The approach and interpretation of probability associated with Bayes theorem; usually used as opposed to the frequentist approach. It can be seen as an extension of logic that enables reasoning with propositions whose truth or falsity is uncertain. A Bayesian probabilist starts with some prior ...

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How to get more profit in stochastic process?

Suppose there is a system, for each step, I cost something but I didn't know how much I cost, and the system return to me something, which follow Guassian distribution and the expectation is what I ...
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1answer
24 views

simplify the division of popular probability density function

This is my first question in Mathematics on Stack Exchange. Please forgive that this is a none sense question... Question I'd like to know a simple form of the division of popular probability ...
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1answer
18 views

How to do continuous-time Bayesian updating?

I am reading a game theory lecture notes. Some parts involve a continuous time Bayesian updating computation which I didn't really understand. There are two states $\{Good,Bad\}$. At time t people ...
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1answer
406 views

Revising probability using Bayes' rule

Let's say $100$ numbers is picked from a set of numbers one by one with replacement, which is between $000$ to $999$. I'm required to give my subjective probability before the experiment and revise ...
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23 views

Bayes Probability computation [on hold]

I received a HW in which i have some problems. I will be very grateful if anyone could help me out. This is the question: Consider a line segment whose length equals 1. We throw a first ball that ...
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1answer
27 views

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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1answer
25 views

Difference between Frequentist and Bayesian approach to Statistics

What is the difference between the Frequentist vs. the Bayesian approach to Statistics? Would someone be so kind to come up with a simple example that shows how the approaches and possibly the ...
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1answer
37 views

Maximum likeliood estimation of variances of transformed variables

I use MATLAB's fminunc function in order to find the minimum of a negative log-likelihood function $f(\overrightarrow{\theta})$, parametrized by 3 parameters lets say ...
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1answer
103 views

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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1answer
20 views

Trouble understanding how Naive Bayes Classifier is derived

I've come across the Naive Bayes Classifier while studying machine learning, but the trouble I'm having is with some of the probability theory used to derive the formula for finding the optimal ...
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2answers
60 views

How can Bayesian and Frequentist approach be different?

Let's say I am trying to add numbers, like say one to ten. I can either add them in order, or I can notice that I can group them into five groups of eleven, so I suppose which method to use depends on ...
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2answers
295 views

Improper Uniform Prior Distribution

In Bayesian method, choosing the prior distribution is an important step when using the Bayesian method. When choosing prior, we consider the prior knowledge to choose which prior distribution is the ...
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2answers
826 views

Calculating Laplace's law for bigrams

Reading this PDF, I encountered a very simple simplification that I can't obtain. Basically, it asks what is the probability of the occurrence of a word $w_{n}$ given that we know another word ...
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1answer
15 views

What prior to use given a Poisson likelihood?

I am trying to incorporate a prior into a model I am working on. From available data, I have found that the likelihood follows a Poisson distribution with $\lambda = 1.5$. I have then used R to ...
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1answer
33 views

Cluster probabilites: Bayesian network (sprinkler example, Russel/ Norvig) as a clustered network

like others here I am also learning with Russel's and Norvig's book about artificial intelligence. My question is about the conditional probability tables of a clustered multiply connected network ...
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1answer
24 views

Bayesian Networks - Probability of variables with a common parent

I'm having some trouble figuring out a homework assignment which requires me to find the probabilities of two different variables that have a common parent. In order to better understand how to do ...
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0answers
22 views

Textbook recommendation for Non-parametric Bayesian?

I am looking for textbooks which include basics as well as advanced models like latent Dirichlet allocation, hierarchical Dirichlet process. The most important thing is that those books should present ...
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0answers
12 views

Inverse-Wishart distribution pdf is different if we derive it directly from Wishart distribution?

According to Wikipedia, there is the following relation between the Wishart and the inverse-Wishart distribution: "If ${\mathbf A}\sim \mathcal{W}({\mathbf\Sigma},\nu)$ and ${\mathbf\Sigma}$ is of ...
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0answers
26 views

Bayesian probability and coin toss

Assume that John and Mary, not knowing anything about fairness of the coin, have common prior of obtaining H (heads) in coin toss equal to $\frac{1}{2}$. Before tossing a coin, each of them is allowed ...
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105 views

The Bayesian approach to parameter estimation [closed]

Can someone show me how to do this question? Suppose that the waiting time in a queue is modeled as an exponential random variable with unknown mean θ, and that the average time to serve a random ...
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1answer
35 views

Inferring poisson rate from interval determined by data

I have a dataset of arrivals, which are from a Poisson process. For the purposes of this question, let's say they're arrivals of cars on a particular road. My goal is to infer the gamma posterior for ...
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3answers
451 views

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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1answer
43 views

Bayes vs frequentism and the fair coin

Suppose I have a coin, which I want to test for bias. My problem is: surely there's a philosophical problem with defining "bias". Let me illustate with an example. Firstly, I use a Bayesian approach, ...
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1answer
118 views

How this equation is derived [closed]

Can any one kindly explain how this equation [1] is derived. Has it something to with Bayesian network? Where this equation is normally used? [1] http://d-nb.info/997569794/34
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1answer
60 views

Uniform lattice sample inside a particular convex polytope

[update]: hardmath suggests using tools from linear programming. This looks like a good idea indeed. I can now tell that my feasible set is described by: $Set = \{d \in \mathbb{N}^c, -B.d\leqslant ...
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7 views

Discriminant Functions of two classes sharing same covariance matrix

How can i find the discriminant functions of two classes having same diagonal covariance matrix with different means? (their feature vector is two dimensional) Thank you!
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19 views

Log likelihood function for binary classification

I need help with this following task. There is a binary classification problem where each observation xn is belong to one of two classes (t = 0 and t = 1). The training data points are sometimes ...
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1answer
25 views

how to calculate crash probability?

