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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Deriving the conditional distributions of a multivariate normal distribution

We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning $\boldsymbol\mu$ and ${\boldsymbol Y}$ into $$\boldsymbol\mu = \begin{bmatrix} \...
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148 votes
15 answers
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Amazon interview question—probability of 2nd interview

I got this question during an interview with Amazon: 50% of all people who receive a first interview receive a second interview 95% of your friends that got a second interview felt they had a good ...
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98 votes
3 answers
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Can someone explain Gibbs sampling in very simple words? [duplicate]

I'm doing some reading on topic modeling (with Latent Dirichlet Allocation) which makes use of Gibbs sampling. As a newbie in statistics―well, I know things like binomials, multinomials, priors, etc.―,...
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64 votes
3 answers
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A generalization of the Law of Iterated Expectations

I recently came across this identity: $$E \left[ E \left(Y|X,Z \right) |X \right] =E \left[Y | X \right]$$ I am of course familiar with the simpler version of that rule, namely that $E \left[ E \...
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59 votes
4 answers
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How to generate correlated random numbers (given means, variances and degree of correlation)?

I'm sorry if this seems a bit too basic, but I guess I'm just looking to confirm understanding here. I get the sense I'd have to do this in two steps, and I've started trying to grok correlation ...
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40 votes
15 answers
10k views

What is the intuition behind the formula for conditional probability?

The formula for the conditional probability of $\text{A}$ happening given that $\text{B}$ has happened is:$$ P\left(\text{A}~\middle|~\text{B}\right)=\frac{P\left(\text{A} \cap \text{B}\right)}{P\left(...
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34 votes
4 answers
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Can anyone explain conjugate priors in simplest possible terms?

I have been trying to understand the idea of conjugate priors in Bayesian statistics for a while but I simply don't get it. Can anyone explain the idea in the simplest possible terms, perhaps using ...
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32 votes
8 answers
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What is the probability that this person is female?

There is a person behind a curtain - I do not know whether the person is female or male. I know the person has long hair, and that 90% of all people with long hair are female I know the person has a ...
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31 votes
5 answers
2k views

Wikipedia entry on likelihood seems ambiguous

I have a simple question regarding "conditional probability" and "Likelihood". (I have already surveyed this question here but to no avail.) It starts from the Wikipedia page on likelihood. They say ...
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31 votes
4 answers
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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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30 votes
8 answers
99k views

Two dice rolls - same number in sequence

I am currently studying Statistical Inference class on Coursera. In one of the assignments, the following question comes up. ...
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29 votes
3 answers
16k views

Why Normalizing Factor is Required in Bayes Theorem?

Bayes theorem goes $$ P(\textrm{model}|\textrm{data}) = \frac{P(\textrm{model}) \times P(\textrm{data}|\textrm{model})}{P(\textrm{data})} $$ This is all fine. But, I've read somewhere: Basically, ...
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28 votes
4 answers
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Problem with proof of Conditional expectation as best predictor

I have an issue with the proof of $E(Y|X) \in \arg \min_{g(X)} E\Big[\big(Y - g(X)\big)^2\Big]$ which very likely reveal a deeper misunderstanding of expectations and conditional expectations. The ...
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26 votes
2 answers
73k views

Definition of Conditional Probability with multiple conditions

Specifically, say I have two events, A and B, and some distribution parameters $ \theta $, and I'd like to look at $P(A | B,\theta)$. So, the simplest definition of conditional probability is, given ...
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25 votes
3 answers
3k views

Is there any difference between Frequentist and Bayesian on the definition of Likelihood?

Some sources say likelihood function is not conditional probability, some say it is. This is very confusing to me. According to most sources I have seen, the likelihood of a distribution with ...
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24 votes
4 answers
6k views

Monty Hall Problem with a Fallible Monty

Monty had perfect knowledge of whether the Door had a goat behind it (or was empty). This fact allows Player to Double his success rate over time by switching “guesses” to the other Door. What if ...
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24 votes
2 answers
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The paradox of i.i.d. data (at least for me)

As far as my aggregate (and scarce) knowledge on statistics permits, I understood that if $X_1, X_2,..., X_n$ are i.i.d. random variables, then as the term implies they are independent and identically ...
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23 votes
4 answers
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Why is P(A,B|C)/P(B|C) = P(A|B,C)?

