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

Filter by
Sorted by
Tagged with
13
votes
2answers
2k views

A problem on estimability of parameters

Let $Y_1,Y_2,Y_3$ and $Y_4$ be four random variables such that $E(Y_1)=\theta_1-\theta_3;\space\space E(Y_2)=\theta_1+\theta_2-\theta_3;\space\space E(Y_3)=\theta_1-\theta_3;\space\space E(Y_4)=\...
13
votes
1answer
402 views

Invalid inference when observations are not independent

I learned in elementary statistics that, with a general linear model, for inferences to be valid, observations must be independent. When clustering occurs, independence may no longer hold leading to ...
13
votes
2answers
4k views

Online resources for learning statistics, exercises (with solutions)?

I'm currently working as a teaching assistant at my university, in an introductory statistics course (for medical students). Offline, there are many books available with information to aid the ...
13
votes
2answers
628 views

How to define a Rejection Region when there's no UMP?

Consider the linear regression model $\mathbf{y}=\mathbf{X\beta}+\mathbf{u}$, $\mathbf{u}\sim N(\mathbf{0},\sigma^2\mathbf{I})$, $E(\mathbf{u}\mid\mathbf{X})=\mathbf{0}$. Let $H_0: \sigma_0^2=\...
13
votes
1answer
899 views

Do likelihood ratios and Bayesian model comparison provide superior & sufficient alternatives to null-hypothesis testing?

In response to a growing body of statisticians and researchers that criticize the utility of null-hypothesis testing (NHT) for science as a cumulative endeavour, the American Psychological Association ...
13
votes
1answer
326 views

Is frequentist conditional inference still being used in practice?

I've recently reviewed some old papers by Nancy Reid, Barndorff-Nielsen, Richard Cox and, yes, a little Ronald Fisher on the concept of "conditional inference" in the frequentist paradigm, which ...
12
votes
4answers
8k views

Can a trend stationary series be modeled with ARIMA?

I have a question / confusion about stationary series required for modeling with ARIMA(X). I am thinking of this more in terms of inference (effect of an intervention), but would like to know if ...
12
votes
3answers
962 views

Are independent variables necessarily “independent” and how does this relate to what's being predicted?

I'm fairly new to statistics. I'm not clear on the meaning of independent and dependent variables and the relationship to what's being predicted. In my text, as an example there is a data set ...
12
votes
4answers
2k views

Why I should use Bayesian inference with uninformative prior? [duplicate]

I am a Ph.D. student and currently I am studying Bayesian inference concerning vector autoregressive models. A lot of researchers when talking about uninformative prior, conclude that the results of ...
12
votes
2answers
1k views

In Bayesian inference, why are some terms dropped from the posterior predictive?

In Kevin Murphy's Conjugate Bayesian analysis of the Gaussian distribution, he writes that the posterior predictive distribution is $$ p(x \mid D) = \int p(x \mid \theta) p(\theta \mid D) d \theta $$ ...
12
votes
2answers
8k views

Formula for 95% confidence interval for $R^2$

I googled and searched on stats.stackexchange but I cannot find the formula to calculate a 95% confidence interval for an $R^2$ value for a linear regression. Can anyone provide it? Even better, let'...
12
votes
2answers
686 views

Behrens–Fisher problem

Is there a good published expository account, with mathematical details, of the various approaches that have been taken to the Behrens–Fisher problem?
12
votes
1answer
2k views

If a tennis match was a single large set, how many games would give the same accuracy?

Tennis has a peculiar three tier scoring system, and I wonder if this has any statistical benefit, from the point of view of a match as an experiment to determine the better player. For those ...
12
votes
3answers
238 views

Researcher 1 runs 1000 regressions, researcher 2 runs only 1, both get same results — should they make different inferences?

Imagine a researcher is exploring a dataset and runs 1000 different regressions and he finds one interesting relationship among them. Now imagine another researcher with the same data runs just 1 ...
12
votes
1answer
8k views

Bayesian network inference using pymc (Beginner's confusion)

I am currently taking the PGM course by Daphne Koller on Coursera. In that, we generally model a Bayesian Network as a cause and effect directed graph of the variables which are part of the observed ...
12
votes
1answer
489 views

Inference on fixed effects in a mixed effects model

I have correlated data and am using a logistic regression mixed effects model to estimate the individual level (conditional) effect for a predictor of interest. I know that for standard marginal ...
11
votes
2answers
896 views

Maximum likelihood parameters deviate from posterior distributions

I have a likelihood function $\mathcal{L}(d | \theta)$ for the probability of my data $d$ given some model parameters $\theta \in \mathbf{R}^N$, which I would like to estimate. Assuming flat priors on ...
11
votes
7answers
20k views

How would you explain statistical significance to people with no statistical background?

