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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How to calculate distribution of Y|X from distributions of X|Y and Y [duplicate]

I'm trying to solve the following homework problem: Let $X$ given $Y=y$ have a normal distribution with mean $y$ and variance one, and let the marginal distribution of $Y$ be normal with mean ...
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1answer
20 views

PDF of sum of independent Gaussian variables

I am looking for deriving the pdf of $Z$ where $Z= (\sum\limits_{i=1}^N a_i X_i +Y_1)^2 + (\sum\limits_{i=1}^N b_i X_i +Y_2)^2$, where $X_i$ and $Y_i$ are independent, zero mean Gaussian random ...
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14 views

Regarding Naive Bayes and conditional independence

We all have been talking about how Naive Bayes may, in some cases, not perform well due to the fact that this assumes conditional independence of features and MOSTLY, this is not true for real world ...
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1answer
41 views

Covariance of a compound distribution

I am trying to find the covariance of a compound distribution. Given $X=x$, where $X \sim \mathrm{Uniform}(0,1)$, $Y$ is (conditionally) normally distributed with mean $x$ and variance $x^2$. I ...
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1answer
25 views

Bayesian Network

I am preparing for midterm exam and need to know what is the step by step solution to this question? Answer is shown in red. Also any external related link is very much appreciated.
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1answer
41 views

Probability distribution of Y in regression?

I'm trying to predict the probability distribution of $Y$ given $X_0, X_1, ...$ with a nonlinear regression. The probability distribution of $Y$ is likely not normal. So far, I've set up and trained ...
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1answer
52 views

Convolution of random vectors

Suppose, I have two random vectors $A=[A_1, A_2, \dots A_k]$ and $B=[B_1, B_2, \dots B_m]$. What could be the joint PDF $f_{\mathbf{y}}(y_1,y_2,\dots y_N)$ where $\mathbf{y}=A \ast B$, here $\ast$ ...
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2answers
74 views

basic probability problem

Problem 2 (page 33 of text) Assume that every patient with a particular type of disease has the probability 0.1 of being cured within a week, if the patient is given no treatment for the disease. Ten ...
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25 views

Proof of alternating conditional expectation base equations

How do we prove the base equations for Alternating Conditional Expectation algorithm. The statement is thus: We define arbitrary mean-zero transformation $\theta(Y),\phi(X_i)$,$1<i<p$ for ...
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0answers
26 views

Conditional independence: conditioning on an empty set of random variables

Is $X \perp\!\!\!\perp Y$ a conditional independence, arguing that the independence is conditioned on an empty set of random variables? If so, does that mean that an unconditional independence is ...
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1answer
22 views

Marginal Distribution from Conditional Distribution

I came across a problem where the marginal distribution of a random variable $Y$, $f(y) = c/y^2$ and $f(x|y) = 1/y$. Can I simply multiply these two to get $f(x,y)$ the joint distribution of $X$ and ...
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22 views

Hypothesis testing and order statistics

I have the following setup. There is a set $S = \{S_1, \ldots, S_N\}$ of $N$ sensors that are probed for readings (once). Each reading is an independent sample from one of the two distributions $r_i ...
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1answer
60 views

Combination of Random Variables Conditional Probability

If A and B are independent discrete random variables and C = A+B, then how should one compute the pmf of P(A|C)? For example, let X be the result from a coin toss(1 or 0 for H and T) and Y be the ...
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1answer
36 views

An example of r.v.s such that their distribution has more (conditional) independencies than their directed graphical model

I was trying to form an example where I had 3 r.v.s such that the distribution describing them had more conditional independencies or independencies than the directed graphical model corresponding to ...
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23 views

In Probabilistic Graphical Models, what does it mean that r.v. X influences r.v Y?

I wanted to pin down what the intuitive phrase: r.v. X influences r.v. Y as precisely and as rigorously as I could, and wanted to check if my interpretation was correct and complete with the ...
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1answer
37 views

Why do we need undirected (Markov) graphical models?

