All Questions
1,933 questions
3
votes
1
answer
136
views
t-test on non normal data: type I/II error vs validity
First, I don't believe this is a duplicate post even though this topic has been brought up a million times. If it is, please point me to the relevant post and I will remove this one.
I am basically ...
- 147
1
vote
0
answers
64
views
Definition p-value and find p-value in practice
I have a problem that I can't solution. Let $\mathbf{X}=\{X_1,X_2,\ldots,X_n\}\sim\mathrm{Uniform}(0,\theta)$ and we have $H_0:\theta=\theta_0$ and $H_1:\theta>\theta_0$. We reject the $H_0$ when $...
0
votes
0
answers
16
views
Monte Carlo Simulation deteriorates model predictions?
I have a model that works well on predicting the target variable $Y$ on time $t$ based on the input variables $X$. However, since I need to predict the $Y$ at $t+1$, for which I don't have the data (...
- 365
3
votes
1
answer
54
views
Estimation Error Calculation
I'm learning about variance reduction for Monte Carlo methods and I am confused about how to calculate the "estimation error" of a given method.
My question is how should I interpret "...
- 133
0
votes
1
answer
133
views
Monte Carlo simulations and sum of normal distributions
I am trying to predict the revenues of a portfolio of items. I want to simulate the revenues in a particular market situation in which they might increase. Each item's revenues is made up of 3 ...
- 21
1
vote
1
answer
41
views
Can I do a meta-analysis by Monte-Carlo synthetic data?
I'm trying to do a meta analysis of ~30 studies (total N = ~2000) on the correlation (X, Y). However, the heterogeneity is soooo high. My hypothesis (and what has been suggested in the literature) is ...
- 23
2
votes
0
answers
169
views
In practice, should I use k-fold cross-validation or repeated random sub-sampling validation as my default choice of evaluating the model performance?
I was wondering if someone can shed some light on which cross-validation method should I, in general, use more often: k-fold cross-validation or repeated random sub-sampling validation.
From Wikipedia,...
- 121
1
vote
1
answer
3k
views
What are correlated errors and why are they important?
I am looking for help on correlated systematic errors, and their meaning. I have some quantities $x,y,z$ which determine a function I need to calculate. These 3 quantities are determined by a ...
2
votes
0
answers
21
views
How to compute a rectangular credible region from samples
Given high-dimensional Monte Carlo samples $\bf{X_1},...,\bf{X_N}$ from a probability distribution $p({\bf x})$ in $\mathbb{R^d}$, I want to estimate a rectangular highest-density credible region for $...
- 31
0
votes
0
answers
34
views
Find maximum of bimodal posterior pdf
can you help find the maximum (analytically) of the following posterior pdf?
$p(\theta|x) = \frac{\alpha}{\sqrt{2\pi}}e^{-\frac{1}{2}(\theta-x)^2} + \frac{1-\alpha}{\sqrt{2\pi}}e^{-\frac{1}{2}(\theta+...
- 1
3
votes
1
answer
244
views
Monte Carlo Integration Results in Heavy Tailed Distribution
I am running a Monte Carlo simulation that results in an heavy-tailed distribution. The image below shows the distribution of 1,200 runs of the Monte Carlo simulation, where each run consists of ...
1
vote
1
answer
32
views
Which are the statistical methodologies to consider when examining study group death rates but without considering time to death?
I have a dataset for a group of 66,000 subjects diagnosed with a dangerous condition, and the time it takes for death to occur (the “event”) or to not occur (survival or “censored”). I am pursuing ...
- 827
1
vote
1
answer
170
views
Question on Equation 5.2 of Reinforcement Learning by Sutton and Barto
I'm currently studying the textbook Reinforcement Learning by Sutton and Barto. I can't seem to understand the derivation in Equation 5.2:
How did (a) become (b)?
In particular, why is the ...
