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)

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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 ...
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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 decides to ...
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40 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 ...
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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 ...
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31 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(...
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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 ...
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30 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 ...
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48 views

conditioning for deep neural networks

What is the best way to do conditioning when working with deep neural nets? For example, say we we want to condition a VAE on the class i.e. CVAE. There exists different ways of adding the class to ...
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What is E[X|Y>c] where Y=X+Z, and X, Z are normally distributed rvs? [duplicate]

Suppose X, Z are normally distributed random variables and independent ($X$ follows $N(\mu,\sigma^{2})$ and $Z$ follows $N(0,\sigma_{z}^{2})$), and that Y=X+Z. What is E[X|Y>c], where c is just a ...
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Find conditional pdf given joint

Let the joint pdf of $X$ and $Y$ be $f(x,y) = 12e^{-4x-3y}, x>0, y>0$. What is the marginal cdf of $X$? of $Y$? Am I just supposed to integrate f(x,y) with respect to $x$ or $y$ to get the ...
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Meaning of “for each value $X = x$, the random variable Y can be represented in the form $Y = \beta_0 + \beta_1 x + \epsilon$” in linear regression

In what follows assume $Y: \ \Omega \to \mathbb R$ and $X: \ \Omega \to E$ The following quoute is from page 700 in DeGroot and Schervish - Probability and statistics, introducing a simple linear ...
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When can we use fixed design regression results for the random design setting? [closed]

Suppose I have an independent vector $X$ and a dependent scalar random variable $Y$ and I wish to construct a regression model to predict $Y$ using $X$ given data $\{(x_i,y_i)\}_{i=1}^{n}$. For ...
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What does “Normal distribution conditioned on $x \mod 1$” mean?

I understand what $Pr[X = x | Y = y]$ means, however, on this paper (Lemma 10 in Appendix A), we have the following: Let $D_r$ denote the continuous Gaussian distribution of parameter $r$,i.e., the ...
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Correct terminology to refer to a collection of conditional probability distributions

I would like your help to find the correct terminology to refer to a collection of conditional probability distributions. Consider two random variables $Y$ and $X$, with supports $\mathcal{Y}$ and $\...
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Consistent estimator for conditional expectation

Take sequence of random vectors $(Y_i, X_i)_{i=1}^N$ i.i.d. $X_i$ has finite support. Let $x$ be a point in the support of $X_i$. Consider $E(Y_i|X_i=x)$. Suppose it exists and is finite. Is it ...
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Implications of conditional mean independence

I have three random variables $Y,X,W$ with supports $\mathcal{X}, \mathcal{Y},\mathcal{W}$, respectively. I assume $E(Y|X,W)=0$ almost surely. Take two functions $z: \mathcal{X}\rightarrow \mathbb{...
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Conditioning and linear MSE

Let $\sigma_{X|Y}^2$ denote the linear mean squared error in estimating $X$ from $Y$. Then is it always true that additional conditioning cannot increase the LLSE? In other words, is this true? $$ \...
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How time series structure can affect the independence of residuals condition for MLR?

I am going through all four conditions for Multiple Linear Regression and stick with this question: what happens with the independence if we have time series data structure?
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Computing Variances by Conditioning

I have trouble with the first part of this problem (Please take a look at the image below). This is an example problem from my old textbook years ago and I have had trouble understanding: How Y is ...
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281 views

Law of Iterated Expectations Example

Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
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Is the conditional distribution of Y given X the most we can know about how X “affects” Y?

In his book "Introductory Econometrics", Jeffrey Woolridge states "The most we can know about how X affects Y is contained in the conditional distribution of Y given X". Is this statement true? Would ...
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RiskMetrics VAR calculations and conditional distribution of sum of log returns

According to Tsay's book in Chapter 7, for the Risk Metrics model: A nice property of such a special random-walk IGARCH model is that the conditional distribution of a multiperiod return is ...
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How can I estimate the confidence interval of correlations possibly dependend with time?

I have a multivariate problem (with solar data from different meteorological stations) that I am working on my engineering master thesis. I would like to estimate the correlations of different ...
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Does conditioning on a random variable yield a random variable?

Let $X$ and $Y$ be random variables and $y\in Im(Y)$ a possible value of $Y$. Is $X|Y=y$ a random variable in the mathematical sense? Or is that just abusing notation?
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Defining “variance” of a partially defined random variable

Elsewhere within CrossValidated the following survey sampling problem was mention. To each member $i$ of a population $\{1,\ldots,N\}$ there is assigned some value $c_i$ whose average $\mu=(c_1+\cdots+...
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Bayesian Statistical Conclusions: We Implicitly Condition On the Known Values of Any Covariates, $x$?

My Bayesian data analysis textbook says the following: Bayesian statistical conclusions about a parameter $\theta$, or unobserved data $\tilde{y}$, are made in terms of probability statements. ...
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Is there an informative term for calling the random elements conditional on which a PDF of a random element is defined?

Let $X_{1}, \dots, X_{n}$ be i.i.d. random elements; suppose the conditional PDF $f_{X_{1} \mid X_{2} , \dots, X_{n}}$ exists. Then I wonder if there is already in literature an informative name for $...
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Is the parameter vector of an indentifiable distribution of a transformed random vector always a subvector…?

