Questions tagged [inference]

Drawing conclusions about population parameters from sample data. See https://en.wikipedia.org/wiki/Inference and https://en.wikipedia.org/wiki/Statistical_inference

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Sufficient statistics for $\mu_1 - \mu_2$

If $ X_1, ..., X_n$ is a random sample from $ X \sim N(\mu_1, \sigma^2)$ and $Y_1,..., Y_n$ is a random sample from $Y \sim N(\mu_2, \sigma^2),$ if the samples are independent and $ \sigma^2$ is known,...
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Dealing with dependent data in a Bayesian model

Background: Consider a series of dependent data points, $$ y_1,y_2,y_3,\cdots,y_N. $$ In cases where the dependence is well described by an exponentially decaying auto-correlation function, it is ...
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What is the difference between a "population," a "sample space," an "underlying probability distribution? and a "model"?

I'm trying to understand an overview of the topic of statistical inference. I have learnt bits and pieces of many of the probability and statistics involved in it but before learning it rigorously it ...
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Can an asymptotically efficient estimator be biased?

In "Theory of point estimation" by Lehmann and Casella (1998) there is the following definition: It is also said that So terms of the asymptotically normal sequence of estimators can be ...
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How to make Bayesian-style inference for a Poisson process?

I am working on a fleet management software recently. Normally, the arrival of merchant request is a Poisson process. That is to say, on average we have a new merchant request every 10 minutes, but ...
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Let $X_1,X_2,\dots,X_n$ be random sample from Poisson($\theta$). Find MVUE of $e^{-2\theta}$

Question: Let $X_1,X_2,\dots,X_n$ be random sample from Poisson($\theta$). Find MVUE of $e^{-2\theta}$ My attempt has been by modifying the answer from this question: The Poisson distribution is a one-...
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How to show a UMVUE exists only if $g(p)$ is a polynomial of degree at most $n$?

Let $X\sim Bin(n,p)$. The problem is to show that a UMVUE can exist for $g(p)$ only if $g(p)$ is a polynomial in $p$ of degree at most $n$. For the case when $g(p) = \frac{1}{p}$ we can show that it ...
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Before using CV-selected Regression model for Inference, shouldn't model performance be evaluated on unused test set?

I just came across a biokinesiology paper that used some Machine Learning methods, but I think there is a flaw in their methodology. The authors had data on stroke patients and used Lasso regression ...
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887 views

Under what conditions will a Bayesian posterior fail to converge to a point mass?

Let's say you have a Bayesian model: $$\theta' \sim g(\theta|\mu) $$ $$ y \sim p(y|\theta')$$ And we have some data ($n$ data points) $\mathbf{y}_n$, which we will use to perform inference on $\...
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How to do inference over two steps in a graphical model simultaneously?

I have observed data $D$ about a physical object described by $M$. I would like to determine the posterior distribution of $M$ given $D$, or $p(M|D)$. Now I can't infer this directly because unknown ...
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Optimal Stopping for Bernoulli One-Armed Bandit with a Fixed, Known Payout

I'm very new to bandit problems (apologies if I've formatted my question incorrectly), but I have to solve the optimal stopping of what I think is a very simple case. Suppose I have two arms $k = {1, ...
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Variance of marginal posterior distribution

Suppose $Y_1,\dots,Y_n\mid\mu,\sigma^2 \sim \text{ iid } N(\mu,\sigma^2)$ and suppose the priors $\mu \mid \sigma^2 \sim N(\mu_0, \sigma^2 / \kappa_0)$ and $1/\sigma^2 \sim \text{gamma}(\nu_0/2, \nu_0 ...
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Would this modification accelerate convergence of generalized linear model, or break it?

This page describes the following iteratively reweighted linear least-squares (IRLS) method for solving a generalized linear model (GLM): let $x_1=0$ for $j=1,2,...$ do linear ...
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Does it make sense to infer a rate (as a probability distribution or upper limits) for a Poisson process if there are "no events"

I have an inhomogeneous Poisson process with a rate $\lambda (\mathbf{t})$ defined on some parameters $\mathbf{t}$. I am trying to infer $\lambda (\mathbf{t})$ from some data, which are events (really ...
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How much of a problem is inference after model selection when few models are manually compared?

tl;dr: I found a better model than the one I first thought of while inspecting the data and performed a few steps of variable selection/model fine-tuning. I assume that this is a (mild) case of ...
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Existence of UMVUE of $\theta$ for sample from $\small{\frac{\ln\theta}{\theta-1}\theta^x\,\mathbf1_{0<x<1}}$?

