is greater than a specific value?

Suppose there are $N$ positive random variables. Each variable follows an exponential distribution with parameter $\lambda_i$. Now, we choose $n$ variables among the $N$ variables. What is the ...
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14 views

Lower bounding the sum of product of two sub-Gaussian variables where one follows an AR(1) process

Suppose we have the sum \begin{equation} \sum_{t=2}^{n}\epsilon_{t-1}u_t \end{equation} where $\epsilon_t$ and $u_t$ are both sub-Gaussian variables. Further suppose that while $u_2,\cdots,u_n$ are i....
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37 views

Predicting limits of bounded dependent variable in Random Forest

I am new to machine learning and trying to use Random Forest to predict a bounded dependent variables (percentage from 0 - 100). The majority of the training data points (~80%) are at the limits of ...
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19 views

why the following implication relationship is true

In a paper(unpublished), I read about the following implication relationship. $g(x)$ is a smooth function. The assumption is that there exist a constant (scalar) $a>0$ such that $E[sup_{|b|\leq a}(...
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1answer
34 views

How to find the bound of a time dependent variable?

I am working on modeling my problem statistically and I need to Know the bound or range of my variable. It is a time series variable position(t) where position(t+1) = position(t) + a.X - (1-a).Y a is ...
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142 views

Bounding sum of quartic deviations from sample mean

[Cross-posted here with no answers for a few days] I came - to the very best of my knowledge from reading the source - across the following statement in The Jackknife and Bootstrap, Shao and Tu, p. 87:...
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115 views

Can we find upper bound for loss functions?

Is it easy to find upper bound for loss functions like 0-1 loss and hinge loss ?!. I always find this sentence, which is "hinge loss is an upper bound of 0-1 loss", Can we compute the upper ...
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18 views

Inferring bounds from joint typicality on three variables

Consider the following exercise from Cover and Thomas: And the given solution from the solutions manual: It is reasonably clear that these bounds are valid (one simply follows the counting argument ...
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118 views

Interpretation of upper bound on the Wasserstein Distance

I am trying to interpret the 2-Wasserstein distance and the upper bound on it. Let's say I have 2-Wasserstein distance between two distributions to be $x$, and I have an upper bound on it which gives ...
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1answer
52 views

How can one show that $\bar{X}$ is the best unbiased estimator for $\lambda$ without using the Cramèr-Rao lower bound?

Assume we have the random sample $X_1, \dots, X_n$ with mean $\mu$ and variance $\sigma^2 < \infty$. We have that $E[S^2] = \sigma^2$, where $S^2 = \sum_{i = 1}^n \dfrac{(X_i - \bar{X})^2}{n - 1}$ ...
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174 views

Upper bound for variance of $\hat{\beta}$ in multiple linear regression

The variance of the beta estimator in an ordinary-least-squares multiple linear regression to express $Y$ as a (linear) function of $X$, $\hat{\beta}$, can be expressed as (knowing $X$ and $\sigma^2$ ...
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1answer
58 views

How to generate samples of ARIMA(p,d,q) model within an interval?

I am want to generate samples from an ARIMA(p,d,q) or ARMA(p,q) model. There is a Python Package to generate ARMA samples. The problem is that I want to generate scenarios for demand which should be ...
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32 views

What Cramer-Rao bound should I use?

I have been researching about the Cramer-Rao bound and I have found two inequalities: $$\text{Var}\left(\hat{\theta}\right)\geq\frac{1}{\text{E}\left[\left[\frac{\partial}{\partial\theta}\ln f(X;\...
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Expected number of hits of decision nodes in decision tree

Given a dataset $X\in \mathbb{R}^{n \times m}, Y \in \{0,1\}^n$. One can fit a decision tree model. Assuming the fitted decision tree have $p$ Decision node. If we a child node is about feature $i$, ...
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26 views

Show $(E|X|^2)/(E|X^2|) \leq P(X \not =0)$

I'm looking to show this inequality is true, and in turn use it to conclude the second moment method's bound. Show that $\frac{E|X|^2}{E|X^2|} \leq P(X \not =0)$. Again, I'm not supposed to use second ...
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49 views

