For questions related to Bayes' theorem, a result about conditional probabilities.

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Bayes' theorem with multiple variables

On the page: https://en.wikipedia.org/wiki/Bayesian_inference#Formal_description_of_Bayesian_inference there is the result: $$p(\theta \mid \mathbf{X},\alpha) = \frac{p(\mathbf{X} \mid \theta) ...
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Having a value, calculating the probability of the number of times a die has been rolled

If I enter a friend's house and he has rolled a dice (d6) on a table which has a value $1$, he asks me the following, "Do you think I rolled this dice once and got $1$ or do you think I rolled it ...
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Probability of pairwise dependent events [closed]

There are $3$ events $X, Y$ and $Z$. Event $Z$ is dependent on both $X$ and $Y$ separately. But $X$ and $Y$ are independent of each other. None of them are mutually exclusive. Probability of $X$ and ...
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Creating pdfs froms Sample Data and Bayes Theorem for Continuous Probability

I am not much of a math guy, but know some basics of pdfs, pmfs, Bayes theorems, probability distribution and stuffs. I am actually trying to build a Bayesian Network that models the personality of ...
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Using Bayes Theorem To Simplify Probabilities

I want to compute P(AB|X) in terms of the (given) probabilities P(B), P(X), P(A|BX), P(AB, !X), P(A|!B,X), P(A|!B, !X) where every letter is some random variable. How would I go about doing so using ...
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Determine probability of better player given indirect measurements

Say you have a two-player game and three, fixed-but-different skill players: A, B and C. Player A only plays against C and player B also only plays against C. You want to to determine if player A is ...
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Understanding different definitions of bayes theorem

I am taking course on probability and reading about bayes theorem. In Sheldon Ross' book, it given as $$P(E) = P(E|F)P(F) + P(E|F^C)P(F^C)$$ with note: Equation above states that the probability of ...
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Baye's Theorem Conditional Probability with multiple conditions

Lets assume I have a supermarket and I track the purchase history of my customers with 2 attributes of each customer - Gender (M/F) & Smiling (Y/N). Assume this is historical data of purchases: ...
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Why do We Refer to the Denominator of Bayes' Theorem the “Marginal Probability”?

Consider the following characterization of Bayes' Theorem: Bayes' Theorem: Given some observed data $x$, the posterior probability that the paramater $\Theta$ has the value $\theta$ is $p(\theta \mid ...
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ANSML - Proof of Naive Bayes Derivation

I was working through one proof of the Naive Bayes and got stuck at the last step. The setup is as follows: Given a dataset $\left\{ (x^{(i)},y^{(i)}), \cdots\right\}$ for $i=1,\cdots,m$, $y$ can ...
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Classification Rule for Bayes Optimal Classifier (continuous x var)

I'm having trouble determining the Bayes classifier where x is a continuous variable. Basically the problem I'm trying to solve is this: There are type 1 and Type 2 components. There are 66 Type 1 ...
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Bayes estimator under squared error loss

Consider one random variable X from the Bernoulli distribution with parameter θ. Let p, the prior density, be equal to 6θ(1 − θ), for θ ∈ (0, 1). Under squared error loss, L(t, θ) = (t − θ)$^2$, the ...
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Stumped on a Bayes Theorem Question

A certain medical syndrome is usually associated with two overlapping sets of symptoms, A and B. Suppose it is known that: P($A|B$) = 0.8 P($B|A$) = 0.9 P($B'|A'$) = 0.85 Find P($A'$|$B'$) From ...
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Statistical/Combinatorial: How to analyze?

I'm currently preparing for my exam and in the process trying to solve some statistical problems. The question goes as follows: Q1: A book consisting of 269 pages contains 40 missprints. Only, you ...
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unintuitive meaning to Bayes theorem

The equality $\frac{\mathbb{P}(Y\mid X)}{\mathbb{P}(Y)}=\frac{\mathbb{P}(X\mid Y)}{\mathbb{P}(X)}$ means that in a supermarket analysis, knowing that a customer bought milk ($X$) multiplies the ...
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Combining probabilities from different sources

Lets say I have three independent sources and each of them make predictions for the weather tomorrow. The first one says that the probability of rain tomorrow is 0, then the second one says that the ...
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Bayesian posterior with a constraining equation (slice-projection?)