Suppose $X_1,X_2,\ldots,X_n$ is a random sample drawn from a distribution with pdf $$f_{\theta}(x)=\small\frac{\ln\theta}{\theta-1}\theta^x\,\mathbf1_{0<x<1}\quad,\,\theta>1$$ Does there ...
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Survival model for an epidemic -- can the observations be treated as independent?

I've been thinking about ways to tackle an epidemic modelling problem I've been working on, and I've come up against a conceptual difficulty over the way survival analysis works. Here's a really ...
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What are necessary & sufficient conditions for exponential family representation to have complete statistic $T(X)$?

My textbook gives the following theorem for exponential families: Let $X_1, \dots, X_n$ be a random sample from an exponential family with pmf/pdf of the form $$f(x|\theta) = h(x) c(\theta) \exp (w(...
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If a statistic can be written as a function of a minimal sufficient statistic almost everywhere, is it minimal sufficient?

I know that if $T(X) = f(W(X))$ for one-to-one $f$, where $W(X)$ is minimal sufficient, then $T(X)$ is also minimal sufficient. But my textbook does not include "almost everywhere" or "almost surely" ...
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Show that there is no efficient estimator for the variance of a normal distribution using properties of the exponential family

I want to prove the statement in the title using the following statement from Wikipedia: it was proved that efficient estimation is possible only in an exponential family, and only for the natural ...
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Dealing with multiple cases per subject

I want to analyze a table with surgery operations data. For some patients, about 10%, multiple operations have been made, so there may be multiple rows for a patient in the table. My goal is to ...
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Is there a principled ANOVA-like approach when a "subject" factor forms a symmetric 2D matrix?

I have a dataset where the dependent variable is a similarity measure between any 2 data channels, lets call it S(i,j), with a few dozen simultaneously recorded channels (so there are quite a few ...
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Interesting application of E-M algorithm

Suppose the following dataset: [3 4 3 4 6 12 12 7 8 9] [2 5 3 4 12 2 2 10 7 6] [3 4 3 4 5 11 10 7 8 9] These numbers are totally random. So this dataset, depicts ...
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148 views

How do I perform Bayesian Updating for a function of multiple parameters, each with its own distribution?

I have a variable that is a recursive function involving other variables with known distributions (see problem below). Let $b(t+1) = b(t) + C \sqrt{b(t)}$ where I know $C \sim N(1.82, .0298)$ and ...
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Finding the uniformly most powerful test

Let $X_1,X_2,...,X_n$ denote a random sample from density, $$f(x;\theta)={1\over 2\theta}, \quad 0<x<2\theta.$$ Find the uniformly most powerful test for testing $H_0:\theta \le \theta_0$ vs ...
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Test of confidence intervals?

In one of my assignments I have to "test" if the confidence intervals (CIs) for a set of parameters in a mixed effect model is accurate. I'm asked to simulate from fitted parameters and after that to ...
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87 views

Hypothesis testing with interval as null hypothesis

I have seen tons of situation where the null hypothesis reads along the lines of $\mu = \mu_0$, but what if I wish to test is $\mu \leq \mu_0$? Or more complicated null hypothesis? For a more ...
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786 views

Comparison of Variational Bayes and Expectation Maximization algorithms

I need to learn both the VB and EM methods for Bayesian Networks. Before going into detail of both algorithms, which I am a bit aware of, I need to EXACTLY understand the basic motivations behind them....
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3 votes
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Policy announcement as a treatment variable (causal inference)

I am using data from sub-reddits like [this][1] or [this][2], where users discuss their thoughts on the Federal government unemployment insurance and its fairness. Specifically, I wonder if it makes ...
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3 votes
1 answer
97 views

Methods for drawing population inferences from multiple sub-population datasets

What would be an appropriate model or method for making inferences about a broader population quantity from multiple quantities representing subsets of the population? Imagine, as an example, that I ...
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Ay (2017) vs Amari (2016) for information geometry

I am interested in learning information geometry in detail with a focus on applications, and am currently considering two main texts: Amari (2016)'s "Information Geometry and Its Applications&...
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ML Estimator parameter of bimodal, polynomial distribution

Let $X_1, \ldots X_n$ i.i.d with density of (using the indicator function $\mathbf 1$) $$f(x|\mu) = \frac 3 2 (x-\mu)^2 \cdot \mathbf 1_{[\mu-1, \mu+1]}(x) = \left\{ \begin{array}{ll} ...
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270 views

