All Questions

0
votes
1answer
19 views

how to multiply two conditional probabilities in general

I am trying to understand how to multiply two conditional probabilities. $P(X|C) \times P(C| P,S)$ seems to equal to $P(X,C | P,S)$. How to understand this? I understand the product rule, but how ...
0
votes
0answers
34 views

Conditional Probability Table in R

I want to perform Bayesian network analysis in R. I have a large network and i am bit confused with defining conditional probability tables! In my network i have a node with in-degree of centrality ...
1
vote
1answer
39 views

Does independence and mutual exclusivity induce impossibility?

Given that we know A and B are independent and they never occur at the same time, one of them must be impossible, no? $$ P(A\mid B)=\frac{P(A \cap B)}{P(B)}\\ \text{if A and B independent, B gives no ...
0
votes
1answer
30 views

Constraints on choice of marginal distribution and likelihood

For some time I have been reading into Bishop's Pattern Recognition and Machine Learning. Coming back to some earlier chapters the following got me confused and I am interested where, formally I go ...
0
votes
0answers
29 views

Naive bayes example by hand

Given the following data ...
1
vote
0answers
14 views

identifying which of $d$ normal distribution generated a given sample

I have $d$ Normal Distributions, $N_1(\mu_1, \sigma_1^2) \cdots N_d(\mu_d, \sigma_d^2)$. We pick one of the $d$ distributions with each distribution having a probability of $\frac{1}{d}$ of being ...
0
votes
1answer
32 views

Computing posterior based on sum of multivariate normal distribution

Currently I am exploring topics for my undergrad thesis. Although I took a course in Bayesian statistics, I am not yet sure how to proceed in finding the posterior in the following case. I have a d-...
0
votes
0answers
27 views

Bayes Theorem in call center routing problem?

I want to route the call to the agent to maximize customer satisfaction. I have following data: Probability that customer is currently happy p(C+), we have this from sentiment analysis of the reviews/...
0
votes
0answers
10 views

Why dependencies would cancel while paramterizing CPD?

Consider that in CPTs of a Bayes Net(structure is fixed) one dependence is fix(parameters are set for $P(X|Y)$) and we are parameterizing other CPDs, is it possible that the other parameters would ...
0
votes
1answer
79 views

Why does the order of events A and B not matter with conditional probability? p(A | B) x p(B) = p(B | A) x p(A)

Please can someone explain why the order of events with conditional probability does not matter? p(A and B) = p(A | B) x p(B) = p(B | A) x p(A)
0
votes
0answers
18 views

Is there an error in those tables?

I recently came across this popular answer on Stack Overflow (1st result for profiling code linux on Google). I have some knowledge of Bayesian Statistics and I ...
0
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0answers
47 views

Why is the conditional distribution from a multinomial a binomial with these parameters?

Say we have a multinational distribution , $$Y=(Y_{1},Y_{2},Y_{3},Y_{4},Y_{5}) \sim multi(n,\frac{1}{2},\frac{\theta}{4},\frac{1-\theta}{4},\frac{1-\theta}{4},\frac{\theta}{4})$$ with observed $X=(...
0
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0answers
35 views

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 ...
0
votes
1answer
55 views

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 $p$ ...
0
votes
1answer
48 views

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 ...
5
votes
3answers
233 views

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 $x$ ...
2
votes
0answers
26 views

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 ...
0
votes
1answer
23 views

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 ...
1
vote
1answer
52 views

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: $$max[D^n] \le \theta \le 10 $$ ...
1
vote
2answers
67 views

clarification on Bayes theorem application

I'm experimenting with bayes rule. I have been invited to last stage interview in highly selective company. I want to use bayes rule to evaluate my chances of getting the post. Let's define the ...
0
votes
1answer
62 views

Basic probability theory

I was recently given the following statistics: On a particular highway, 18% of drivers are black, 63% of drivers searched by the police are black. So, a black driver is 7.7 times more likely to be ...
0
votes
1answer
146 views

How to create a distribution and sample?

