Questions tagged [bounds]
The bounds tag has no usage guidance.
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0answers
5 views
On the practical usage of generalisation error bounds
(This questions is based on a question that I've posted previously here, but I would like it to get more exposure)
In many practical scenarios, one would like to answer how much more data is needed ...
11
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6answers
953 views
Linear regression when Y is bounded and discrete
The question is straightforward: Is it appropriate to use linear regression when Y is bounded and discrete (e.g. the test score 1~100, some pre-defined ranking 1~17)? In this case, is it "not good" to ...
3
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2answers
38 views
Bounds on Expectation $E[A(B-C)^2]$
[This question has been edited for more given conditions].
Given possibly correlated random variables $A,B,C$, I want to find the best upper bound for $E[A(B-C)^2]$ given the following:
$E[A(B-C)]$
$...
2
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0answers
19 views
Bound for type of correlation measure
Assume you have a finite, discrete probability distribution for a joint random variable and such that $P(X=i,Y=j) = p_{i,j}$ for $i \in \{1, \dots, |X|\},j \in \{1, \dots, |Y|\}$. The marginal ...
1
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0answers
19 views
Bounds on quantiles of the minimum of summations of (possibly dependent) random variables
Suppose I have $2N$ continuous random variables $X_1, \ldots, X_N, Y_1, \ldots, Y_N$ and that I can evaluate the quantiles of the respective distributions. Given a value $w \in [0, 1]$ I would like to ...
1
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1answer
26 views
Is there an upper bound on number of logistic regression responses that yield infinite estimates
Suppose a logistic regression problem has N observations of {0, 1} and that there are p parameters. Also assume the design matrix, X, is full rank with p < N. We know that there will be certain ...
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0answers
25 views
Approximation of the upper bound on the expectation of log sum of exponentials
I am having some trouble replicating the results in Guillaume Bouchard's paper, Efficient Bounds for the Softmax Function and Applications to Approximate Inference in Hybrid Models, where he discusses ...
1
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0answers
51 views
Finding Chernoff bounds maximum estimators
I am currently trying to resolve the following exercise about Chernoff bounds:
Let $X_{1}, X_{2}, \dots, X_{n}$ be independent, identically distributed (i.i.d) random variables with distribution $N(0,...
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0answers
18 views
Reference request: Least Squares Concentration Bounds
I've recently become interested in parameter estimation for regression and time-series type models and I often encounter, and indeed need to understand, results using concentration of measure-type ...
1
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0answers
44 views
Tight upper bound on the expectation of a concave function
N is a random variable whose sample space is [0,$\infty$). I have an expression in terms of the expectation of this variable and I want to find a tight upper bound on the whole expression. The ...
0
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1answer
28 views
Bounded functions when x tends to infinity
Please help me understand the below:
The notation $g(x) = O(f(x))$ denotes that $\left|\frac{f(x)}{g(x)}\right|$ is bounded as $x \to \infty$.
For instance if $g(x)=3x^2 + 2$, then $g(x) = O(x^2)$
...
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1answer
52 views
Find bayesian credible Interval
I am taking this example from here. They have given the steps but i cannot understand them so a little dumbing down of answer is necessary or you can just explain the answer given there. its on page 5
...
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0answers
81 views
Is it true that normalizing the output of a ReLu feedforward Neural Network that its Rademacher Complexity becomes a constant?
I was trying to understand what happened with the Rademacher Complexity:
$$ R_S(F) = \frac{1}{m} \mathbb E_{\sigma} [\sup_{f \in F}\sum^m_{i=1} \sigma_i f(z_i)] $$
or
$$ R_{P,m} = \mathbb E_{s \...
0
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1answer
156 views
ELBO interpretation in Variational Autoencoder (VAE) for anomaly detection
How to interpret different ELBO values when checking anomaly detection possibilities of VAE model on different "testing" datasets?
The higher the ELBO value of the model when testing it on different ...
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0answers
12 views
Upper bound on single frequency error based on SSE of the full signal
I have an empirical signal:
$y(t) = A_0 + \sum_{i=1}^{M}A_i \cos(\omega_i t + \phi_i) + \epsilon(t)$
The signal is tidal, and so dominated by a small number of frequencies and in particular there is ...
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1answer
108 views
Fréchet Hoeffding bounds for symmetric random variables
(Edited to clarify the question).
The Hakan & Demirtas (2012 doi: 10.1198/tast.2011.10090) approach to approximating Pearson correlation bounds uses the concept of the Fréchet-Hoeffding bounds by ...
2
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0answers
47 views
What's the reasoning behind non-seasonal ARIMA model lag-order bounds?
I read some papers on the non-seasonal ARIMA model, and the consensus I've seen is that for ARIMA(p, d, q), p and q should not be greater than 3, maybe 5. What's the reasoning for that? Is it for ...