Geometric distribution and entropy

According to wikipedia, among all discrete probability distributions supported on $\{1, 2, 3, ... \}$ with given expected value $\mu$, the geometric distribution X with parameter $p = \frac{1}{ \mu} $ ...
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Finding the UMVUE of $e^{3\lambda}$ in Poi($\lambda$)

Let $ X = (X_1, ... , X_n)$ iid variables coming from Poisson distribution with mean $\lambda$. Find the UMVUE of $e^{3\lambda}$. I tried understanding the solution below (in the possible duplicate ...
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Does this "transpose block bootstrap" make any sense?

Say we have the following data Say we are concerned with a linear regression model that is $\boldsymbol{y} = \alpha + \boldsymbol{d} \beta + \boldsymbol{u}$ and that we are interested in the ...
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How can I infer the precision of my sample?

Let's say I have a survey about the quality of my product. But only 30% of my customers answered. If, for instance, the average rate for my product quality is 7, what can I infer from the real rate? I ...
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Bernoulli distribution and Strong Law of Large Numbers

I'm trying to find concrete examples of the SLLN theorem. Before, let's see the statement of this theorem precisely from this book, page 81: Definition: We say that $X_n$ converges almost surely to $...
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Finding the p-values for one sample t-test in different cases

Calculate and interpret the $p-$ value in the following situations: (a) one-sample $t$ - test for testing (i) $H_{0}: \mu=\mu_{0}$ vs $H_{1}: \mu \neq \mu_{0}$ (ii) $H_{0}: \mu \leq \mu_{0}$ vs $H_{1}:...
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Estimating standard deviation of underlying distribution

I need to estimate the standard deviation of an underlying normal distribution. I have the following statistics of one sample: mean, min and max, standard deviation, and sample size; however, I do not ...
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Using calculus of variations to deduce lower bound on Pitman efficiency (asymptotic relative efficiency) between $t$-test and sign test

Let $Y_1,...,Y_n$ be iid draws from a location family $\{f(\cdot - \theta) : \theta \in \mathbb{R}\}$. $f$ is a symmetric density w.r.t. the Lebesgue measure on $\mathbb{R}$ with finite variance. We ...
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Causal inference method for analyzing randomized control trial with covariates / pre intervention observations

I've got a seemingly easy situation, which turns out to be a little more complex than originally thought. Here's the Setup: We have a randomized controlled trial. Test and Control groups are the same ...
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Statistical inference for morphogenesis models

I work developing morphogenetic models of plant tissues. In short, a typical model of this kind would be a single starting cell that grows and divide, while exchanging molecules with other cell based ...
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Testing percentages when I may or may not have the original values that give the percentages

I've found tons written about testing proportions between two groups (Fisher's exact test, $\chi^2$-test, logistic regression, etc) but very little on what happens in a t-test-like setting where the ...
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191 views

Factorization of Proper Scoring Rules

Suppose that we have a joint probability distribution $P(X_1,X_2,...,X_n)$. Given a sample $x = (x_1,x_2,...,x_n)$, the proper scoring rule log score can be computed as follows: $$S(P,x) = \log P(x_1,...
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Possible to work backward from Convolution of Distributions?

So, having discovered distribution convolution, which is a method for deriving the density of a sum of individual probability distribution densities, $$S = X_{first\_distribution} + Y_{second\...
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Deductive reasoning and artificial intelligence

AI has proven to be extraordinary effective for solving certain types of intellectual problems that we thought before only our brains could solve. The number of applications is tremendous: engineering,...
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What Statistical principles are being violated by comparing specific Trainer Fatality Rates to Race Track Fatality rates?

A Hall of Fame Trainer of Thoroughbred Racehorses has been banned from a Race Track because 7 horses under his care have have been injured while training or racing in the past year. Critics cite a ...
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Determine statistical difference of slopes of quadratic relationship in a Poisson regression

I'm looking for a statistical or mathematical way to test the difference between two slopes. Others have asked related questions but my problem is quite particular. I'm running a Poisson regression ...
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Inference from small sample size with aggregated country data

I have a data set with country aggregate patient data for yearly total number of diseases, treatments provided by health services, and some other covariates for two countries over 10 years. The ...
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Who invented train/validation/Test method and when?

I can't seem to find here or in other places the earliest source for this method. it seems the holdout method was separately proposed by Highleyman in 1962, and cross validation was separately ...
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