Prior and signals: Let $y_1$ and $y_2$ be iid signals on $Y$. The intial prior is $Y \sim N(\bar{Y}, \sigma^2_Y)$, where $N(\cdot, \cdot)$ is the normal distribution The signals are independent and ...
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How am I misusing the Bayes' rule?

I just started studying probability. I am trying to solve this exercise: When coin A is flipped it comes up heads with probability 1/4, whereas when coin B is flipped it comes up heads with ...
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Bayes theorem: Can some one explain in plain words how argmax is used here?

According to Bayes theorem $$p(y |x) = \frac{p(x |y) p(y)}{p(x)}$$ So, to find the maximal value for $p(y |x)$, we want to compute: \begin{align} &\arg\max_{y\in Y} p(y|x) \\ ...
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How to use Bayes's rule with mixed distributions?

On page 81 of The Likelihood Principle by Berger and Wolpert (1988) I find the following claim (which references example 20 on page 75). We consider a certain statistical problem from a Bayesian ...
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Probability Urn problem - application of Bayes thm

I am learning statistics and I am trying to solve this problem: Players A and B draw balls in turn, without replacement from and urn containing three red and four green balls. A draws first. The ...
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Bayes theorem - is it applicable in any case?

I'm studying the Bayes' Theorem and I have a doubt. In this wikipedia page there's an example of application for the following events: ...
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In Bayesian Statistic how do you usually find out what is the distribution of the unknown?

To estimate the posterior we have $$p(\theta|x) = \frac{p(\theta)*p(x|\theta)}{\sum p(\theta ')*p(x|\theta ')}$$ $x$ is usually the experimentally sampled data, and $\theta$ is the model, but both ...
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Validity of conditional CDF proof via PDF integral

Given the question: $$\text{Show that}\ F_X(x\mid A) = \dfrac{\Pr(A\mid X\leq x)}{\Pr(A)}\cdot F_X(x)$$ I have seen the solution via probabilities 'directly'. My question is whether the following ...
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Proof using Bayes rule?

In Statistical analysis of randomized experiments with non-ignorable missing binary outcomes:an application to a voting experiment by Kosuke Imai a proof is given referring to Bayes rule. Let: ...
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Bayes Rule in 2 Fair and 1 Biased coin

I was watching this video on Khan Academy about condition probability where they demonstrated a problem using a tree. I tried to solve that problem using Bayes rule, but my answer doesn't match the ...
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Bayes' rule with 3 variables

I have been using Sebastian Thrun's course on AI and I have encountered a slightly difficult problem with probability theory. He poses the following statement: $$ P(R \mid H,S) = \frac{P(H \mid ...
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Conditional probability, Baye's rule, prisoner Ural / Siberia + coat exercise

I am currently statistics and probability course. One of the questions in the textbook is following: A prisoner will be sent to either Urals or Siberia, but he does not know where. He knows, ...
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Probability of coin flips conditioned on an assistant lying about the results

During a project researching coin flips, an assistant is asked to flip a fair coin twice. But the assistant is feeling lazy, and instead of following the directions, he does the following: He starts ...
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Probability question for economics that I'm struggling with. Please help.

(There are 4 districts in the land of Oz. At home, the inhabitants of each region wear ties of a special colour, Munchkins (M) wear blue, Scarecrows (S) wear purple, Tin Men (T) wear red and Wizards ...
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Please help me to answer this conditional probability question

Question: So, for part (a), my answer is: $$ {10 \choose 8} (0.6)^8 (0.4)^2 + {10 \choose 9}(0.6)^9 (0.4)^1 + {10 \choose 10 } (0.6)^10 (0.4)^0 = 0.1673$$ I am not sure how to answer part ...
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Probability of balls drawn with replacement

We have two bags, Bag A has 40 red balls and 15 blue balls, Bag B has 40 blue balls and 10 red balls. One of these bags is selected at random and from it five balls are drawn at random, replacing each ...
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Normalizing factor for product of Gaussian densities - interpretation with Bayes theorem

The normalizing factor for the product of two multivariate Gaussian densities, $f(x)$ and $g(x)$ with mean vectors $a$ and $b$ respectively, and covariance matrices $A$ and $B$ respectively, is itself ...
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Simple Probability of Playing Cards