I understand the modular nature of directed models, and that each node captures a conditional probability. But why do we need undirected models? As far as I can see they lack intuition in that the ...
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84 views

Bayesian modeling using multivariate normal with covariate

Suppose you have an explanatory variable ${\bf{X}} = \left(X(s_{1}),\ldots,X(s_{n})\right)$ where $s$ represents a given coordinate. You also have a response variable ${\bf{Y}} = ...
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23 views

Bayesian Updating Process, 3 signals and 2 states of the world

Suppose Nature chooses a state $\omega = \{X,Y\}$ at $t=0$. Long-lived agents observe a signal $s_t$ at every period $t$, where $s_t = \{ x,y,z \}$. Agents all hold a common prior $\mu_0 \in (0,1)$ ...
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19 views

Determine which variable or variables is/are the most efficient to predict the outcome

I have a small dataset (n=74) with a +/- 50 variables, not the best data but I have to work with it. The variables are used to select a product. I want to determine which variable or variables is/are ...
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0answers
21 views

Fusing probability scores from different independent sources

Consider a situation where two independent sources, $s_1$ and $s_2$ are giving probability estimates regarding the occurrence of a event $e$. I tried to model this as a bayesian network but that ...
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26 views

Word probabilities in a Naive Bayes filter

While implementing a Naive Bayes filter, I stumbled across a problem with the calculation of the conditional probabilities $p(w|c)$ of a word $w \in \mathcal{W}$ given a class $c \in \mathcal{C}$. ...
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1answer
27 views

How to work with conditional expectations and variances?

I have seen this statement in a lecture note : $$E[(Y-f(X))^2\,|\,X]=Var[Y|X]+E^2[(Y-f(X)\,|\,X]]$$ If $Z,X$ are random variables, then shouldn't be $Var[Z|X]=E[Z^2|X]-E^2[Z|X]$?In which case ...
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16 views

Conditional Forest - Calculate Overfitting, OOB Error

My name is Abhi and I am trying to teach myself conditional forests. I have got a basic example working myModel <- cforest(Survived ~ .,data = ...
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3answers
113 views

Conditional Probability with Normal Distributions?

Let's say that I have $3$ independent random normal variables, $A$, $B$ and $C$. They all have a standard deviation of $17.526$, while $A$ has a mean of $143$, $B$ of $139$, and $C$ of $129$. My ...
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2answers
141 views

Probability of two aces given at least one ace (out of two cards) vs probability of two aces given one card is the ace of spades (out of two cards)

So, in the video lectures from Harvard's Statistics 110:Probability course that can be found on iTunes and Youtube, I encountered this problem. https://www.youtube.com/watch?v=JzDvVgNDxo8#t=566 I ...
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2answers
108 views

Conditional expectation of $X$ given $Z = X + Y$

Suppose I have two independent normal variables $X$ and $Y$ with known mean and variance. Defining $Z = X+Y$, what is the most straightforward way to compute $\mathbb{E}\left[X|Z\right]$? I am ...
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1answer
23 views

Modelling a probability distribution on different feature sets

I have a binary classification problem, and I use method A and method B to extract features, F1 and F2, for this problem from dataset X. Now, I train two models, y1 and y2, separately on the two ...
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1answer
70 views

about the definition of bayesian network

In this PDF http://people.csail.mit.edu/yks/documents/classes/mlbook/pdf/chapter2.pdf page 5 says: Given a set of functions $f(x_i,pa(x_i))$ non-negative and sum to 1, we define a joint probability ...
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119 views

how to calculate Root Mean Square Error (RMSE) for predicted Probability Density Function (PDF) in Matlab

I have used Mixture Density Networks for probability density function prediction. I am wondering how I can calculate Root Mean Square Error (RMSE) of predicted pdf in MATLAB. Thanks.
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1answer
140 views

How to compute this conditional probability in Bayesian Networks?

I met a problem related to conditional probability from the article "Bayesian Networks without Tears"(download) on page 3. According to the Figure 2, the author says $$P(fo=yes|lo=true, ...
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85 views

Binomial random variable conditional on another one

On the Wikipedia page for the Binomial distribution, the following property is mentioned (under the related distribution section): (paraphrased) If $X\sim \text{Bin}(n,p)$ and $Y|X \sim ...
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1answer
42 views

simply demostration on conditioned probability

I don't find the answer of this simply problem: how can I algebrically demonstrate that, with 3 variable A,B,C $ \sum\limits_C P(B|C)P(C|A)=P(B|A)$ under condition that $P(A,B,C)=P(A|C)P(B|C)P(C)$ ...
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0answers
25 views

Probabilities in a Markov Model

I am reading a paper on Markov Models and I am trying to figure out how to compute the probabilities for the $\alpha$-pass. I am given an $N\times N$ matrix $A$, that has the probabilities of ...
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1answer
26 views

What is full conditional distribution in my case?