Background: I had to perform a data analysis for a client (some kind of lawyer) who was an absolute beginner in statistics. He asked me what the term "statistical significance" means and I really ...
11
votes
1answer
8k views

Maximum Likelihood estimator - confidence interval

How can I construct an asymptotic confidence interval for a real parameter, starting from the MLE for that parameter?
11
votes
2answers
178 views

Why aren't “error in X” models more widely used?

When we calculate the standard error of a regression coefficient, we do not account for the randomness in the design matrix $X$. In OLS for instance, we calculate $\text{var}(\hat{\beta})$ as $\text{...
11
votes
1answer
3k views

Estimating probability of success, given a reference population

Suppose you have the following situation: You observed over time 1000 bowling players, who each played a relatively small number of games (say 1 to 20). You noted the strike percentage for each of ...
11
votes
1answer
1k views

What does “variational” mean?

Does the use of "variational" always refer to optimization via variational inference? Examples: "Variational auto-encoder" "Variational Bayesian methods" "Variational renormalization group"
11
votes
1answer
3k views

Neg Binomial and the Jeffreys' Prior

I'm trying to obtain the Jeffreys' prior for a negative binomial distribution. I can't see where I go wrong, so if someone could help point that out that would be appreciated. Okay, so the situation ...
11
votes
1answer
806 views

In general, is doing inference more difficult than making prediction?

My question comes from the following fact. I have been reading posts, blogs, lectures as well as books on machine learning. My impression is that machine learning practitioners seem to be indifferent ...
11
votes
2answers
472 views

Are bayesian methods inherently sequential?

Ie, to do sequential analysis (you don't know ahead of time exactly how much data you will collect) with frequentist methods requires special care; you can't just collect data until the p-value gets ...
11
votes
1answer
302 views

Hypothesis Testing and the Scientific Method

Reading the answers to this thread, I started wondering about how Hypothesis Testing relates to the Scientific Method. While I have a good understanding of both, I am having a hard time drawing the ...
11
votes
1answer
357 views

What is the difference between VAE and Stochastic Backpropagation for Deep Generative Models?

What is the difference between Auto-encoding Variational Bayes and Stochastic Backpropagation for Deep Generative Models? Does inference in both methods lead to the same results? I'm not aware of any ...
11
votes
1answer
201 views

Should degrees of freedom corrections be used for inference on GLM parameters?

This question is inspired by Martijn's answer here. Suppose we fit a GLM for a one parameter family like a binomial or Poisson model and that it is a full likelihood procedure (as opposed to say, ...
11
votes
2answers
954 views

Why is using cross-sectional data to infer / predict longitudinal changes a Bad Thing?

I'm looking for a paper which I hope exists, but don't know if it does. It could be a set of case studies, and / or an argument from probability theory, about why using cross-sectional data to infer / ...
10
votes
2answers
2k views

Why is an estimator considered a random variable?

My understanding of what an estimator and an estimate is: Estimator: A rule to calculate an estimate Estimate: The value calculated from a set of data based on the estimator Between these two terms, ...
10
votes
3answers
822 views

The concept of 'proven statistically'

When the news talk about things been 'proven statistically' are they using a well-defined concept of statistics correctly, using it wrong, or just using an oxymoron? I imagine that a 'statistical ...
10
votes
4answers
1k views

Implications of current debate on statistical significance

In the past few years, various scholars have raised a detrimental problem of scientific hypothesis testing, dubbed "researcher degree of freedom," meaning that scientists have numerous choices to make ...
10
votes
1answer
17k views

How to interpret Cochran-Mantel-Haenszel test?

I'm testing the independence of two variables, A and B, stratified by C. A and B are binary variables and C is categorical (5 values). Running Fisher's exact test for A and B (all strata combined), I ...
10
votes
1answer
136 views

Is there any real statistics behind “the Pythagorean theorem of baseball”?

