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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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What method to use for conditional density querying

I have a dataset of 3d poses each represented by 40 points (all relative to the central point). So my data has dimensionality 120. What is needed is to learn how build realistic pose, when positions ...
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Wavenet joint probability

As presented in the first article of Google Wavenet (https://arxiv.org/pdf/1609.03499.pdf) the model can approximate the joint probability of the whole sequence (raw audio waveform) using the chain ...
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Question regarding conditional expectation [duplicate]

In Larry Wasseman's lecture notes(lecture 4, page 4) I found this statement $\mathbb{E}[Y|X=x] = \sum_y y f_{Y|X}(y|x)$ or $=\int_y y f_{Y|X}(y|x)dy.$ An important point about the conditional ...
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Conditioning the probability obtained from a machine learning model

I have developed a random forest classifier to predict whether a customer will churn. The data used to produce this model has the following form ...
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Conditional distribution with Jeffreys’ Prior [on hold]

If $\pi(\mu,\sigma)$ corresponds to $N(\mu,\sigma^2)\times\mu^{-1/2}$, what is $\pi(\mu|\sigma)$?
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Conditional Probability on Disease

A man living in a country where only 1 out of 1000 people has the virus A. There is a test available that gives a positive result 5% of the time when the patient does not have virus A and a negative ...
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consistency of an estimator in real data

as we know in statistics we are interested in the properties of an estimator , as my estimator is consistent , in probability sense it can be shown as $P(|estimator - true| <- positive) = 1$ as ...
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Conditional Independence in Bayes Network

I have just started working through Michael Jordan's notes on Probabilistic Graphical Models and seem to be stuck on the exercise on page 5. I summarize the question here: Suppose $G = (V,E)$ is a ...
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Help examining predicted probabilities from nested logistic models

I am having trouble explaining a recent finding to myself. I'm not sure if I'm asking a redundant question, but here is my situation: 1) Suppose I want to model the estimated probability of some ...
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$k$-th order statistics when the value of $j$-th one is known

Suppose there are $n$ random variables $X_i,~i\in\{1,\cdots,n\}$ which are independently drawn according to a CDF $F$ and pdf $f$. Suppose also that we know one of the realization, say $X_{(j)}=x_{(...
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Conditional Probability - Drawing balls from an urn

I have that question from a past exam (without answer): There are two urns, say I and II. Urn I contains 1 white ball and 1 black ball. Urn II containts two white balls and 3 black balls, and suppose ...
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Bayes Rule Bayesian Risk and Decision

Good day, When attempting this problem I came across some difficulties. A humanitarian charity wishes to classify a village as being at either high or low risk of flooding. The following ...
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Bayesian Inference Toy Problem

Problem statement: Consider a probabilistic model where there are two states of the world, framed as complimentary events: $A$: All chocolates are black and $A^C$: 50% of chocolates are black. Let $p$ ...
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relationship between correlation, covariiance and conditional distribution

What are the relationships between correlation and conditional distribution. For instance, given three dependent variables, X1, X2 and ...
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An algorithmic fairness / equal calibration lemma

I'm having trouble with the lemma and proof on page 7 on this paper. Mitchell, S., Potash, E., & Barocas, S. (2018). Prediction-Based Decisions and Fairness: A Catalogue of Choices, Assumptions,...
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Using conditional probability to calculate sentiment score probability

Sorry, maybe this is a bit of a rookie question, but I would like to find out the probability of A(tweet sentiment being negative or positive) based B (the length of the tweet). This to me sounds ...
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How to calculate the probability that white will win a chess game? [closed]

I am developing a chess engine, and wrote four classifiers (logistic regression) for: 1) detecting which part of the game it is (opening, middlegame, endgame) based on current board / position A) ...
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Conditional expectation of an exponential variable

I have trouble with conditional probabilities, therefore I am wondering if the following derivation is correct: Both X and Y are exponential random variables, with $\lambda_x$ and $\lambda_y$ ...
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Probabilistic model: what's the probability of this model?