Suppose we are given some small set of data on bundles of electrical wires and increasing voltages run through them, and we note how many of the individual wires fail. So for example, a large data ...
0
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0answers
32 views

How do I prove: the posterier is proportional to the product of the prior distribution and the likelihood function? [duplicate]

In the book of pattern recognition and machine learning equation (1.66) says: ...
2
votes
1answer
37 views

P(X=x|Z=z) given Z=X+Y are all rv's

Let $X,Y,Z$ be random variables where $Z=X+Y$ and $X,Y$ are independent. By Bayes' Law, $$ \begin{align} P(X=x|Z=z) &= \dfrac{P(Z=z|X=x)\ P(X=x)}{P(Z=z)}\\ &= \dfrac{P(Y=z-x)\ P(X=x)}{P(Z=z)} ...
0
votes
1answer
77 views

Why Does the $\propto$ Symbol Replace the $=$ Symbol When Using Bayes' Rule to Convert Posterior Density to Unnormalised Posterior Density?

My textbook says the following: In order to make probability statements about $\theta$ given $y$, we must begin with a model providing a joint probability distribution for $\theta$ and $y$. The ...
2
votes
1answer
118 views

A confusion about Bayes's theorem

I am reading a paper on the differences between bayesian outlook and frequentist outlook. The exact pic from the paper is: I have read a decent amount about what likelihood is and how it is not a ...
4
votes
2answers
50 views

Bayesian Statistical Conclusions: We Implicitly Condition On the Known Values of Any Covariates, $x$?

My Bayesian data analysis textbook says the following: Bayesian statistical conclusions about a parameter $\theta$, or unobserved data $\tilde{y}$, are made in terms of probability statements. ...
1
vote
1answer
45 views

Bayesian statistics: probability of next point

I am reading the Deep Learning book and having some difficulties with the following formula (page 134): $$ p(X^{m+1} | x^1, \dots, x^m) = \int p(X^{m+1} | \theta) p(\theta | x^1, \dots, x^m) d\theta. ...
1
vote
2answers
90 views

How to sample for conditional probability from unknown populations

I am providing the full question as well my solution below. I'm looking for help with part (d), a simulation question. Q - Suppose there are two species of Pandas, $T_1$ and $T_2$ which are ...
1
vote
1answer
78 views

What is the Bayesian Prior Predictive distribution from two normal populations?

The question goes as follows: A shoe factory produces brown shoes and black shoes. They look the same but differ only in their weight characteristics. Brown shoes have their weight distributed as ...
1
vote
0answers
17 views

How to understand the following Bayesian schema?

My knowledge of probability is basic, and I understand the point of Bayesian interpretation most roughly. The following is part of this paper. It is about how p can be rational for person 1 and not-p ...
0
votes
3answers
109 views

what does p( y | μ,σ²) really mean?

Just started to study Bayesian Statistics. I am very confused the concept of having a conditional probability on a distribution. Specifically: I understand what p( A | B ) where A="I am sick" and ...
0
votes
1answer
30 views

What can I conclude about the distribution of wrong phone numbers?

Let's say I have a list of 100 phone numbers. I call them all. Nobody picks up for 70. I get someone on the line for 30. Of those, 10 are wrong numbers. What can I conclude about the distribution of ...
1
vote
1answer
58 views

Objective function of Bayesian Model Averaging

I am quite confused about the objective function of the bayesian model averaging in the paper "Bayesian Averaging of Classifiers and the overfitting Problem".1 On the section 2, here is the first ...
5
votes
1answer
502 views

Posterior Predictive Distribution as Expectation of Likelihood

Say we have a posterior predictive density: $$p(\tilde{y}|\mathbf{y}) = \int p(\tilde{y}|\theta)p(\theta|\mathbf{y})d\theta$$ In Hoff's Bayesian Statistical Methods text, he suggests that to obtain ...
4
votes
1answer
97 views

Uniform distribution with Gaussian Priors

Let's say i've got a uniform distribution defined as follows $$X \sim U[\min (\theta_1,\theta_2),\max (\theta_1,\theta_2)]$$ I've also got that $\theta_1,\theta_2$ are i.i.d zero-mean normal ...
3
votes
1answer
71 views

Is it OK to update binomial probability with new data?

