Questions tagged [conditional-probability]
The probability that an event A will occur, when another event B is known to occur or to have occurred. It is commonly denoted by P(A|B).
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A question about "do" operator under unobservable confounder
The original question is here. Suppose we have a DAG in the figure.
If there is no confounder $U$, as chang_trenton point out since $S$ and $W$ happen before $X$, we have
$$
P(Y \mid do(X), S) = ...
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Conditional Distribution of Multivariate Gaussian given Linear Inequalities
Consider a multivariate Gaussian $Y\sim\mathcal{N}(\mu,\Sigma)$ of dimension $n$. For fixed $c\in\mathbb{R}^n, A\in\mathbb{R}^{m\times n}$ and $c\in\mathbb{R^m}$, what is the conditional distribution ...
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Do operator in a given DAG
The Original question is here Suppose we have a DAG in the figure.
The question to is find the decomposition for $P(Y \mid do(X), S)$. If the backdoor criteria can be applied here, then the following ...
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A question about do calculus
Suppose we have a DAG in the figure.
I want to find the formula for $P(Y \mid do(X), S)$. What I think is that: Since $W$ is the parent of $X$ then we should have
$$
P(Y \mid do(X), S) = \sum_{W} ...
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Conditional expectation - comparing the definitions [migrated]
Let $X: \Omega \to U$ and $Y: \Omega \to \mathbb{R}^n$ be random variables. Let $$E[Y | X] := E(Y | \sigma(X )): \Omega \to \mathbb{R}^n$$ be the conditional expectation with respect to the sub-$\...
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How to compute the ethnicity distribution percentage from American Community Survey Data Tables at the block group level? [closed]
Hello I'm doing research to study the ethnicity distribution percentages from American Community Survey Data Tables. Initially, I used the table below to compute percentages of white, Black or African ...
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Incorrect reasoning on sampling from marginals
Suppose we want to sample $X$ from a density $p(x) = e^{f(x)}$. My thought was that maybe we can introduce an auxiliary variable $Y$ such that $p(y|x) \propto e^{-\|(x,y)\|^2/2 - f(x)}$. Then the ...
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how to calculate conditional probablity when one even'ts occurance is dependent on mutiple events
I have X, Y, and Z all as binary variables, values either 0 or 1. Y and Z are and got values of P(Y =
1), P(Z = 1), P(X = 1|Y = 1, Z = 1) , P(X = 1|Y = 1, Z = 0) and P(X = 1|Y = 0). here I need to ...
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Should I account for email opens when modeling link clicks embedden in the email?
I'm trying to build a model that predicts whether an email recipient will click on a link inside an email. As input, it takes member-related email click/open/unsubscribe history, as well as that ...
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If $P(A|D) > P(A)$ and $P(B|D) > P(B)$, then is $P(A \cap B|D) > P(A \cap B)$?
There are 3 events $A, B, D$ such that $D$ makes $A$ more likely and $D$ makes $B$ more likely. Does this mean that $D$ makes it more likely that both $A$ and $B$ occur? How can you prove this using ...
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Probability of an image containing a specific object, by combining the results of multiple dependent tests
I am trying to assign to images the probability of them containing a metal building. The images can contain either metal buildings, non-metal buildings or no building.
Let $B$ be the event that an ...
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Implementing Quantile Loss function
I have been reading about Quantile Regression and the Quantile Loss function, but I have to admit I am a bit lost as how to practically implement it. I would like to use it to calculate the prediction ...
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Implications of $\Pr\big(X_n=x \, | \, X_{1},\dots, X_{n-1}\big) \geq a$
Let $(X_n)_n$ be a sequence of discrete random variables and assume that
\begin{equation}
\Pr\big(X_n=x \, | \, X_{1},\dots, X_{n-1}\big) \geq a \quad \text{ for all } n=1,2,\dots
\end{equation}
...
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Does P(Y|X)=const. in the whole P(Y,X) imply that P(X|Y) also remains constant? Or is covariate shift the same as label shift?
I research covariate shift solutions for ML models. Some papers/books (e.g. "Probabilistic Machine Learning" by Kevin Murphy) claim that one needs different solutions for covariate and label ...
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Distribution of transformation of conditional random variables
Let $(\Omega,\mathcal{F},P)$ be a probability space and let
$Y,S: \Omega \to \mathbb{R}$ be random variables with a density $p:\mathbb{R}^2 \to \mathbb{R}$ wrt. the Lebesgue measure with the property ...