The above model is from Berkeley cs294-112 Page 23. It is said that $$ p(O_{1:T},s_{1:T}, a_{1:T}) =p(s_1)\prod_{t}p(s_{t+1}|s_t,a_t)p(O_t|s_t,a_t)\tag 1 $$ I'm quite confused about this solution: ...
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Conditional distribution of $\exp(-|x|-|y|-a \cdot |x-y|)$

I am trying to use Gibbs sampling or Metropolis-Hastings to draw samples from the joint distribution$$f(x,y)\propto\exp(-|x|-|y|-a \cdot |x-y|)$$ For this I need the conditional distributions of $x$ ...
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Conditional probability given only the converse conditional probability, and the average of one variable

I’ve been working on this question for a few days now. Full disclosure: this is from a homework problem set. This is one of the exercises of Barnett's book on quantum information. A particle counter ...
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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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Theorem of total probability with basketball players

Suppose you have a player from team A and a player from team B. You know that one of them has a 60% chance to make a shot and other has 40% chance to make a shot, but you are not sure which is which. ...
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Putting into words conditional probabilities and the chain rule

Consider events $a, b, c, d$. $p(a,b,c,d) = p(a)p(b|a)p(c|a,b)p(d|a,b,c)$ ; by the chain rule. We derive this by repeatedly applying the definition of conditional probability. This feels a bit dry, ...
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probabilistic distribution of a variable which are based on random imputs

In practice, I have a variable x, which is based on (b,c,d). We may have a physics based math formula to describe the relationship between x and (b,c d), i.e., x=f(b,c,d). Beforehand, we may know the ...
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Combining probabilities from multiple observations

Given a number of probabilities taken from a large number of recorded football games. A: Chance of WIN given 6 corners: 0.6 B: Chance of WIN given 2 goals: 0.6 C: Chance of LOSS given 2 red cards: 0....
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Expected value of last Gamma RV in a sum

I've got a sum of $X_i \sim \text{Gamma}(k, \theta)$ i.i.d. random variables. I'm trying to find the expected value of the final $X_i$ that takes the sum above a certain value, i.e., to find the value ...
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What does this notation $p(s', r \mid s, a)$ mean in reinforcement learning?

I was reading a book on reinforcement learning, and came across the following notation: Possibly stupid question, but I cant seem to google how to read this. Is it A) Probability of (s' and r) GIVEN ...
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33 views

Joint conditional density of two iid exponential random variables conditioned on their sum

I have the following question: Suppose $X_1, X_2$ iid $\sim f_X(x)=\theta e^{-\theta x}1\{x\ge 0\}$ and $S=X_1+X_2$. What is $f_{X_1,X_2}(x_1,x_2|S=s)$? The solution from our exercise class to this ...
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Conditional probability questions

Three dice are rolled. If no two show the same face, what is the probability that one is an ace? Given that a throw with ten dice produced at least one ace, what is the probability p of two or more ...
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Conditional Probability and Expectation for Poisson Process

To solve part (a) I have $P(X_2 = k\mid X_1 = 1)= \dfrac{P(X_2 = k \cap X_1 = 1)}{P(X_1 = 1)} = \dfrac{e^{-2}}{e^{-1}}=e^{-1}$. Then for part (b), for simplicity, I let $X_2=X$ and $X_1=Y$, then $$E(...
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Mutual info between continuous and discrete variables from numerical data

I am looking for references/measures to estimate the mutual information between a continuous (C) and discrete (D) variable, given a real-world (i.e. finite sample) data set. C is uniformly distributed ...
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How do I intuitively understand that independence is always symmetric?