- 23
0
votes
0
answers
48
views
Effect of Simulation Error on the Monte Carlo Estimator
Given a random variable $X$ with known pdf $f$ and some computer simulation model $g(x): \mathcal
R \rightarrow \mathcal R$, mapping samples $x \sim f$ to a scalar metric $g$, we can estimate the ...
- 43
1
vote
0
answers
65
views
Averages in Monte Carlo error propagation
I'm doing a Monte Carlo error propagation, so for $10^5$ iterations I:
generate a curve $y(x)$ choosing the parameters using a multivariate gaussian distribution.
generate a random point $x_k$ using ...
- 33
2
votes
2
answers
221
views
Calculating probability related to maximum of random variables
Let $X_1, X_2, \cdots, X_n$ be non-negative continuous iid random variables. The goal is to find the probability:
\begin{align*}
\Pr(\max_{k+1 \leq i \leq j } X_i < \max_{1 \leq i \leq k }X_i)
\end{...
0
votes
0
answers
62
views
How to choose priors for bounds on circular truncated distributions?
I am considering choices of priors for truncated distributions on a circle. Let's take the truncated normal distribution on the unit circle as an example. It has parameters $\mu \in [-\pi, \pi]$ and $\...
- 9,680
3
votes
1
answer
435
views
How to use Monte Carlo simulation to get the conditional mean
Given the following assumptions:
$Z,Z'\in\mathbb{R}^4$ where $(Z,Z')\sim N(0,\Sigma)$, for some known $\Sigma\in\mathbb{R}^{8\times 8}$.
$Y=f(Z,u,\epsilon)=Z_1\boldsymbol{1}\Big[u<\frac{\exp(Z_3)}{...
- 229
0
votes
0
answers
75
views
What can be concluded when standard deviation plus mean exceeds largest value?
The sum of the mean and standard deviation of a non-normal distribution can exceed the value of the largest sample. For a good explanation of why, see Can mean plus one standard deviation exceed ...
- 101
4
votes
2
answers
170
views
Goodness-of-fit for Lomax distribution
I have some data n > 3000 https://drive.google.com/file/d/1gwB_U_TOX-IQHZJJDX-WeErLzrZZFoXu/view?usp=sharing (Third column) that I believe based on my physical theory should follow a Lomax ...
- 65
1
vote
1
answer
176
views
How to choose the wanted root of the maximum likelihood function when there are multiple roots?
I need to estimate a parameter of a distribution but I don't have an explicit estimator. I decided to do a partition of the interval range for the parameter and use the newton-raphson method to find ...
2
votes
1
answer
178
views
CDF of max of $n$ cauchy variates
Suppose $X_1,X_2,\cdots,X_n$ are iid copies of a standard cauchy variate with pdf
$$ f(x)=\frac{1}{\pi(1+x^2)},0<x< \infty. $$
Define:
$$ Y=1+ \underset{1 \leq i \leq n}\max X_i.$$ I want to ...
- 447
2
votes
0
answers
35
views
How to empirically check a PDF
Intro
Let $Y$ be a random variable whose PDF is $p_Y(\cdot)$. Let's say that $Y$ is a function $g(\cdot)$ of another random variable $X$ whose PDF $p_X(\cdot)$ is given. Then, you do your calculation ...
- 473
0
votes
1
answer
405
views
Why does First-Visit Monte Carlo Prediction (Policy Evaluation) converge?
In Barto and Sutton's "Introduction to Reinforcement Learning" book, in Section 5.1 (Monte Carlo Prediction), they describe the First-visit (and every-visit) Monte Carlo (MC) methods for ...
- 121
2
votes
0
answers
84
views
Calculating confidence Interval for a return time curve, via non-parametric bootstrapping
I have some precipitation data (yearly extremes), which I have fit with a Gumbel distribution (CDF), from which I have calculated a return time distribution. I want to calculate the 95% confidence ...
- 21
1
vote
1
answer
37
views
How to handle physically meaningless values in sampling?