7
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1answer
194 views
Bounds on the difference of correlated random variables
Given two highly correlated random variables $X$ and $Y$, I'd like to bound the probability that the difference $ |X - Y| $ exceeds some amount:
$$ P( |X - Y| > K) < \delta $$
Assume for ...
0
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0answers
36 views
Is max. Eigenvalue of k-sparse PCA always $\leq$ max. Eigenvalue of normal PCA on same dataset?
Is max. Eigenvalue of k-sparse PCA always less than or equal to the max. Eigenvalue of normal PCA on same dataset? K refers to the number of non zero eigenvalues when the dataset is of dimension n <...
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0answers
25 views
Expected value of (F(x)/x(1-F(x)) where F(x) is CDF?
I was solving a problem and encountered the following:
\sum (f(x)/x) (F(x)/(1-F(x))
Here f(x) is pmf and F(x) is CDF. Is expected value of F(x)/x(1-F(x)) a known function? I found out that f(x)/(1-F(...
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0answers
143 views
regression with given upper and lower bounds for the target value
I am using several regressors like xgboost, gradient boosting, random forest or decision tree to predict a continuous target value. I have some complementary information like I know my prediction (...
3
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0answers
43 views
Concentration inequalities for weighted sums of gaussians
Suppose that $x \sim \cal{N}(0,I_d)$ be a $d$-dimensional standard Gaussian vector and let $x_1,\ldots,x_n$ denote $n$ i.i.d. samples drawn from the same distribution.
For some fixed vector $\theta \...
2
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2answers
78 views
Understanding the concept of “Bounded in probability”
My statistics book defines the concept of "bounded in probability" in the followng way:
..But doesn't this mean that any sequence of R.V.'s that does not include any R.V.'s with a pdf with infinite ...
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0answers
44 views
Bounds on probability that a random variable is neither the maximum nor the minimum of a set of random variables
Suppose I have $n$ independent random variables $X_1,\dotsc,X_n$ which are Poisson distributed with $X_i \sim Poi(\lambda_i)$. Without loss of generality, an additional condition $\lambda_1\le\...
1
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0answers
30 views
Comparing bound between lasso and OLS
Assume fixed design regression model $Y = \mathbf{X} \beta + \epsilon$ with common assumptions. Lasso estimator $\hat{\beta}_\lambda$ can be shown to have the following bound.
For any $\delta > ...
2
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0answers
58 views
Bounds on variance and mean of maximum of difference of independent random variables
Suppose $X_1,\dotsc,X_n$ are independent but not necessarily identical random variables. $$Y = \max_{1\le i,j\le n}(X_i-X_j)$$
What upper and lower bounds can be derived for expectation and variance ...
1
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0answers
336 views
Upper bound on KL divergence
Is there a maximum (unique?) to the KL divergence between discrete distributions p & q, with the restriction that q is a proper probability distribution?
I know KL is unbounded from above when q ...
3
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0answers
38 views
Correlation bounds for uniformly distributed matrix?
For a uniformly or Guassian distributed $M\times N $ matrix. Is there any analytical expression in terms of $M$ and $N$ to estimate the maximum and minimum bounds of correlation between the columns of ...
3
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0answers
61 views
Tail bounds for F-distribution (not using $\chi^2$ bounds)
Are there any sharp tail bounds for an $F_{p,q}$ distribution? That is, if $X \sim F_{p,q}$, then for a $t_1,t_2 > 0$, what are the sharpest $\delta_1$ and $\delta_2$ known such that
$$P(X > ...
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0answers
47 views
Bounds on tail conditional expectation of random variable given variance
Given a random variable $X$ with CDF $F(X)$, mean $E(X)=0$, and variance $Var(X) =\sigma^2$, I would like to bound the tail conditional expectation where $X$ is in the tail with probability $1-p$:
$E(...
3
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1answer
58 views
Minimum of Poissons
Let $X_i\sim\text{Pois}(\lambda_i)$ for $i=1,2,\ldots,n$ and $Y = \min X_i$. Can we show that, for example $\mathbb{E}[Y] \leq f(\lambda,n)\min\lambda_i$ for some $f : (\mathbb{R}^n,\mathbb{N}) \to [0,...
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0answers
32 views
Is there a method to fit a bound to the plot of an linear inequality?
I have a physical dataset that is bounded by several different processes, and thus the plot takes the form of a linear inequality:
I'm specifically interested in studying the upper bound. Is there a ...
3
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1answer
115 views
Tightest bounds on sample variance given sample size, mean, minimum, and maximum
For real-valued samples (possibly known to lie in some interval, but without further constraints on them), I am interested in the tightest possible bound on the sample variance $\sigma^2$, given the ...
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0answers
35 views
How to estimate the bound on a random variable with semi-infinite distribution?