I understand $P(A\cap B)/P(B) = P(A|B)$. The conditional is the intersection of A and B divided by the whole area of B. But why is $P(A\cap B|C)/P(B|C) = P(A|B \cap C)$? Can you give some intuition?...
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22 votes
2 answers
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difference between conditional probability and bayes rule

I know the Bayes rule is derived from the conditional probability. But intuitively, what is the difference? The equation looks the same to me. The nominator is the joint probability and the ...
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21 votes
5 answers
13k views

A meeting has 12 employees. Given that 8 of the employees are female, what is the probability that all employees are female? [closed]

If I use Bayes' theorem here , event A denoting that 12 employees are female and event B denoting that 8 employees are female, (assuming that an employee has equal chances of being a male or female) I ...
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20 votes
3 answers
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How can I calculate the conditional probability of several events?

Could you inform me please, how can I calculate conditioned probability of several events? for example: P (A | B, C, D) - ? I know, that: P (A | B) = P (A $\cap$ B) / P (B) But, unfortunately, I ...
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19 votes
3 answers
6k views

Can a posterior probability be >1?

In Bayes' formula: $$P(x|a) = \frac{P(a|x) P(x)}{P(a)}$$ can the posterior probability $P(x|a)$ exceed 1? I think it is possible if for example, assuming that $0 < P(a) < 1$, and $P(a) < ...
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19 votes
2 answers
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How should I mentally deal with Borel's paradox?

I feel a bit uneasy with how I've mentally dealt with Borel's paradox and other associated "paradoxes" dealing with conditional probability. For those who are reading this that aren't familiar with it,...
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18 votes
2 answers
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Expected value of x in a normal distribution, GIVEN that it is below a certain value

Just wondering if it is possible to find the Expected value of x if it is normally distributed, given that is below a certain value (for example, below the mean value).
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18 votes
1 answer
591 views

simulating random samples with a given MLE

This Cross Validated question asking about simulating a sample conditional on having a fixed sum reminded me of a problem set to me by George Casella. Given a parametric model $f(x|\theta)$, and ...
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17 votes
4 answers
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How to develop intuition for conditional probability?

In the video lectures from Harvard's Statistics 110:Probability course that can be found on iTunes and YouTube, I encountered this problem. I tried to summarize it here: Suppose we are given a ...
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17 votes
1 answer
6k views

Intuitive explanation of contribution to sum of two normally distributed random variables

If I have two normally distributed independent random variables $X$ and $Y$ with means $\mu_X$ and $\mu_Y$ and standard deviations $\sigma_X$ and $\sigma_Y$ and I discover that $X+Y=c$, then (assuming ...
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16 votes
1 answer
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Sampling from marginal distribution using conditional distribution?

I want to sample from a univariate density $f_X$ but I only know the relationship: $$f_X(x) = \int f_{X\vert Y}(x\vert y)f_Y(y) dy.$$ I want to avoid the use of MCMC (directly on the integral ...
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15 votes
5 answers
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Is "not independent" the same as "dependent" in English?

When I learned about conditional probability, I found this statement: if A is not independent of B then also B is not independent of A. Formally, if P(A) ≠ P(A|B) then P(B) ≠ P(B|A). I think "...
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15 votes
2 answers
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Why is posterior density proportional to prior density times likelihood function?

According to Bayes' theorem, $P(y|\theta)P(\theta) = P(\theta|y)P(y)$. But according to my econometric text, it says that $P(\theta|y) \propto P(y|\theta)P(\theta)$. Why is it like this? I don't get ...
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15 votes
3 answers
10k views

What is the relationship between event and random variable?

I've been told that an event is just a random variable that has been assigned, and that random variables are a generalisation of events. However, I can't relate that to the definition of an event as a ...
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15 votes
3 answers
3k views

Conditional probability of continuous variable

Suppose that random variable $U$ follows a continuous Uniform distribution with parameters 0 and 10 (i.e. $U \sim \rm{U}(0,10)$ ) Now let's denote A the event that $U$ = 5 and B the event that $...
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15 votes
4 answers
557 views

If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$

Question If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$. Attempt: Please check if the below is correct. Let ...
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14 votes
6 answers
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The more important statistic: '90 percent of all women survived' or '90 percent of all those who survived were women'?

Consider the following statements w.r.t the Titanic: Assumption 1: Only men and women were on the ship Assumption 2: There were a large number of men as well as women Statement 1: 90 percent of all ...
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14 votes
2 answers
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Why are density functions sometimes written with conditional notation?

