Questions tagged [conditioning]
Conditioning is a probabilistic operation that consists in examining the probabilistic properties of a random variable (or of an event) given the realised value of another random variable (or of an event)
132 questions
0
votes
0
answers
9
views
Why can't one define a complete statistic from a minimal sufficient one by accounting for all first-order ancillaries?
According to this: "If a minimal sufficient statistic $T$ is not complete, then there is a non-trivial first order ancillary statistic $V(T)$, and there does not exist any complete statistic.&...
- 642
0
votes
0
answers
49
views
Conditional expectation of order statistics when observing ratio/difference
Suppose two independent ordered draws from some continuous and bounded distribution $F_{[0,\bar{x}]}$ with $f>0$ everywhere, represented by the order statistics $$X_{(1:2)}=\min(X_1,X_2)\text{, and ...
- 65
1
vote
0
answers
13
views
Can including too many continuous variables in Little's MCAR result in Type 1 error?
When examining missing data at the individual item level (before computing scale scores), my missing data ranges from 0% - 1.7% for each individual item.
Is there a threshold at the individual item ...
- 11
0
votes
0
answers
35
views
When conditioning a Gaussian process, why is the conditional mean always trending upwards?
I have a Gaussian vector with mean $\mu$ and covariance matrix $\Sigma$, both estimated. The vector $\mu$ represents a process and for each entry starting with the second, I want to find the ...
2
votes
1
answer
53
views
How to choose what to integrate out or what to condition on for marginal distributions?
I am trying to work out the Bayesian posteriors of $\theta$, $\tau$ and the $\varepsilon$ in the following model:
$$y(t) = \phi(t,\tau)\theta+v(t),$$
where $\{v(t)\}$ is an iid sequence of random ...
- 21
4
votes
1
answer
161
views
When and how of stratification?
I understand stratification at a novice level. For example, if we want to condition on gender in post experimentation inference, we might stratify or block on gender. As I understand it, we take each ...
- 3,520
2
votes
1
answer
41
views
conditional probabilities in chess
Let's say I am analyzing certain chess positions of a chess player (Magnus Carlsen, Lichess games when playing as white).
We know that the 'unconditional probability' (empirically speaking) of this ...
- 304
1
vote
2
answers
149
views
Distribution of a random variable conditional on its being a maximum or not
Consider the random variables $\epsilon_1,\dots, \epsilon_D$ defined on the probability space $(\Omega, \mathcal{F}, P)$. Assume they are continuous. Let
$$
Y=\sum_{d=1}^D d\times \mathbb{1}\{\...
- 935
2
votes
1
answer
108
views
Chain rule conditional entropy
A textbook I am reading states that$$H(X,Y)=H(X)+H(Y|X)$$where $H(X,Y)$ is the joint entropy of random variables $X,Y$, $H(X)$ the entropy of $X$, and $H(Y|X)$ is conditional entropy. It then states ...
- 187
0
votes
0
answers
59
views
Expectation of the product of two random variables
I recently tried to derive a formula that I saw in a paper. The scenario was a follows:
Let $X\in\lbrace 0,1\rbrace $ a.s. be a binary random variable and $Y$ be a continuous random variable. Let $a,b\...
- 61
0
votes
0
answers
28
views
How to perform conditional poisson regression for a 1:5 matched cohort study?
I have a matched cohort population using data that spans the years 2016-2021. My exposure of interest is continuous dialysis treatment, and my outcome is e.coli infections (count outcome). Each time ...
- 1
7
votes
2
answers
334
views
When to use fixed effects or multi level models in regression?
Suppose you run an experiment where the treatment is Gatorade and the outcome is one-mile runtime. You’ve stratified on variables such as sex, height and weight so they’re well randomized and have no ...
- 3,520
3
votes
1
answer
41
views
Converse of pairwise Markov property
Random vector $X$ follows a pairwise Markov property on graph $G=(V,E)$ if for any $(i,j) \notin E$, $X_i$ and $X_j$ are conditionally independent given $X_{V \setminus \{i,j\}}$.
My question is, why ...
- 656
1
vote
0
answers
44
views
Constant conditional variance
Let $X$ and $U$ be two independent random variables, and let $Z= X+U$. Under which conditions is $Var(X|Z)$ constant? I know this holds for instance when $X$ and $Z$ are jointly normal. I'm ...
- 305
2
votes
1
answer
643
views
How does the QR-code monster model for SD controlNet work?
Background
Recent trends show cool QR-codes made by the QR-code monster model used in Stable-Diffusion (SD) with ControlNet.
Those are also used to create images containing hidden text/hidden images (...
- 121
1
vote
1
answer
50
views
How to obtain marginal density [x], given [y|x] and [y]
I came across a problem knowing density of Y, conditional density of Y given X, how would I obtain density of X? Or would this even unique?
- 13
0
votes
1
answer
61
views
Maximum change in entropy when conditioning on an event
Let $P_{XYZ}$ be the joint distribution of discrete RVs $X,Y,Z$ where $Z$ is binary-valued. Let $Q_{XY}=P_{XY|Z=0}$, i.e. the distribution of $XY$ conditioned on $Z=0$. Are there lower/upper bounds on ...
