Questions tagged [bounds]

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Bound on the expectation of a function of random variable having a strictly log-concave probability density

let $\theta \in \mathbb{R}^d$ be a random variable having a strictly log-concave probability density function, i.e \begin{equation} p(\theta) = e^{-\phi(\theta)} \end{equation} where $\phi(\theta)$ is ...
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Regression left limited dependet variable

My scope is to analyze the impact of certain variables on the change in sales. As you can see, my dependent variable is a proportion of two variables and is limited to -1 (-100%). On the other hand, ...
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21 views

The lower bound of K-L divergence of a mixture

I'm wondering if there is a lower bound for a mixture when each single component K-L divergence in the mixture is lower bounded by some constants. Let $$D(p||q)=\int p(x)\log \frac{p(x)}{q(x)}dx$$ If $...
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Fixed-leg Kalman filter smoother (Rauch–Tung–Striebel) error bounds

Although very intuitive and with plenty of results that talk about the asymptotic convergence of the estimate I wasn't able to track down any paper stating explicitly convergence bounds based on the ...
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Bound over sample probability

I have a discrete random variable $x \sim \text{Cat}(\textbf{p})$. What I'm trying to compute is the probability that any sample $x$ has an associated probability of at least $\alpha$. I would compute ...
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2 votes
1 answer
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Is this derivation in Manski (1990) correct?

Consider the following setting. There are two treatments, $A,B$. Individuals in the population are described by a tuple $(y_A,y_B,z)$ where $z \in \{A,B\}$ denotes the treatment received. Only $y_A$ ...
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2 answers
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Given a correlation between A and B, what are the best bounds on the product of the AC and BC correlations?

Let's say I have three vectors (or random variables) $A, B,$ and $C.$ I can of course calculate the correlation between any of them (and have these numbers). However what I'm interested in is if there'...
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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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1 answer
37 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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1 vote
1 answer
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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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1 answer
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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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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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1 answer
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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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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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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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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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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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6 votes
2 answers
150 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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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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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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What code lengths can optimal prefix codes assign to the symbols in a given probability distribution?

(Notation) Consider a finite alplhabet $\Sigma\equiv \{x_1,...,x_n\}$, corresponding to a probability distribution $\{p_1,...,p_n\}$. I want to encode this using a uniquely decodable binary code. Let $...
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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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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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1 vote
1 answer
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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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0 votes
1 answer
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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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1 vote
1 answer
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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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1 vote
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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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1 vote
1 answer
87 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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"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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3 votes
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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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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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7 votes
2 answers
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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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0 votes
1 answer
53 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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1 answer
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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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4 votes
1 answer
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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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2 votes
0 answers
191 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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1 vote
0 answers
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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2 votes
0 answers
76 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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1 vote
0 answers
25 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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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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5 votes
2 answers
545 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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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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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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3 votes
1 answer
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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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-3 votes
1 answer
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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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2 votes
1 answer
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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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1 vote
1 answer
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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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2 votes
1 answer
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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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2 votes
1 answer
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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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2 votes
4 answers
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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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