Upper Bound for 2nd Raw Moment of Positive Random Variable

Let $X$ be a random variable with support $(0,\infty)$. All I know about $X$ is the support, finite higher moments, and $\mathbb{E}(X)=\mu$. I am trying to come up with a more tractable upper bound ...
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1answer
74 views

What does knowing two pairwise copulas tell us about the third

Say we have three random variables, which are all standard uniforms: $$ X \sim U(0,1), \\ Y \sim U(0,1), ~\text{and}~~~ Z ~ U(0,1) $$ If we know two of the pairwise copulas, $C_{XY}$ and $C_{YZ}$, ...
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17 views

"Z-value" equivalent for sample variance

For a random variable $X$ (mean $\mu$, variance $\sigma^2$, kurtosis $\kappa$), I take $n$ i.i.d. samples $X_1,\dots,X_n$ and find their mean, $\hat \mu^{(n)}$. By linearity of expectation, I know it ...
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23 views

Bounding values of a Dirichlet distribution

Consider $k$ random variables $X_1, X_2, \ldots, X_k$ such that $(X_1, X_2, \ldots, X_k)$ follow a $\text{Dirichlet}(1, 1, \ldots, 1)$ distribution. For a large enough $k$, I am trying to bound/find ...
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34 views

How to derive Chernoff Bounds for Sample Variance?

I was reading a paper on Bandits where I encountered this: After searching around on the internet I found and understood the first set of bounds quite well. However, I could not find any explanation ...
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131 views

Tail probability bounds on $P(|Z| > t)$ tend to be useless for small $t>0$. Why is that?

Background I am taking an introductory course on probability and inference. We recently covered several useful inequalities which I will list below: Markov's Inequality Let $X$ be a non-negative ...
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1answer
47 views

Conjectures regarding EM approximations of mixtures of multivariate normal distributions

Consider $X\in\mathbb{R}^{N\times d}$ containing data for $N$ points in $d$ dimensions drawn from a bimodal multivariate normal distribution, where any row $x$ of $X$ follows the mixed multivariate ...
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35 views

Measures of correlation / influence for predictors with bounded outcome

I'm doing a systematic review of epidemic models that project "the % reduction in incidence ($Y$) after K years" given a particular simulated intervention. The models include various ...
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65 views

Which regression model distribution or transformation for data bounded between -1 and 1?

It seems quite common in studies of plant interactions to find response variables that are bounded between -1 and 1, such as this relative interaction index (from Armas et al 2004, Ecology 85, https://...
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139 views

How does maximising ELBO for a Gaussian mixture model fit the model to data?

I am following along in Bishop's Pattern Recognition and ML chapters 9 and 10, and I understand that the EM algorithm works by iteratively updating model parameters using equations derived from ...
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35 views

Bounds on distance between two independently variables drawn from the same distribution

Suppose $X_1$ and $X_2$ are iid from an arbitrary distribution with variance $\sigma^2$. How can we derive an upper bound for: $$P(|X_1-X_2|\ge\delta)$$ One simple idea is Chebyshev's Inequality, ...
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56 views

Are there supposed to be bounds on parameters in 2PL Item Response Theory models?

Recently I've been studying Item Response Theory (IRT) and have come across some issues with the application side of it. I currently have a dataset of ~200 respondents x 7405 questions (quite ...
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18 views

How to find upper and lower bound

Let $\Sigma \in S_{++}^n$ be a symmteric positive definte matrix with all diagonal entries one. Let $U \in R^{n \times k_1}$, $W \in R^{n \times k_2}$, $\Lambda \in R^{k_1 \times k_1}$ and $T \in R^{...
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38 views

Bounding the norm of the difference between two related probability densities

Suppose we have a continuous random variable $X$ and two continuous functions $f$ and $g$ such that $f(X)$ and $g(X)$ are continuous random variables. Let $p_A$ be the probability density function of ...
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442 views

Is there a statistical distribution whose values are bounded $[-1,1]$ and sum to 1?

The Dirichlet distribution contains values that are bounded $[0,1]\in \mathbb{R}$ and sum to $1$. Is there a parametric distribution or similar method whose values do the same but reach as low as $-1$?...
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29 views

What is a common-sensical approach to setting the boundaries of an interval?

