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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1answer
174 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 ...
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1answer
84 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. ...
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32 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 ...
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1answer
46 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: $...
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1answer
16 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 ...
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1answer
95 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 ...
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1answer
161 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 ...
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1answer
118 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 ...
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1answer
40 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 \...
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60 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 ...
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How to prove equivalence between models, and show that one (Bayesian hierarchical) model is an extension of another?

Is there such a thing as equivalence between statistical models (in my particular case, Bayesian hierarchical models) ? If so, how to prove it ? Let me explain myself with an example. Consider a ...
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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 ...
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31 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 ...
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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? $Cov(X, Y | X + Y \geq Z)$
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1answer
57 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 ...
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1answer
40 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 ...
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1answer
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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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1answer
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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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1answer
52 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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32 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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1answer
281 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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1answer
95 views

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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68 views

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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732 views

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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1answer
248 views

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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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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91 views

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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1answer
191 views

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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1answer
397 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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1answer
83 views

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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121 views

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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2answers
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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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107 views

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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2answers
207 views

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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1answer
141 views

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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1answer
55 views

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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33 views

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 ...
2
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1answer
56 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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143 views

Suppose $X,Y,Z$ are random variables such that $Y,X$ are perfectly correlated. Does it hold that $P(Z|X,Y) = P(Z|Y)$?

Suppose $X,Y,Z$ are random variables such that $Y,X$ are perfectly correlated. I am wondering if it holds that $P(Z|X,Y) = P(Z|Y)$? It would seem that $X$ would then be a substitute of $Y$. Is there a ...
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172 views

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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50 views

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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68 views

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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1answer
423 views

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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1answer
125 views

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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300 views

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 ...