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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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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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27 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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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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91 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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50 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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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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Law of total covariance with variables conditioned on different variables

Suppose I have $X_1|\theta_1$ and $X_2|\theta_2$ such that $X_i \sim D(\theta_i)$ and $D$ is some known distribution with parameter $\theta$. $X_1|\theta_1$ and $X_2|\theta_2$ are assumed independent. ...
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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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Compare health spending per capita of 5 countries conditional on their age structure, health index and GDP per capita

I am interested to answer the following question: How does the health spending per capita in 2010 of 5 countries different if they have the same age structure in their population, health index and GDP ...
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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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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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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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100 views

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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468 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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393 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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107 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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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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350 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 ...
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Intuition for Conditional Expectation of $\sigma$-algebra

Let $(\Omega,\mathscr{F},\mu)$ be a probability space, given a random variable $\xi:\Omega \to \mathbb{R}$ and a $\sigma$-algebra $\mathscr{G}\subseteq \mathscr{F}$ we can construct a new random ...
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Conditioning within definition explanation

I have a doubt on the meaning of a conditioning within a definition. In a book I've found the following definition of upper tolerance limit: $P(P(X<\bar X+kS|\bar X, S)>p)=1-\alpha$ where $X$ ...
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Calculating percentile conditional on continuous variable

I have a collection of datasets with (x,y) - pairs where the value of y is conditional on x. Neither the distribution of x nor of y is known, n is ranging from 1000 to 10000. Here is the scatterplot ...
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Computing conditioned probability of $X$ by $U=\min(X,Y)$

Let $X$ and $Y$ be independent random variables with $P(X\leq x)=F_x(x)$ and $P(Y\leq y)=F_y(y)$. Let $U=\min(X,Y)$. I know that $F_u(u)=1-(1-F_x(u)(1-F_y(u))).$ By definition: $P(X \leq x |U=u)= \...
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162 views

P(X<Y|Z=t) where Z=min(X,Y)

Lets X and Y be uniform random variable where $x \in [0,a]$ and $y \in [0,b]$ where a < b. We design $Z=\min(X,Y)$. I know that the CDF of Z is $P(Z<z)=1-\frac{(a-z)(b-z)}{ab}$ And by ...
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Methodology to generate conditionally independent data correct?

I am trying to generate samples from continuous distributions that are conditionally independent. More specifically, I would like to generate samples from the following joint distribution $f(x,y,z)$ ...
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73 views

Implications of independence of several random variables

Consider 4 real-valued random variables $X,Y,Z,V$ defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assume that $X$ is independent of $Y,Z,V$, i.e. the probability ...
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383 views

Maximum Likelihood Formulation for Linear Regression

I have seen the following for maximum likelihood estimation (MLE) for linear regression in multiple sources, e.g. here: $$ \mathcal{D} \equiv \{(x_1, y_1), ..., (x_n, y_n)\} $$ I do not understand ...
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Estimate point in metric variable where probability of success of a conditionally Bernoulli distributed variable changes (independent observations)

There is a metric variable X and a conditionally Bernoully distributed variable Y, where the probability of success of Y changes at a threshold x of variable X. The obervations are independent. I want ...
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3k views

iterated expectation conditional on two variables

How to prove that $E[Y]=E[E[E[Y|X_1, X_2]]]$ ? PS. I don't see how $E[E(Y|X_{1},X_{2})|X_{1}]=Y[Y|X_{1}]$ and $E[Y]=E[E(Y|X_{1})]$ can be used here. But it feels close. Please help, I'm stuck PPS. ...
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How do you interpret the condition number of a correlation matrix

I have two correlation matrices, one with a condition number of 9 and the other with a condition number of 70. From what i have read, it will appear that the first matrix is better conditioned than ...
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Kullback-Liebler's divergence on a conditioned function

Let $q$ be a conditioned pdf over $\mathbf{X}=X_1,\dots,X_n$ binary r.v.s in the form $$q(\mathbf{X})=\begin{cases}q_{0}(\mathbf{X}_{\setminus i}) \text{ if } X_{i}=0\\q_{1}(\mathbf{X}_{\setminus i}) \...