Take the baseball world series, which is out of 7 games. If the favored team has p(s)=.6 (take this as given), then the probability of winning 4 or more games is 70%. Basic binomial probability. If ...
4
votes
1answer
107 views

How many natural parameters are really in the exponential family conjugate prior?

The exponential family with natural parameter $\theta$ can be written $$ p(x|\theta)=h_\ell(x)\exp(\theta^Tt(x)-a_\ell(\theta)) $$ with conjugate prior $$ p(\theta|\lambda)=h_c(\theta)\exp(\lambda_1^T\...
3
votes
2answers
197 views

Bayes theorem confusion with likelihood [duplicate]

I learned that Bayes theorem was defined as follows : $$p(\theta\mid y)=\frac{p(y\mid\theta)p(\theta)}{p(y)}$$ But then today I came across definition with likelihood: $$p(\theta\mid y)=\frac{L(\...
0
votes
1answer
11 views

Applying probabilities to events

I recently thought about an example for a probabilistic graphical model that involved an alarm system for burglars. It roughly was a follows (it included an earthquake as well which is not relevant to ...
0
votes
1answer
53 views

Writing a conditional probability as a marginal

So I understand that using sum-rule one can write a probability as a marginal: \begin{equation*} P(x) = \int{P(x,\theta)d\theta} = \int{P(x|\theta)P(\theta)}d\theta \end{equation*} But how is this ...
0
votes
0answers
101 views

Odds ratio formulation of bayes rule for joint conditional distribution

I'm wondering how we can express the marginal conditional odds $O_{h|e}$ when $h$ and $e$ are part of a large joint distribution $P(h,e,z)$. I know that Bayes rule can be re-expressed in terms of ...
1
vote
1answer
77 views

Assuming n variables are conditionally independent given y, how do I compute p(y | x_1,…,x_n)?

Referencing this question, I know that if $x_1$ and $x_2$ are conditionally independent given $y$ (big assumption), then $$P(y | x_1,x_2) = \frac{P(x_1,x_2 | y)P(y)}{P(x_2 | x_1)P(x_1)}$$ $$ = \frac{...
2
votes
1answer
32 views

Estimating incidence rate for subpopulation from test calibrated on whole population

Formal problem: I'm given random variables $X\in\{0, 1\}$, $T \in \mathbb{R}$ and $S \in \{0, 1\}$ such that 1) $S$ and $T$ are independent given $X$, i.e. for both $x\in\{0, 1\}$: $$P(S\text{ and }...
21
votes
3answers
1k 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 ...
1
vote
1answer
1k views

Posterior vs conditional probability

When talking about events, there is the following formula called Bayes' rule, where $A$ and $B$ are random events: $$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$ Now let's say that for now only $A$ happened. I ...
2
votes
0answers
62 views

How would a frequentist solve this?

I think i understand what bayesian viewpoint is and what frequentist viewpoint is. But i always feel like i am missing something. I think there is a blind spot. so as an attempt :- Can somebody ...
2
votes
2answers
114 views

Applying Bayes's Theorem When Evidence is Uncertain

My limited understanding of Bayes's theorem, $$P(H|E)=\frac{P(E|H)P(H)}{P(E)}$$ is that—even though one of it's terms is $P(E)$—it's applied when $E$ is definitely known to be true. (You went to ...
0
votes
3answers
29 views

conditional probability calculation in terms of number of events

In trying to compute a discrete probability of some event $E$, call it, $P(E)$, one typically takes $P(E) = n(E) / n(S)$, where $n(E)$ is the number in the event, and $n(S)$ is the number in the ...
18
votes
3answers
3k 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) < ...