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Show that $\Pr\big(X_t=x \, | \, X_{1},\dots, X_{t-1}\big) \geq \nu_x$ implies $\\Pr\big(X_t=x \, | \, Y_{1},\dots, Y_{t-1}\big) \geq \nu_x$
Let $\{X_t\}_t$ be a sequence of discrete random variables and assume that
\begin{equation}
\\Pr\big(X_t=x \, | \, X_{1},\dots, X_{t-1}\big) \geq \nu_x \quad \text{ for all } t=1,2,\dots
\end{equation}...
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Monty Hall Permutation over undefined rounds
This is just for curiosity's sake, but I found this problem, and I don’t know how to solve it. This is not the regular Monty Hall Problem!
You're a participant in yet another version of Let's Make a ...
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Instrument variable analysis using randomized data
I am analyzing the effect of two randomized interventions on 1) first-time and, 2) second-time event attendance. All variables are binary (0 vs. 1). Second attendance was conditional on first ...
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Find the conditional PDF of a multivariate normal distribution given a constraint [duplicate]
Problem to solve
We have a vector of random variables $\textbf{X}=(X_1,X_2)$ issued from a bivariate normal distribution. In particular, $\mu = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$, $\Sigma = \begin{...
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KDE-like technique to learn a continuous distribution from samples subject to specific noise
There's a continuous-valued random variable $X$ with distribution $f_X$. Normally, we're given a bunch of i.i.d. samples $X_1, \ldots, X_n$, and we try to give an estimate $\hat{f}_X$ of the ...
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Interview Question: Expected value of dice game where each subsequent roll determines number of dice sides
Consider the following game:
You are playing a game where you roll a die and earn $1 each time you roll higher than 1. For the first roll you get a 4 sided die, and for each subsequent roll you get a ...
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Questions about conditional probabilities
Some people claim that 90% of suicides are committed by people who grew up in single-mother homes – the implication being that that kind of upbringing leads to suicide.
I'm not sure one can validly ...
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Slight inconsistency between probability estimated from logistic regression and manual calculation from cross table
I want to estimate (using Titanic data as an example) the probability of being in each of the classes (class 1, class 2, class 3) by sex (male, female) and port of ...
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Equivalent Formulations of Thompson Sampling
I am studying Chapter 36 Thompson Sampling of the book Bandit Algorithms by Lattimore and Szepesvari. The authors present two equivalent formulations of Thompson Sampling on page 460, and I am having ...
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Product of normal multivariate distributions [duplicate]
I read a book about statistics and machine learning, and can't understand assertion that: let $P(y) \sim N(y|\mu^*, \Sigma^*)$, i.e. multivariate normal distribution $p(y_1|\mu, \Sigma)$, where $\mu_1 ...
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What is the expected number of children until having the same number of girls and boys?
A couple decides to keep having children until they have the same number of boys and girls, and then stop. Assume they never have twins, that the "trials" are independent with probability 1/...
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Two notions of independence?
I would like some clarification as to if independence as referred to in both of the following cases, is the same or different. I would assume they are different?
Scenario 1: For any SINGLE trial of a ...
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Interview question - Over the long term what fraction do I expect to spend in the left lane?
The following is an interview question:
A certain highway has two lanes in my direction. Whenever traffic moves, if I'm in the left lane I switch to the right with probability 60%.
If I'm in the ...
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Cumulative distribution of Gaussian conditional independent random variables
Suppose X, Y, Z are three jointly Gaussian random variables and X and Z are independent given Y. For example, take three r.v. from a OU process. Here is some R code:...
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Best estimate of conditional probability P(C|A and B) from P(C|A) and P(C|B)?
Assume I have three events A, B, and C, and I know the following probabilities:
Scenario 1:
$P(A)$ and $P(B)$
$P(C|A)$ and $P(C|B)$
Scenario 2:
I additionally know $P(C)$.
I am looking for $P(C|A\...
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How to mathematically express the fact that the conditional probability $P(Y|X)$ can be independent of $P(X)$?
Mathematically, $P(Y|X) = \frac{P(X,Y)}{P(X)}$ and so $P(Y|X)$ must depend on $P(X)$. Since $P(Y|X)$ will change when $P(X)$ changes.
However, consider this scenario:
X = amount of red meat consumed ...
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Independence and conditional independence in probability
In a book, to explain independence and conditional independence in probability, the author gave an example that, consider the following situation:
F={ a hospital patient is smoker}
G={ a hospital ...
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standard error of conditional survival probability
Let $$S(t|t-1) := \frac{P(T>t)}{P(T> t-1)} = \frac{S(t)}{S(t-1)}$$ be the conditional survival probability. I have two questions.
Assuming $\text{SE}(S(t))$ and $\text{SE}(S(t-1))$ (as well as $...
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About the requirement of iids to form a martingale
Most problems about martingales start with an assertion similar to the following one:
Suppose that $X_n$ are iid such that $\mathbb{E}[|X_n|]<\infty$ and $\mathbb{E}[X_n]=0$. Then $S_n = \sum_i^n ...