An ordinary deck of playing cards has four suits: hearts, spades, diamonds, and clubs. Suppose you have a reduced deck of eight playing cards, consisting of four aces and four kings. I draw two cards ...
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interpretting prior and posterior

I am bit puzzled on how we can interpret the posterior. Assume a coin which is 0.1 probable to be unfair. So our prior probability on the coin being unfair is 0.1, and being fair is 0.9. Also by ...
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Conditional Probability of Sinking Ship Question

Question: Two ships. Ship A's missiles have an 80% probability of hitting its target, ship B's missiles have a 50% probability of hitting the target. It only takes one hit from a missile to sink a ...
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Proving Conditional Probability Statement

Question: Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Prove or disprove that $P(A|B \cup C) = P(A|B) + P(A|C) ~~~\forall~~ A,B,C~ \in \mathscr F$ where $B \cap C = \emptyset$. ...
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Interpretation of Bayes Formula Question

This is a simple application of Bayes Theorem, I'm just confused about the labeling. If i let T = test outcome is positive, D = patient has disease, then, I am trying to find $$ P ( D | \bar{T} ) ...
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Bayesian update multivariate normal based on one-dimensional signal: simple rule

Is there a simple rule to update the linear combination of normal distributions based on a one-dimensional signal? The unconditional joint density of $(\eta,\theta)$ is multivariate normal ...
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1answer
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In Bayes' theorem, what is little $p$?

In Wikipedia's conjugate prior article, Bayes theorem is given as: $$p(\theta|x) = \frac{p(x|\theta) \, p(\theta)} {\int p(x|\theta') \, p(\theta') \, d\theta'}.$$ What is $p$ here? Is it the ...
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Subpopulations of an island (Bayes theorem?)

Help appreciated here. An island with 2 regions, I and II, has 4 types of individuals: AX, AY, BX and BY, for which we know their exact total nos. Here A-B-X-Y are simply traits, e.g., A=Male, ...
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Bayesian Updating with 1 Signal but 2 Unknowns

Suppose I have an unknown variable $X_i = \alpha_i + \beta_i$ where $\alpha$ is one of 2 different values {${\alpha_1, \alpha_2}$} such that $\alpha = \alpha_1$ with probability $p_1$ and $\beta$ is ...
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Why does Bayes' theorem work?

Why does Bayes' theorem work? I'm not looking for a cryptic math demonstration. Rather, what I'm interested in is the intuition behind the theorem that allows to obtain the a posteriori probability ...
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Bayes: markov chain, serial connection, marginalization

Goal is to check if p(a) is unconditionally independent to p(c) in the markov chain - serial connection. $$ p(a,b,c) = p(a) p(b|a) p(c|b) $$ $$ p(a,c) = \sum_b p(a) p(b|a) p(c|b) = p(a) p(c|a) \neq ...
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Conditional probability relationship among $\Pr(A|B), \Pr(A|C), \Pr(A|B, C)$

I need some help in deriving some relationships between $\Pr(A|B,C)$ and $\Pr(A|B), \Pr(A|C)$. Specifically, I want to have a proportional relationship between the former and the two in the latter. I ...
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Prove that $P(G \mid E_i) > P(G) \land P(G \mid E_1, E_2) < P(G), i \in {1, 2}$

Let G be the event that a certain individual is guilty of a certain robbery. In gathering evidence, it is learned that an event E1 occurred, and a little later it is also learned that another event ...
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Bayes factor for fair and biased coin

There is the following task: Suppose we toss a coin $ N = 10$ times and observe $m = 9$ heads. Let the null hypothesis be that the coin is fair,and the alternative be that the coin can have any bias, ...
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Bayes Theorem chaining

Is this statement true $P(A|B)=P(A|C) \times P(C|B)$ If yes, how to prove it? If no, what should be minimal conditions on A, B and C for this to hold true? Thanks for help.
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Calculate CPT of bayes net

I have a Bayes net of a pretty simple construction. I need to find the expressions that the CPT's represent and also the number of entries. A--B--C .....| ....D A is the parent node of B. B is ...
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Setting up Posterior Probability and Bayes Theorem

I would just like to ask some questions with regards to Bayesian probabilities and an exercise I am trying to solve. Here is the problem: A footprint was found at the crime scene, which is known to ...