As I am trying to present my results to a non-expert audience, I am wondering about what the most commonly used boundaries are for intervals. I mean specifically, which of the four versions explained ...
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52 views

How to bound a regressor function?

I've seen similar questions on here, but none seem to quite apply to my use case. I want to predict Metacritic scores bases on a number of features. Metacritic scores are bounded to a 0-100 scale, ...
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295 views

Use Chebyshev's inequality to find a lower bound of a Chi-Square Distribution

I'm trying to solve the following exercise but I'm not sure if what I'm doing is right. "Let $X$ be an r.v. distributed as $\chi_{40}^{2}$. Use Tchebichev’s inequality in order to find a lower ...
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63 views

Positive or negatively bounded CDFs [closed]

If $X\in\mathbb{R}^n$ is a continuous random variable whose cumulative distribution function is ordinarily $$F_X(x) = \int_{-\infty}^{\infty} f_X(x) dx $$ what is the meaning of $$F_X(x) = \int_{0}^{\...
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1answer
96 views

Does a generalization bound that holds with high probability imply a bound that holds in expectation?

I am interested in generalization bounds, for example PAC bounds (Probably Approximately Correct). In particular, I wonder if a high probability bound implies a bound in expectation (or vice versa). ...
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1answer
39 views

How to deal with training models on data where the examples are highly dependent on each other?

Say you have a dataset of products sold at a store with the special condition that each day there is only one of each product in stock. That is, if there are multiple orders for a given product on a ...
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1answer
97 views

A tail bound for an unknown distribution via sampling

I know that many results exist for making an argument about the tail of a distribution, i.e., for a random variable $X$, one can find a bound $\epsilon$ such that $\Pr[X \geq a]<\epsilon$. Some ...
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1answer
180 views

Is a bounded real-number random variable discrete or continuous?

A discrete random variable is countable (such as integers and natural numbers), whereas a continuous r.v. is not countable (like the real numbers $\mathbb{R}$). If I have a dataset whose observations ...
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1k views

How to generate random numbers normally distributed in R or any software with limitations (bounds)?

I am working on a project where I need to generate random numbers for a given task time which is normally distributed with mean = 40, and standard deviation = 150. Because of the high SD, I will get ...
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93 views

On the difference between the main effect in a one-factor and a two-factor regression

Consider a linear regression (based on least squares) on two predictors including an interaction term: $$Y=(b_0+b_1X_1)+(b_2+b_3X_1)X_2$$ $b_2$ here corresponds to the conditional effect of $X_2$ when ...
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71 views

Cramer-Rao Lower Bound Proof (fuzzy step)

The following is the derivation of the Cramer-Rao lower bound as detailed on p.336 of Casella and Berger's Statistical Inference: $\frac{d}{d\theta}E[W(\bf{X})|\theta] = \int_{\chi}W(\bf{x})\left[\...
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116 views

Symmetrization in Proof of Hoeffding's Lemma

This alternative proof of a slightly weaker version of Hoeffding's Lemma features in Stanford's CS229 course notes. What's notable about this proof is its use of symmetrization. However, I find this ...
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1answer
25 views

Why $Pr[X-\mu \geq t]= Pr[e^{\lambda(X-\mu)} \geq e^{\lambda t}]$ for all $\lambda> 0$

I hope everyone is having a nice day. I don't know why this inequality holds. $$ Pr[X-\mu \geq t]= Pr[e^{\lambda(X-\mu)} \geq e^{\lambda t}] $$ For $\lambda >0$. I guess it has something to do ...
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35 views

Equivalence testing: Is it appropriate to set the equivalence bound such that I can reject H0 at alpha=0.05?

I have conducted a survey. One sample answered a binary question (answer A or B), once with and once without treatment. Now there does not seem to be a treatment effect as the proportions of answers ...
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68 views

Causal AR Model?

This questions is about necessary conditions (in form of inequality on coefficients) for the causality of autoregressive models. For instance, $|\phi_1| < 1$ is a necessary condition for an AR(1) ...

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