- 71
2
votes
0
answers
28
views
Wishart conditionned by the determinant
I am interested in the density of the Wishart distribution under the constraint that the determinant of the outcome is 1.
It suffice do divide the density of the Wishart by the marginal density of the ...
- 151
1
vote
1
answer
58
views
In probability, if we have an equality, can we condition on an arbitrary event and guarantee that the equality still holds?
I have a question regarding equalities and conditioning probabilities. I was reading through this proof for conditional independence, which can be found here. In the proof, they first state that $p(A|...
- 145
1
vote
2
answers
173
views
MANOVA (SPSS) test - determine control variables in moderation analysis
I am examining the moderating effect of IV1 in the relationship between IV2 and DV using hierarchical regression analysis. The study has 2 demographic variables: gender and age. To determine if it is ...
- 11
1
vote
0
answers
85
views
Where does the random come in for conditional expectations $\mathbb{E}[X | \mathcal{F}]$?
For continuous random variables $X, Y$ the conditional expectation $\mathbb{E}[X | Y]$ is itself a random variable. I understood this in the sense that for a realisation of $Y$ we can say
$$
\mathbb{E}...
- 212
1
vote
1
answer
30
views
how to view conditioning on a random variable rather than a particular value of that variable
The question is about $P(Y|X)$ versus $P(Y|X=x)$. Is there an alternate way to write $P(Y|X)$
that makes its meaning more clear?
I believe these are correct equations for conditional entropy:
$$
H(Y|...
- 283
1
vote
1
answer
348
views
Condition on two random variables
I'm trying to set up the proper assumptions for a proof I'm working on:
Given that $P(A|e) = P(A)$ and $P(A|c,e) = P(A|e)$, can we prove that $P(A|c)=P(A)$?
I understand that A is independent of e and ...
- 13
2
votes
1
answer
57
views
Conditional Tolerance Interval
With the generous help of people here, I've recently learned about the notion of a Tolerance Interval (Confidence in a range estimate, and +- 2sigma rule of thumb).
I am now working on an application ...
- 405
2
votes
1
answer
479
views
Probability Conditioned on Inequality
Assume that $A \sim \mathcal{N}(0, 1)$, $B \sim \mathcal{N}(0, 1)$. I am trying to calculate $P(A \,|\, A < B)$.
For the sake of this problem, we can assume that $A \perp B$, but (for obvious ...
- 145
0
votes
1
answer
228
views
Is it possible to calculate a bivariate normal density using only the univariate normal density function? [duplicate]
Suppose $(X,Y)$ are two jointly normally distributed random variables. Suppose further that we want to calculate the density of $(X=x,Y=y)$. Is it possible to calculate this density if we do not have ...
- 268
0
votes
1
answer
138
views
If P(A) + P(B) = 1, does P(A|C) + P(B|C) = 1?
Just to explain where this is coming from. I was working on question 5(b) from stat 110 on conditional probabilities. I'll put a picture of the question and its solution below
I worked on question 5(...
2
votes
0
answers
103
views
What’s a textbook covering similar content to “Introduction to Probability Models” by Sheldon Ross?
I’m taking a class with a instructor using said textbook, and I find the explanations in it lacking.
It’d be great if anyone can offer an alternative book covering similar content (i.e. conditioning ...
2
votes
1
answer
97
views
A regressor performs better under a certain regime -- how to condition that regressor to make regression better?
I ran a regression $Y \sim X_1 + X_2 + .. X_n$. I find out what one regressor , $X_1$'s performance (or correlation with $Y$) depends on another variable $t$ (not in the regression). So basically if I ...
- 220
7
votes
1
answer
303
views
Variance of Bernoulli when success probability varies
Say the success probability $X$ is a random variable with mean $\mu$ and Variance $\sigma^2$ which takes values in $[0,1]$. How can I compute the variance of a random Variable $Y$ which is 1 with ...
- 87
3
votes
1
answer
105
views
If $e_0$ are the OLS residuals, what is random in $\hat{\beta}_{OLS}|f(e_0) < \hat{\beta} < f^*(e_0)$?
This is a follow up question to the question I've posted here.
Suppose $Y \sim N(X\beta, \sigma^2I)$, where $y \in \mathbb{R}^n$. Let $X \in \mathbb{R}^{n \times p}$ denote a full rank design matrix. ...
- 2,683
1
vote
0
answers
231
views
Test for conditional dependence
I am trying to find out whether two variables are conditionally dependent on each other. I would like to have a test for B implies A (B->A) or not and one for A implies B (A->B) or not. It is ...
- 11
2
votes
1
answer
286
views
Find conditional PMF of a multinomial distribution
I am trying to find the conditional PMF of a multinomial analytically and though I know my result is wrong I can't seem to pinpoint where my argument is wrong. Seeking help to find my mistake.
Given:
$...