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Semantics: Conditional Distribution vs Marginal Distribution
This is something which has always confused me.
Suppose we take a standard statistical regression model:
\begin{equation}
Y_i = \beta_0 + \beta_1 X_i + \epsilon_i
\end{equation}
where:
$Y_i$ is the ...
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Probability of next flip being heads given I have seen h heads and t tails
I am currently attempting to understand "Question 2" at this link but having many difficulties. The problem is as follows:
A coin has a chance of landing heads with an unknown probability ...
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Why is this integral not Lebesgue measurable? (and how to rectify)
Suppose I have the Bernoulli random variables $S_t$ for $t=1,\cdots, T$ and for an issue related to my problem, I want to express $P[S_t=s_t\mid S_{t-1} = s_{t-1}]$, by using the law of total ...
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Probability of heads given p is less than some value
Say I have a coin with a chance $p$ of landing heads where $p$ is drawn from the uniform distribution from 0 to 1. Now let's say I have some fixed number $x \in (0,1)$. My question is what is the ...
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distributions in "A Note on the Inception Score" [closed]
I'm reading the following paper A Note on the Inception Score. IS is a evaluation metric for GAN. While reading the paper I got very confused about the notation of so called distributions.
I will ...
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Why: $y | x , \theta \sim N(\theta^T x , \sigma^2)$ [closed]
Given:
$$\begin{align*}
&\epsilon \sim N(0, \sigma^2),\\\\
&\theta \sim N(0, \eta^2I), \\\\
&\text{And } y^{(i)} = \theta^T x^{(i)} + \epsilon^{(i)} \\\\
\end{align*}$$
Why:
$$y^{(i)} | x^{...
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What is going on? Contradictory results on the variance of random vector with random mean and covariance
Suppose $f\mid\mu, F\sim N(\mu, F) \in \mathbb{R}^n$, where $\mu , F$ are both random (random vectors and random matrix respectively).
What is the correct way to derive $Var(f)$?
First, let $\tilde{f} ...
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Converting a Conditional Expectation into an Unconditional one
Consider two random variables $V_1$ and $V_2$, where $V_1$ is continuous and $V_2$ is binary.
In a paper, I found that the following equality holds:
\begin{align*}
\mathbb{E}\left[V_1|V_2=1\right]=\...
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How to correctly compute conditional probabilities by considering the probability of each possible outcome?
So I've been trying to understand the
boy-girl paradox, and I reframed it in terms of coin flips to help it make more sense to me and I get now why the probability is different depending on whether we'...
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Is a nested expectation equivalent to expectation on the joint? $\mathbb{E}_{p(x)}[\mathbb{E}_{p(y \mid x)}[f(x, y)]] = \mathbb{E}_{p(x, y)}[f(x, y)]$
Is it always true that nested expectations equal expectations on the joint?
$$
\mathbb{E}_{p(x)}[\mathbb{E}_{p(y \mid x)}[f(x, y)]] = \mathbb{E}_{p(x, y)}[f(x, y)]
$$
Something along these lines ...
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What is the correct way of expressing this probability mass function?
Suppose I have a discrete random variable $S$. Now further suppose that I have a random variable $X$ which is a continuous random variable on the real line. I wanted to express the p.m.f. $P[S=s]$ for ...
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Maxdiff Approach - Claims comparison , how to compare product claims of two different surveys Maxdiff gives Preference shares or count based analysis
My business objective: I want to create a MAX diff approach where I will have multiple surveys my output will be Claims and its Posterior Probability , count of best and worst selection.
For more ...
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Conditional CDF given one dimension equals derivative of joint CDF towards that dimension divided by the density at that dimension?
So I am familiar with the following:
$$P\left(X<x|Y=y\right) =\int_{-\infty}^{x}f\left(X=u|Y=y\right)du=\frac{1}{f\left(Y=y\right)}\cdot\int_{-\infty}^{x}f\left(X=u,Y=y\right)du$$
But during a ...
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Conditional distribution of product of two random variables given one of the random variables
Let's say I have two independent random variables $X$, $Y$. Let's define a new random variable $Z = XY$. I know that the distribution of $Z$ follows: $f_{Z}(z) = \int_{-
\infty}^{\infty}f_{X}(x)f_{Y}(...
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Conditional density of a linear transform of a random variable
Given a random variable $Y \sim D$ and a linear transform $g(Y) = a + bY$, we define a new random variable $W = g(Y)$. The probability density of $W$ is $f_{W}(w) = \cfrac{1}{b}f_{Y}(\cfrac{w-a}{b})$.
...
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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 ...
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