I'm working on stochastic optimization for optimal energy dispatch, where the uncertainty of photovoltaic power output should be considered with monte carlo sampling and scenario reduction technique. ...
- 113
-1
votes
1
answer
160
views
Evaluate integral using R [closed]
I need to evaluate $\displaystyle{\int_{1}^{1}\int_{1}^{1}\int_{1}^{1}(y)e^{x+}}dxdz}$ using in R.
Here is my attempt:
...
- 99
2
votes
1
answer
150
views
Generate a point cloud distributed according to a Boltzmann-Gibbs distribution with prescribed marginals
Let $p$ be a probability density on $\Omega\in\mathcal B(\mathbb R^d)$ for some $d\in\mathbb N$ (I'm primarily interested in $\Omega=[0,1)^d$). We can approximate $p$ by $$A_x(y):=\sum_{i=1}^k\varphi_{...
- 213
1
vote
0
answers
84
views
Monte Carlo Dropout as surrogate model for Bayesian Optimization
I am interested in using Monte Carlo Dropout as a surrogate model for Bayesian optimization. I noticed that the paper states:
The use of dropout (and its variants) in NNs can be interpreted as a ...
- 11
1
vote
0
answers
49
views
Is the variance of the maximum of a set of variables higher than the variance of the other variables?
Does the maximum of a set of random variables have high variance compared to the other variables in the set? If so, can someone give an intuitive explanation of why?
Some details about the motivation:
...
- 41
1
vote
0
answers
32
views
Suppose $f: \mathbb{R}^n \to [0, 1]$ is known, how to sample $x \in \mathbb{R}^n$ such that $f(x)$ follows uniform distribution? [closed]
Suppose $f: \mathbb{R}^n \to \mathbb{R}$ is known, where evaluation and gradient computation is easy. How can I sample $x \in \mathbb{R}^n$ such that $f_x \in \mathbb{R}$ follows uniform distribution? ...
- 211
1
vote
0
answers
144
views
Is there a variant of the Metropolis-Hastings algorithm with proposal and/or acceptance function depending on the history?
Is there a version of the Metropolis-Hastings algorithm where either
the proposal kernel; or
the acceptance function
might depend on the whole history (or at least a part of it) of the chain so far?
...
- 213
1
vote
1
answer
321
views
Robustness of Quantile Regression
Is the 99th Quantile Regression model a robust model?
From my understanding, Quantile Regression is supposed to be robust in nature, but removing some outliers using IQR, the results obtained by 99th ...
- 41
0
votes
0
answers
23
views
How to sample from a predictive distribution of a transformed quantity
Typically, it's easy to simulate from the predictive distribution for next observation, say $\tilde{y}$, after obatining the posterior distribution for $\theta$. We may first simulate a large number ...
- 21
1
vote
0
answers
32
views
Estimating fit parameter variance when the true distribution is unknown and systemic errors exist
I have a novel model for the errors that affect many types of qubits (quantum bits) and want to show my theory is correct. Visually it is great, but that is not quantitative.
I'm a theorist, while my ...
- 11
2
votes
0
answers
133
views
Is there any intuitive explanation for MoM in estimating parameters?
I found from some literature that when we use the method of moments to fit the Gumbel distribution, the estimated
(On page 24) A comparison of the variance formulas in (1.66) with the CramBr-Rao ...
- 747
2
votes
1
answer
100
views
Is there a variant of the Metropolis-Hastings algorithm where the acceptance probabiltiy can depend on all states generated so far?
I wasn't able to find anything on google, but is there a variant of the Metroplis-Hastings algorithm where the acceptance probability (not the proposal kernel) in the $i$th iteration might depend on ...
- 213
0
votes
0
answers
20
views
What is the expression for covariance in the context of Monte-Carlo estimator? [duplicate]
I am trying to calculate the variance:
$$
\langle(\bar{O}-<O>)^2\rangle
$$
of the Monte-Carlo estimator
$$
\bar{O}=\frac{1}{M}\sum_{m=1}^M{O_m}
$$
For uncorrelated samples.