If $\text{P}(M|D)$ is posterior, $a$ is the proportionality constant, $\text{P}(M)$ is the prior and $\text{P}(D|M)$ is the likelihood. I have the the prior distribution, and I know the function that ...
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1answer
39 views

What is the correct definition for completness in d-separation in directed graphical models?

I was reading Koller's book of probabilistic graphical models and in section 3.3.2 she discusses what properties should hold for d-separation as a method for determining independence. She tries to ...
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1answer
52 views

Maximum expected difference between players when using 4d6 drop lowest

Players of a certain TRPG have characters with 6 ability scores, each ability score ranging from 3-18. One method of generating those is by rolling 4d6 drop lowest for each of the scores. That means ...
2
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1answer
94 views

Probability for completely unfair stats when rolling ability scores using 4d6 drop lowest for D&D

Players of a certain TRPG have characters with 6 ability scores, each ability score ranging from 3-18. One method of generating those is by rolling 4d6 drop lowest. That means four six-faced-dice are ...
2
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2answers
122 views

What does the notation $(\textbf{X} \perp \textbf{Y} , \textbf{W}\mid \textbf{Z})$ mean?

I was reading Koller's and Friedman's Probabilistic Graphical Models book and became confused about some of its notation because of a set of notes that either contradict it or express it differently. ...
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2answers
61 views

Negating conditional probability

I'm refreshing on bayes theorem and conditional probability and I ran across these practice problems. I was trucking along until problem 9, which states: ...
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2answers
78 views

If $X \sim$ unif$(0,1)$, what is the distribution of $U=\max \{X,1-X \}$?

Let $X$ be uniformly distributed in $(0,1)$ and set $U=\max \{X, 1-X\}$. How can I find the distributiopn of $U$? My first thought was to consider this a mixture distribution problem and use the CDF ...
2
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1answer
23 views

Estimating workers within Industry Sector using commuter data (combining probabilities?)

I have some detailed data about the commuting patterns of workers. If I know the following 3 things... The total number of workers commuting from one block to another. The total # and % of ...
4
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1answer
71 views

Verifying independence for two Random Variables

Could you please give me some hints for the exercise below? Suppose we toss a coin once and let $p$ be the probability of heads. Let $X$ denote the number of heads and let $Y$ denote the number of ...
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22 views

Non specific order of probabilities in a time step

I'm simulating a random collision process. At each time interval I calculate the probability of a collision occurring between each object and all other objects in proximity. Currently if I wish to ...
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1answer
42 views

Greater than 1 Naive Bayes Probabilities?

I am trying to train a Naive Bayes classifier. In addition to getting the most likely class as an output from the Naive Bayes classifier, I would also like to compute the probabilities associated with ...
2
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1answer
59 views

Bounded expectation implied bounded conditional or vice versa?

If $\mathrm{E}\left(X\right)<\infty$ does that imply $\mathrm{E}\left(X|Y\right)<\infty$? How about vice versa? I'm thinking if we condition on an event (say $Y>2$) then if we have ...
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20 views

How to find conditional distribution for Rao-Blackwellizing an estimator?

Let's say I have an unbiased estimator $u(\underline x)$ for function $v(\theta)$ where $\theta$ is a parameter of the distribution of $x$, and $T(\underline x)$ which is a sufficient statistic for ...
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28 views

Creating statistically balanced teams

Suppose that you have a tournament for a game with four players on each team. We also have a table that tells us overall statistics for each player. This table includes things like each player's # ...
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64 views

Some doubt in reading Machine Learning A Probabilistic Perspective ( chapter 3.2 )

When I am reading Murphy's Machine Learning A Probabilistic Perspective. In chapter 3.2. I have some doubt. I think the author want to express is two things. First, we can use Bayes formula to ...
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19 views

How to compute the marginal probability form conditional probabilities in logspace?

Normally the marginal probablity is computed as $p(x) = \sum_y p(x | y) \cdot p(y) $ Now, suppose I have all these probabilities at the right-hand side in logspace (so as logprobabilities). How do ...
2
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1answer
37 views

How to combine distributions

If I have two continuous distributions $f(x)$ and $g(x)$, there are several mathematical ways to combine $f$ and $g$ to get new distributions. Which correspond to what statistical interpretation? For ...