I'm reading a book about sabermetrics, specifically Mathletics by Wayne Winston, and in the first chapter he introduces a quantity that can be used to predict the win rate of teams: $$\frac{\text{...
10
votes
1answer
331 views

Find UMVUE of $\frac{1}{\theta}$ where $f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$

Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$$ where $\theta >0$. Give the UMVUE of $\frac{1}{\theta}$ ...
10
votes
2answers
805 views

UMVUE of $\frac{\theta}{1+\theta}$ while sampling from $\text{Beta}(\theta,1)$ population

Let $(X_1,X_2,\ldots,X_n)$ be a random sample from the density $$f_{\theta}(x)=\theta x^{\theta-1}\mathbf1_{0<x<1}\quad,\,\theta>0$$ I am trying to find the UMVUE of $\frac{\theta}{1+\...
10
votes
2answers
233 views

Reference request: Classical statistics for working data scientists

I'm a working data scientist with solid experience in regression, other machine learning type algorithms, and programming (both for data analysis and general software development). Most of my working ...
10
votes
1answer
802 views

On the existence of UMVUE and choice of estimator of $\theta$ in $\mathcal N(\theta,\theta^2)$ population

Let $(X_1,X_2,\cdots,X_n)$ be a random sample drawn from $\mathcal N(\theta,\theta^2)$ population where $\theta\in\mathbb R$. I am looking for the UMVUE of $\theta$. Joint density of $(X_1,X_2,\...
10
votes
3answers
169 views

Measuring some of the patients more than once

I'm conducting a clinical study where I determine an anthropometrical measure of the patients. I know how to handle the situation where I have one measure per patient: I make a model, where I have a ...
9
votes
13answers
5k views

If 'B is more likely given A', then 'A is more likely given B'

I am trying to get a clearer intuition behind: "If $A$ makes $B$ more likely then $B$ makes $A$ more likely" i.e Let $n(S)$ denote the size of the space in which $A$ and $B$ are, then Claim: $...
9
votes
2answers
1k views

Can we reject a null hypothesis with confidence intervals produced via sampling rather than the null hypothesis?

I have been taught that we can produce a parameter estimate in the form of a confidence interval after sampling from a population. For example, 95% confidence intervals, with no violated assumptions, ...
9
votes
1answer
835 views

Checking if a coin is fair

I was asked the following question by a friend. I could not help her out but I hope someone can explain it to me. I could not find any similar example.Thanks for any help and explanation. Q: Results ...
9
votes
3answers
2k views

gaussian process regression for large datasets

I've been learning about Gaussian process regression from online videos and lecture notes, my understanding of it is that if we have a dataset with $n$ points then we assume the data is sampled from ...
9
votes
1answer
239 views

Sufficiency or Insufficiency

Consider a random sample $\{X_1,X_2,X_3\}$ where $X_i$ are i.i.d. $Bernoulli(p)$ random variables where $p\in(0,1)$. Check if $T(X)=X_1+2X_2+X_3$ is a sufficient statistic for $p$. Firstly, how ...
9
votes
1answer
373 views

Example of how Bayesian Statistics can estimate parameters that are very challenging to estimate through frequentist methods

Bayesian statisticians maintain that "Bayesian Statistics can estimate parameters that are very challenging to estimate through frequentist methods". Does the following quote taken from this SAS ...
9
votes
3answers
3k views

KL Loss with a unit Gaussian

I've been implementing a VAE and I've noticed two different implementations online of the simplified univariate gaussian KL divergence. The original divergence as per here is $$ KL_{loss}=\log(\frac{\...
9
votes
1answer
1k views

Proof of Pitman–Koopman–Darmois theorem

Where can I find a proof of Pitman–Koopman–Darmois theorem? I have googled for quite some time. Strangely, many notes mention this theorem yet none of them present the proof.
9
votes
1answer
984 views

Bayesian online changepoint detection (marginal predictive distribution)

I am reading the Bayesian online changepoint detection paper by Adams and MacKay (link). The authors start by writing the marginal predictive distribution: $$ P(x_{t+1} | \textbf{x}_{1:t}) = \sum_{...
9
votes
2answers
275 views

Statistical inference under model misspecification

I have a general methodological question. It might have been answered before, but I am not able to locate the relevant thread. I will appreciate pointers to possible duplicates. (Here is an excellent ...
9
votes
1answer
307 views

Observed Fisher information under a transformation

From "In All Likelihood: Statistical Modeling and Inference Using Likelihood" by Y. Pawitan, the likelihood of a re-parameterization $\theta\mapsto g(\theta)=\psi$ is defined as $$ L^*(\psi)=\max_{\{\...