In order to do so, I ...
- 1
1
vote
0
answers
346
views
What is a mathematic rigorous definition of "blue noise"?
Let $d\in\mathbb N$, $I$ be a finite nonempty set, $(x_i)_{i\in I}\subseteq[0,1)^d$, $(w_i)_{i\in I}\subseteq[0,\infty)$ with $\sum_{i\in I}w_i=1$ and $$\sigma:=\sum_{i\in I}w_i\delta_{x_i}.$$
I ...
- 213
0
votes
0
answers
39
views
Is it possible to calculate the SD of observations for a RR based solely on the confidence interval and mean?
Set up:
I have a epidemiological study with a dose-response curve with a series of relative risk estimates (risk ratio of mortality risk exposed compared to mortality risk unexposed) along a curve. ...
1
vote
1
answer
523
views
Time complexity of Metropolis-Hastings and potential speed-up?
The MH algorithm essentially involves generating a sample destination state from a proposal distribution, computing the acceptance probability as a function of that sample, and checking whether a ...
- 203
0
votes
1
answer
76
views
How can the author get the following conclusion from the QQ plot?
In this paper: https://www.tandfonline.com/doi/pdf/10.1080/02664763.2021.1940109, the authors have two actual datasets (e.g., 59 observations showing continuous annual flood data) and the authors want ...
- 747
0
votes
0
answers
170
views
Help analysing Mean Residual Life Plot for GPD
I'm trying to fit a GPD for a set of time dependant data. I have two columns, data which is a value on the negative real line where values closest to zero are considered extremes, and time. Using only ...
2
votes
1
answer
153
views
Is there a Quasi-Monte Carlo variant of the Metropolis-Hastings algorithm?
If we run the Metropolis-Hastings algorithm for a target distribution $\mu$ with proposals from a quasi-Monte Carlo sequence $(y_n)_{n\in\mathbb N}$ (such as a Sobol sequence) and the generated chain ...
- 213
0
votes
1
answer
198
views
Fitting Gumbel distribution based the maximal observation
Assume that we only consider $$G(x)=\exp(-\exp(\frac{x-\mu}{\sigma}))$$ is the Gumbel distribution.
Question: Suppose we have a set of maximum values $\{Y_i\}_{i=1}^m$, why can the article directly (...
- 747
2
votes
1
answer
566
views
Generate data from posterior predictive distribution [closed]
I am new to Bayesian. I want to draw data from the posterior predictive distribution p(y|D).
Do we need to find the CDF of the posterior predictive distribution and use the monte Carlo method or is ...
- 21
0
votes
0
answers
392
views
How many Monte Carlo simulations must I run to get a 95\% confidence interval for some error $E$
Suppose I want to use Monte Carlo to compute some probability $p$. A single MC simulation will run for $R$ iterations and calculate $p$ as the fraction of 'successes'.
Say I want to compute $p$ within ...
- 9
8
votes
4
answers
1k
views
Linearity of maximum function in expectation
I was solving an exercise for a probability theory course and stumbled upon the following problem.
Given a continuous random variable $X$, and $\max(a,b) = a$ if $a > b$ and $b$ otherwise, is
$$
E[\...
- 193
2
votes
1
answer
186
views
The Monte Carlo of the mean square error of the maximum likelihood estimates
I try to get mean square error of the maximum likelihood estimators in R (using Monte Carlo).
I can write the calculation for the MLE that is repeated once, but I need to repeat the Monte Carlo ...
- 747
1
vote
1
answer
218
views
How do I use MLE for non-iid actual data?
In this paper, the author try to fit the Gumbel distribution based on the r largest value of each year using the maximal likelihood estimators: the likelihood function for r largest values $X_{n1},\...
- 747