I keep seeing density functions that don't explicitly arise from conditioning written with the conditional sign: For example for the density of the Gaussian $N(\mu,\sigma)$ why write: $$ f(x| \mu, \...
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14 votes
1 answer
5k views

How to define a distribution such that draws from it correlate with a draw from another pre-specified distribution?

How do I define the distribution of a random variable $Y$ such that a draw from $Y$ has correlation $\rho$ with $x_1$, where $x_1$ is a single draw from a distribution with cumulative distribution ...
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14 votes
1 answer
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Bayes Theorem with multiple conditions

I don't understand how this equation was derived. $P(I|M_{1}\cap M_{2}) \leq \frac{P(I)}{P(I')}\cdot \frac{P(M_{1}|I)P(M_{2}|I)}{P(M_{1}|I')P(M_{2}|I')}$ This equation was from the paper "Trial by ...
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13 votes
5 answers
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What are the differences between "Marginal Probability Distribution" and "Conditional Probability Distribution"?

While studying probability, I am kind of having difficulties in understanding marginal probability distribution and conditional probability distribution. To me, they look much the same and cannot find ...
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13 votes
5 answers
7k views

What are the differences between stochastic and fixed regressors in linear regression model?

If we have stochastic regressors, we are drawing random pairs $(y_i,\vec{x}_i)$ for a bunch of $i$, the so-called random sample, from a fixed but unknown probabilistic distribution $(y,\vec{x})$. ...
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  • 451
13 votes
3 answers
4k views

Naive Bayes feature probabilities: should I double count words?

I'm prototyping my own Naive Bayes bag o' words model, and I had a question about calculating the feature probabilities. Let's say I've got two classes, I'll just use spam and not-spam since that's ...
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12 votes
5 answers
928 views

Confidence interval and probability - where is the error in this statement?

If someone makes a statement like below: "Overall, nonsmokers exposed to environmental smoke had a relative risk of coronary heart disease of 1.25 (95 percent confidence interval, 1.17 to 1.32) ...
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12 votes
2 answers
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Explanation of I-map in a Markov/Bayesian network

I am finding the concept of an I-map (Independency-map) in the context of Markov networks and Bayesian networks difficult to understand. From Probabilistic Graphical Models, Koller and Friedman, 2009: ...
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12 votes
2 answers
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How to compute conditional expectations with respect to a sigma field?

Example: Toss a coin twice. Letting $\mathbb P$ be a probability measure, suppose $\mathbb P(HH)=p^2,\mathbb P(HT)=\mathbb P(TH)=p(1-p), \mathbb P(TT)=(1-p)^2.$ I would like to answer the following ...
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12 votes
1 answer
8k views

what is the semicolon notation in joint probability?

I see this kind of notation often $$ p_{\theta} (x|z, y) = f(x; z, y, \theta) $$ I understand the conditional prob noation on the left. What is the significance of the ; on the joint prob on the ...
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12 votes
3 answers
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How is $\Pr(X=x|Y=y)$ defined when $Y$ is continous and $X$ discrete?

Say that $Y$ is a continuous random variable, and $X$ is a discrete one. $$ \Pr(X=x|Y=y) = \frac{\Pr(X=x)\Pr(Y=y|X=x)}{\Pr(Y=y)} $$ As we know, $\Pr(Y=y) = 0$ because $Y$ is a continuous random ...
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12 votes
1 answer
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What is "Likelihood Principle"?

While I was studying "Bayesian Inference", I happen to encounter the term, "Likelihood Principle" but I don't really get the meaning of it. I assume it is connected to "...
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12 votes
4 answers
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Expected number I will be on after drawing cards until I get an ace, 2, 3, and so forth

I am having some trouble solving the following. You draw cards from a standard 52-card deck without replacement until you get an ace. You draw from what is remaining until you get a 2. You continue ...
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12 votes
2 answers
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How do programs like BUGS/JAGS automatically determine conditional distributions for Gibbs sampling?

Seems like full conditionals are often quite difficult to derive, yet programs like JAGS and BUGS derive them automatically. Can someone explain how they algorithmically generate full conditionals for ...
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  • 2,504
11 votes
5 answers
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Conditional probabilities - are they unique to Bayesianism?

I wonder whether conditional probabilities are unique to Bayesianism, or whether they are more of a general concept that is shared among several schools of thought among statistcs/probability people. ...
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11 votes
3 answers
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How Conditional Random Fields and Logistic Regression could be the same?

If you have read this tutorial about CRF, on page 4 under the section Classification, it wants to relate ...
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