Independence between two events, $A$ and $B$, is a symmetric relation, that is, if $P(A \mid B) = P(A)$, then $P(B \mid A) = P(B)$. The proof is very simple and can be found at the ProofWiki. ...
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Conditional probability of Negative Binomial R.V. given the SUM of its values

Suppose $\{z_{ij}\}$ are independent Negative Binomial random variables with means $\{\mu_{ij}\}$, with $i=1\dots I$ and $j=1\dots J$. How do you find the (expectation of) conditional probability ...
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Proof of probabilities that may not be independent

I am given the problem: Given $P(A) = \frac{3}{4} $, $P(B) = \frac{3}{8} $, show that: a) $P(A or B) > \frac{3}{4} $. b) $\frac{1}{8} < P(A and B) < \frac{3}{8} $. The problem does not ...
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differences between conditional probability and dependency

Sometimes, I read articles about conditional probabilities and other articles about conditional dependency. My question what is the main differences between them? For example, "https://en.wikipedia....
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Question regarding posterior and prior distribution relation

I am currently reading the book Machine Learning and Pattern regocnition by Bishop. They state in (1) or (1.66) in the book (relating how to derive regularized SSE with posterior and prior ...
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Expectation of standard exponential squared given sum of two standard exponentials

So I have been working on this question for a while and made some progress , but I run into a problem about the normalizing constant. The question is, for $X$ and $Y$ i.i.d. standard exponential, find ...
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Writing down a conditional probability / finding the underlying probability space

This is a notational issue: Let's say we have a partition of $\mathcal X$ given by $\{\Omega,\Sigma\}$ and we define $P(\omega\in\Omega) = p$, $P(X(\omega)=x|\omega\in\Omega) = \pi$ and $P(X(\...
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Recursive Bayes Learning

I'm trying to work through an example from Richard Dudas Pattern Classification on Recursive Bayes Learning. My main question is why do we choose the $max[D^n] $ in: $$max[D^n] \le \theta \le 10 $$ ...
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$V(X|Y)=\Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}$

We know that the conditional variance of a multivariate normal vector $(X,Y)$ is equal to the Schur complement: $$V(X|Y)=\Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}$$ However, $\Sigma_{XX}-\...
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1answer
114 views

Credibility evaluation - how to model conditional continuous density from multiple variables of various types?

I recently got dataset for 37000 households with declared income and a few dozens of other variables of various types: continuous, discrete, binary. The task is to automatically (unsupervised) ...
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What is the probability that at least three guilty parties are caught at the same time and at least four of the innocent are released?

A lie detector will be used by police to investigate 10 suspects of involvement in a particular crime. Admit that among them, five are guilty (but will plead innocence) and the other five are really ...
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Deriving the joint probability density function from a given marginal density function and conditional density function

We have a multivariate normal vector $\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, where $\mathbf{X} = \left[ \begin{matrix} X_1 \\ X_2 \\ \dot\\\\ \dot\\\\ X_n \end{matrix} \...
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67 views

Which book has the right conditional independence formula? [closed]

I'm getting crazy. I've just started to learn probability and, after it, Bayesian networks. I don't know so much about probability, that is why I'm getting crazy. I'm using this book to study a ...
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37 views

Conditional probabilities involving random variables and functions of these variables

I have that $Z = X + 2Y$. $X, Y$ are independent. I know $f_X(x), f_Y(y), f_{X,Y}(x,y)$ and $f_Z(z). $ How can I find $f(x,y|z)$? I know that $f(x,y|z) = f(x, y, z)/f(z) = f(z| x, y)*f(x, y)/f(z)$ ...
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Proof of Pearson-Aitken selection formula

I am trying to understand the proof of the Pearson-Aitken selection formula, widely used in statistical genetics. A proof that the formula is general is given by Aitken (1936). However, I failed to ...
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What is the probability that the drug synthesized is effective

An experimental protocol developed by NuGenCanPharm Inc to test if a cancer drug is effective is correct 99% of the time, on both effective and ineffective drugs. NuGenCanPharm Inc synthesizes 10,000 ...
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120 views

variance of conditional multivariate gaussian

I was playing around with Gaussian Distributions on my machine and I was interesting in making a pretty plot. I wanted to show the distribution of $x_1$ if $x_2$ was given if $x_1,x_2$ were ...
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1answer
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Probability of complement events over time

The probability of catching fish in one hour is 0.64? What is the probability of catching fish in half an hour? Regular solution follows the principle for an opposite event: a probability of NOT ...