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 ...

learn more… | top users | synonyms

1
vote
2answers
39 views

Is conditional probability $P(A\mid B)$ proportional to $P(B\mid A)$?

It feels a bit odd but since $$P(A\mid B) = \frac{P(A,B)}{\sum_A P(A,B)} \propto P(A,B)\text{ and }P(B\mid A) = \frac{P(A,B)}{\sum_B P(A,B)} \propto P(A,B)$$ can we say that $P(A\mid B) \propto ...
0
votes
0answers
10 views

Under what assumptions is the following first moment monotone?

I'm working on an economic model and am encountering the following mathematical issue. Let $x\sim \mathcal{N}(\mu,1)$, and define $$V(\mu)=\int_0^{\hat x(\mu)}x ...
0
votes
0answers
18 views

Is there any closed form for the integration of multiplication of two multivariate normal probability distributions?

I already computed the following integration but its a messy thing. I wonder if there is any easy way to compute it? or it has any closed form? V and p are known where V and p (p<1) are positive. ...
0
votes
0answers
15 views

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 ...
0
votes
0answers
24 views

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 ...
0
votes
1answer
27 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 ...
0
votes
1answer
28 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 ...
2
votes
1answer
26 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 ...
0
votes
1answer
40 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 ...
2
votes
1answer
21 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 ...
1
vote
1answer
21 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 ...
3
votes
2answers
63 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 ...
0
votes
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 ...
0
votes
1answer
37 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 ...
4
votes
1answer
106 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 ...
0
votes
1answer
26 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 ...
0
votes
0answers
13 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 ...
0
votes
0answers
28 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 ...
4
votes
3answers
456 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 ...
1
vote
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 ...
1
vote
1answer
44 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, ...
0
votes
0answers
8 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!
0
votes
0answers
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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
1answer
28 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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
10 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 $exp(1,\eta)$ ...
1
vote
0answers
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 ...
1
vote
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, ...
1
vote
3answers
24 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 ...
2
votes
1answer
34 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 ...
0
votes
0answers
13 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 ...
0
votes
1answer
42 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 ...
0
votes
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 ...
1
vote
0answers
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 ...
1
vote
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 ...
1
vote
0answers
32 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 ...
2
votes
1answer
56 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 ...
1
vote
1answer
40 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 ...
1
vote
0answers
27 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 ...
1
vote
2answers
31 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 ...
1
vote
1answer
24 views

Loss functions for regression

[From PRML Bishop, p:46] The average or expected loss function is given by $$E[L] = \int\int (y(x)-t)^2 p(x,t)\ \ dx\ \ dt$$, where, the loss function $L = (y(x)-t)^2$, given x and the ...
1
vote
0answers
17 views

Bayesian shrinkage doesn't affect eigenspace

The book Machine Learning a Probabilistic Perspective by Kevin Murphy on page 130 states following fact without proof: Consider the MLE estimate of covariance matrix $\Sigma_{\text{MLE}}$. The ...
0
votes
0answers
9 views

LDA with fixed topics?

Suppose I have a collection of "topic" probability distributions $\{\phi^{z}\}$ for LDA (Latent Dirichlet Allocation) that I have found via alternate methods; is there a closed form MLE for the ...
3
votes
2answers
62 views

Poker odds: Chances of a straight flush, given H4,H5

I'm trying to learn Bayes's formula, and am coming up with some poker problems to learn this. My problem is as following: given a $H4,H5$ ($4$ of hearts, $5$ of hearts) hand, what are the odds that ...
0
votes
1answer
28 views

Which is the difference between $P(A \mid B)$ and $P(A=t \mid B)$ in a Bayesian Network?

Which is the difference between $P(A \mid B)$ and $P(A=t \mid B)$ in a Bayesian Network, where $A$ and $B$ are boolean values?
0
votes
0answers
6 views

Gaussian Process: Using partitions of a choleky decomposition solution for conditional deduction.

If I define a GP over observed values, $y$, of a sensor reading over time, $t$, as (for simplicity assuming discrete time series e.g lets say readings after every 5 mins) : $y=f(t)+\epsilon$ where ...
0
votes
2answers
52 views

Why would a uniform prior distribution give a different result than a purely frequentist approach?

I would expect a uniform prior to be a good example of an uninformed prior and get the same result as the frequentist approach. However, this is not the case. As an example, let's look the classical ...