I would like, after further considerations about this problem, to reformulate this question of mine again. I kept a record of the past words and remarks as the appendix below. I think the question ...
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Question about sufficiency

I learned in my (classical) statistics class that (if we have densities) $T(X)$ is sufficient iff $$f(x)= g(T(x))h(x)$$ I am reading "the Bayesian Choice" and there the factorization-lemma is quoted ...
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Modelling resampling as conditioning on sum for independent discrete variables

I am trying to model a discrete data generating process where I first draw $Y = (y_1, ..., y_N), y_n \sim F(\theta_n)$ independently from some family of discrete distributions $F$ (e.g. negative ...
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55 views

Using posterior in an expectation

I am studying (myself, not in class) the book of Rogers & Girolami, A First Course in Machine Learning. In working through a logistic classifier, I found the equation $$ p(t_{new} = 1| \mathbf{...
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Distribution of conditional expectation?

Let $X,Y$ be random variables with pdf $f_{X,Y}$. I would like to find the distribution of the random variable $\mathbb{E}(Y\mid X)$, conditional expectation of $Y$ given $X$. If a specific form of $\...
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Conditional matrix normal distribution

Suppose $\epsilon$ is a $n\times p$ with independent rows $\epsilon_i\sim N(0, \Sigma)$. $Y$ is a matrix of size $n\times p_1$ and $X$ a matrix of size $n\times p_2$ constructed as $$ Y=XA+\epsilon B\\...
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Expressing the likelihood of the multivariate normal

I have two multivariate normal samples: $\boldsymbol{X^{(1)}=\left( \begin{matrix} \boldsymbol{Y^{(1)}}\\\boldsymbol{Z^{(1)}} \end{matrix}\right)}$ and $\boldsymbol{X^{(2)}=\left( \begin{matrix} \...
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Are normal prior and posterior random variables dependent?

Consider the usual normal Bayesian model. The prior $X=\mu_0+\sigma_0\epsilon_0$, where $\epsilon_0$ follows a standard normal distribution. The data $Y=X+\sigma_1\epsilon_1$, where $\epsilon_1$ ...
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notation: precedence of conditional when multiple variables

In expressions such as $ P(X,Y|Z) $ and $I(X; Y|Z)$ (mutual information) there are two interpretations for a student, and the correct one does not seem to be mentioned in textbooks. "joint ...
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How does conditional expectation relate to sufficiency?

In what follows, I will disregard all "measure-theoretic niceties about conditioning on measure-zero sets", as my professor calls it. I just want to know if the following general idea, or even just ...
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428 views

W is normal conditional on a normal variable Z. Does it necessarily follow that W is normal unconditionally?

$Z$ is normal with mean $\mu_z$ and standard deviation $\sigma^2_z$. Contitional on $Z = z$, $W$ is normal with mean $z$ and variance $\sigma^2_w$. Does it follow from these hypotheses that W is a ...
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631 views

UMVUE of location parameter (shifted exponential)

Let $X_1,...,X_n$ be a sample from a distribution with pdf, $f_X(x) = e^{-x + \theta}, x \geq \theta$. Let $x_0 \geq \theta$ be given. I'm trying to find the UMVUE of $f_X(x_0) = e^{-x_0 + \theta}$. I ...
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Simulate Gaussian variables conditional on their sum of squares

Consider a $d$-dimensional Gaussian random vector $\mathbf{Z}=[Z_i]_i$ with mean $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$. What would be the more efficient method(s) to simulate $...
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Purpose of scatterplots

This is a beginner question, hopefully this is allowed here. I understand that scatterplots can be useful in showing the relationship between two variables, so I generated several plots of the ...
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622 views

Conditioning on continuous random variables

Conditioning is a tool I have used a lot in the discrete setting: usually this takes the form $$P(X=k) = \sum_{i=0}^\infty P(X=k | Y=i) P(Y=i).$$ I'm a little confused about the analogous situation ...
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Find the distribution of $Y-X|X=x$

The conditional distribution of $Y|X=x$ is Normal$(x,1)$. The marginal distribution of $X$ is Normal$(0,1)$ Find the distribution of $Y-X|X=x$. I notice that since the conditional distribution ...
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Sample random variables conditional on their sum

Let $(X_1, \dots, X_n)$ be an iid sample of random variables with a known continuous distribution. I would like to simulate such a sample, conditional on the value of its sum, that is: $$ X_1, \dots, ...
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543 views

What is the physical significance of cumulative correlation coefficient?

Say, I have 2 parameters, and based on my dataset, I have iteratively calculated the correlation coefficients between them by taking the correlation of the first i terms, where i ranges from 1 to the ...
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197 views

Investigate correlation conditional on a threshold

I have 3 variables in my data set. (i) My gut feel says variable1 and variable2 are correlated, only when variable3 >= threshold3. What is the technique I can use to see if this holds true, to ...
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123 views

Square of conditional variables

I thought that the following would hold but my classmates doubt me unfortunately: $Z = X|Y \rightarrow Z^2 = X^2|Y$ The original problem was an expectation of this and I tried to go back to the ...
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544 views

Non-parametric conditional variance estimation

Let $(x_i,y_i)_{1\leq i\leq n}$ some dataset. I want to estimate the conditional expectation $E[Y\mid X=x]$ and the conditional variance $V[Y\mid X=x]$. I used Nadaraya-Watson's estimator to estimate ...
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Conditional Distribution of uniform random variable given Order statistic

I have the following question at hand: Suppose $U,V$ are iid random variables following Unif$(0,1)$. what is the conditional distribution of $U$ given $Z:=\max(U,V)$ ? I tried writing $Z=\Bbb{I}\...
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Computation of Conditional Expectation on $\sigma$-algebras

I have not really seen any probability books calculate conditional expectation, except for $\sigma$-algebras generated by a discrete random variable. They simply state the existence of conditional ...