Given samples of a random variable which appears to have a continuous unimodal probability density function that is zero on one side of a bound, what statistical methods are there for estimating that ...
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0answers
140 views
Relationship between big-Oh and little-Oh
I'm reading Wu, J. (2014), Restoring monotonic power in Wald/LM-type tests.
At a certain point in his derivations, an expresion turns out to be $=O_p(\frac{1}{T^{3/10}h^{5/10}}+T^{2/10}h^{20/10})$, ...
0
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1answer
56 views
Price Analytics - Price range for a product
I have data on weekly, total number of units sold, at what price, any discount, at which store, product details.
like below
For over 3 years for the Product details across 10 stores.
I would like to ...
0
votes
1answer
83 views
About the expectation of a posterior [closed]
I am interested in finding an upper bound for
$\mathbb{E}\big(
\int_B
\frac{f(X| \theta)}{f(X| \theta_0)} \pi(\theta) d \theta\big)$,
the $\frac{f(X| \theta)}{f(X| \theta_0)}$ is the likelihood ...
2
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0answers
67 views
Constraints on the moments of a bounded probability distribution
Consider a probability distribution with support on $[0,1]$. Suppose the first $n$ raw moments $m_1,...,m_n$ are given. What are the constraints on the $(n+1)^\text{th}$ moment? Obviously we have the ...
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2answers
488 views
disadvantages variational inference
A lot of methods utilize variational inference for hyperparameter calculation.
What are the advantages and disadvantages of variational inference ?
(ex: does it guarantee a global optimal ? )
0
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1answer
86 views
Need to know pdf of “x/z+sqrt(y^2-x^2)/z” , or any idea about its upper/lower bounds
I need to know the pdf of the following equation or any upper/lower bound would help. Let $X, Y, Z \sim N(0, \sigma^2)$. Then what is the distribution of:
$$A=\frac{x+ \sqrt{\mid y^2 - x^2 \mid}}{z}$$...
4
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1answer
559 views
Gaussian Multi-Armed Bandits and the UCB Algorithm
I've implemented in MATLAB the UCB algorithm for gaussian bandits with zero mean and unit variance (these means were themselves sampled from a gaussian prior of zero mean and unit variance).
Now I ...
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1answer
161 views
Probability distribution for a limiting value
I have a bunch of measurements $D_1, D_2, ..., D_N$ and their associated uncertainties $\sigma_1, \sigma_2, ..., \sigma_N$. Suppose that the (unknown) true values of the data points are $T_1, T_2, ...,...
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0answers
56 views
Solve Multivalued function as a Multilevel model?
I have a dependent variable (y) that is bounded between 0 and 2. The ideal value of the variable is 1. Values closer to 0 or 2 are theoretically bad (not outliers). I also have a set of potential ...
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0answers
76 views
Finding a tighter bound for weighted sum of random variables
Here is the setup of my problem:
Let $X_k$ be a sequence of independent random variables in $\mathbb{R}$ with $E(X_n) = 0$ and Var($X_n$)<$\infty$ for all $n$. Let $\{a_k\}$ and $\{A_k\}$ denote ...
3
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0answers
117 views
Upper bound for randomly weighted sum of independent random variables
I have a sequence of independent random variables {$\epsilon_j$} with mean 0. I also have another sequence of Bernoulli random variables $\delta_1, \delta_2,\dots$ which are dependent on the previous ...
0
votes
1answer
230 views
log-sum-exp bound to be used in variational inference
I have been reading several papers on using a bound on log sum of exponentials to be used in variational inference. One example of this case that can happen is in correlated topic models. Where to ...
4
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1answer
885 views
KL divergence bounds square of L1 norm
In Cover & Thomas, Elements of Information Theory, at the section on Conditional Limit Theorem (11.6), it is proved that the KL divergence bounds the $\cal{L}_1$-norm from above,
$\frac{1}{2\ln2}\...
3
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1answer
514 views
Lower bound on smallest eigenvalue of covariance matrices
Assume that a class of $p\times p$ covariance matrices is characterized by a parameter $\theta$, i.e,
$$\mathbb{F} = \left\{\Sigma(\theta), \theta\in R\right\}$$
Also suppose we know the following
...
4
votes
3answers
162 views
Tails of products of random variables
Let $X$ be a non-negative random variable, and let $Y \sim \chi^2_n / n$ (so that $E(Y) = 1$). $X$ and $Y$ are independent. Note that $X$ and $X\cdot Y$ have the same mean, while $X\cdot Y$ has larger ...
1
vote
1answer
30 views
Bounding cdfs of two distributions with same mean
Let $X$ and $Y$ be two independent non-negative random variables such that $E[X] = E[Y] = \mu$ but $\operatorname{Var}[X] < \operatorname{Var}[Y]$. Can we use only these two moments to show that ...