A plane crashes with probability 0.95 if both of its engines fail. On each flight each engine has a probability of failure of $10^{-5}$. Both engines fail with probability of $10^{-9}$ a) Are the ...
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1answer
22 views

using bayes' rule

Women carrying a certain gene are ten times more likely to develop breast cancer. Only 1 out of 100 women carries this gene. If a woman has breast cancer, what is the probability that she carries this ...
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1answer
26 views

Bayesian networks: What's wrong with my answer?

Consider The following four random binary variables: Given the following Bayesian network: With the following conditional probability tables: I want to calculate the probability that ...
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1answer
47 views

Posterior distribution of $\theta$

Let $X_{ij} ~ N(\theta_i,\sigma^2)$ with $\sigma^2$ known, i = 1,... k, and j = 1, ... ,$n_i$. The prior distribution of $ \theta_i$ is $N(\phi,\tau^2)$, independently for i = 1,...,k and ...
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9 views

Finding a bayes estimator

Let $X_1,...,X_n|\eta~\exp(1,\eta)$ and $\eta$~$N(\mu,1)$, where $\mu\epsilon\Re$. Find the Bayes estimator $\eta$ under the squared error loss. After finding the joint likelihood of $exp(1,\eta)$ ...
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50 views

Baye's Classifier for recovering a signal from a measurement

Below is the question i am having trouble with: Independent and identically distributed symbols s(n) = ±1 are transmitted over the channel C(z) = 1 + z −1 . Symbols s(n) = +1 occur with probability p ...
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1answer
33 views

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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1answer
32 views

Bayesian hypothesis testing

Let $x_1,\ldots,x_4$ be a sample taken from the uniform dstribution with the density $$ f_{\theta}(x)=\theta^{-1} \cdot 1_{(0,\theta)}(x). $$ Assume that $\theta$ is a random variable with the ...
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3answers
23 views

Conjugate priors: wht not binomial-binomial?

Citing from Kevin Murphy's machine learning book: When the prior and the posterior have the same form, we say that the prior is a conjugate prior for the corresponding likelihood. Conjugate ...
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12 views

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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1answer
82 views

Parental Markov Condition Example

I'm currently reading a text on Bayesian networks and the text is giving some very crude interpretations of what appear to be some of the most important foundations of the subject. It states the ...
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1answer
39 views

How to prove Laplace distribution is scale mixture of Gaussians?? [closed]

How does one prove the Laplace distribution is a scale mixture of Gaussians? I.e, how does one show that $X \sim \text{Laplace}(\lambda)$ is a scale mixture of Normal $Y \sim N(0,\tau)$ and ...
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0answers
55 views

How do I put together a set of modified conditional distribution into a single joint distribution?

I am abstracting my original problem to a simple scenario. Consider a bivariate multi-modal mixture of gaussian distribution, $P(x,y)$. When we slice through $x$ or $y$ we get a univariate multimodal ...
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141 views

Conditional probability given two variables from conditional probabilities of one variable

I have a question which comes from an example of a textbook, but all I am concerned with is how we go from having probabilities $P(X|A)$ and $P(X|B)$ to having $P(X|A,B)$. In the example events $A$ ...
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34 views

Evidence propagation in bayesian network

I'm currently trying to wrap my head around evidence propagation in bayesian network (simple tree propagation) but I'm having trouble finding information about the process. As an example, let's take ...
2
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1answer
55 views

Brownian motion and posterior distribution

I am a bit stuck on this question: Suppose that $X_t = W_t + \alpha t$, where $W$ is a standard Brownian motion, and let $\mathcal{F}_t = \sigma ( X_u: 0 \leq u \leq t)$. The drift is constant in ...
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2answers
33 views

How do I combine assertions of experts based on trustworthiness?

5 friends have come up to me and asserted that "Fred is coming to visit tomorrow". The more people I hear it from, the more I believe it to be true. How do I model this probabilistically? I think I ...
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31 views

complicated posterior distribution

I have a question concerning a rather specific posterior. It should be a simple application of Bayes' Theorem. However, I am always confused here. I try my best to describe the situation. There are ...
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0answers
25 views

Rate of convergence of Bayesian posterior

Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let ...
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1answer
35 views

Terminology: Probability “with respect to a measure”

The following excerpt is taken from Shen and Wasserman (2001). I have difficulty understanding some terminologies. On line 4, [...] each $P_\eta$ is a probability on $(\mathscr Y,\mathscr ...
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2answers
28 views

Bayes theorem with multiple variable question

The below formula is from an article that i red for my work. The author said he used Baysian theorem to get this, but I have no idea why this is true! Can someone please clarify how the first ...
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1answer
34 views

How many numbers for the full joint?

Suppose you have 3 binary nodes A, B, C. A and B are independent given C. How many numbers do we need for the full joint? How many numbers do we need for the Baysesian Net? I know the answers to ...
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1answer
47 views

Conditional probabilities given the evidence(Bayesian network) [closed]

Let's say we have a Bayesian network: How can I compute P(A | F, E) ? I have all the probabilities for each node. Thanks!