- 2,693
1
vote
1
answer
18
views
Distribution of R|R+T for R,T binomials with the same probability
$$R \sim \mathrm{Bin}(N_R, p)$$
$$T \sim \mathrm{Bin}(N_T, p)$$
What is the distribution of $R$ given particular value of $R + T$?
My guess would be that $R | R + T \sim \mathrm{Bin}(R + T, {N_R \over ...
- 6,187
1
vote
1
answer
1k
views
Gaussian Processes as weighted averages?
I've been wondering if a "weighted average" is a valid means to consider the Gaussian Process, specifically in the context of GP Regression. The kernel (I'll be referring to the common ...
- 3,520
3
votes
1
answer
2k
views
Expected value of max of two discrete random variables
I'm reading this paper An Efficient PTAS for Stochastic Load Balancing with Poisson Jobs. Which is solving a makespan minimizing job-shop problem for Poisson job sizes. Basically, schedule the minimum ...
- 4,379
0
votes
1
answer
3k
views
Markov transition matrix row sums to 1
I am trying to learn a little bit of Markov Chains through Dobrow's "Introduction to Stochastic Processes with R", but i am struggling with the following:
The entries of every Markov ...
- 175
4
votes
1
answer
101
views
conditional expection gaussian vector
I have got an question on computing conditional expection
I was working on the following conditional expectation problem find
$$\mathbb{E}[X-Y|2X+Y]$$
where $\begin{bmatrix}X \\ Y \end{bmatrix} \sim \...
- 43
1
vote
0
answers
120
views
How conditioning happens?
F. Elwert and C. Winship in the paper "Endogenous Selection Bias: The Problem of Conditioning on a Collider Variable" (content available here) discuss conditioning on different types of ...
1
vote
0
answers
23
views
Observational study comparing 2 products with different (but overlapping) feature coverage
I have two software products $A$ and $B$ which form treatment $X$.
$B$ is a new version of $A$ and in development. So $B$ does not have all the features that $A$ has. However, $B$ is designed to make ...
- 11
0
votes
0
answers
82
views
Graphical representation of unconditional expected value
First, let tell you that I've being struggling with the concept of unconditional expectation for linear regression. For conditional expectation is easier:
We know that the conditional expectation ...
1
vote
0
answers
50
views
Covariance of X and Y conditional on X+Y>Z? [closed]
Suppose that $X$, $Y$, and $Z$ are three independent random variables. Is there a way to compute the following conditional covariance?
$Cov(X, Y | X + Y \geq Z)$
- 21
1
vote
1
answer
193
views
How to notate joint conditional probability
Given $X = \{ x_1, x_2, \dots, \}$ and $Y = \{ y_1, y_2, \dots \}$ let $P(X,Y)$ be their joint probability. Conditioning $P(X, Y)$ on $y \in Y$ corresponds to looking at the distribution of the ...
- 667
4
votes
1
answer
149
views
Definition of covariate-specific effect: why after, not before intervention?
Pearl et al. "Causal Inference in Statistics: A Primer" (2016) p. 70 contains the following text regarding conditional interventions and covariate-specific effects:
[S]uppose a doctor ...
- 69.5k
3
votes
1
answer
914
views
Gibbs sampling for Multivariate: how to update?
In this page of Murphy's 'Machine Learning: a Probabilistic Perspective' it's explained how to do Gibbs sampling on a Gaussian Mixture Model.
Reading this, I was trying to understand when to update ...
- 587
4
votes
1
answer
64
views
Typo in Deepleariningbook.org or am I misunderstanding Bayesian stats?
This is on page 133 of the book: https://www.deeplearningbook.org/contents/ml.html#pf10
In the above, it says that
the data set is directly observed and so is not random
If that data we observe ...
- 3,263
1
vote
1
answer
164
views
generating process of acceptance-rejection algorithm
The acceptance-rejection algorithm is described as follows:
suppose you have RVs $X$ and $Y$ with densities $f_X$ and $f_Y$, respectively, and there exists a constant $c$ such that $\frac{f_X(t)}{f_Y(...
- 1,325
7
votes
2
answers
173
views
Is there a difference in interpretation between $Y|X = m(X) + \epsilon$ vs. $Y = m(X) + \epsilon$?
I understand that $E(Y|X)$ and $E(Y)$ are different, but difference sources, when $Y$ is a function of other random variables such as $X$, use $Y|X$ and $Y$ to describe this relationship. I'm not sure ...
- 1,209
1
vote
1
answer
52
views
Why condition on either the r.v. $X$ or $Y$ and integrate over a product of pdfs rather a single pdf to find this probability density?
Let $X$ have the probability density $f_{X}(x)=\lambda e^{-\lambda x},
\;\; x>0$ and let $Y$ have the probability density $f_{Y}(y)=\lambda
e^{-\lambda x},\;\; y>0.$ Find the probability ...
user271077
20
votes
3
answers
4k
views
What does the assumption of the Fisher test that "The row and column totals should be fixed" mean?
As this source states, one of the assumptions to perform Fisher's exact test of independence is that the row and column totals should be fixed. However, I find the explanation coming with it pretty ...
- 405