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 derive the posterior predictive distribution?

I often seen the posterior predictive distribution mentioned in the context of machine learning and bayesian inference. The definition is as follows: $ p(D'|D) = \int_\theta p(D'|\theta)p(\theta|D)$ ...
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Bayes' rule where the realization of random vector is a subset of the realization of a different random variable?

I have realizations of two different random vectors, where one is a subset (is that proper terminology here?) of the other $$s^t = (x_1,x_2,x_3,\dots x_\tau, x_{\tau +1},\dots x_t)$$ and $$ s^\tau = ({...
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Expected utility of action, given probability model

We record measurements of an appartus every day. If apparatus doesn't break (it has probability equal to $1-p_2$), it will measure zero with probability $p_1$. If apparatus breaks (probability $p2$), ...
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31 views

Find a conditional probability of a Bayes' Net knowing only the prior probability of the root.

Given three nodes A,B,C that form a Bayes Network as the following: (A)-->(B)-->(C) If we know the prior probability of A is 0.3, i.e. P(A)=0.3, is this ...
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30 views

Understanding Bayes' theorem through an example

Suppose I have three nodes A,B,C such that A and B are independent and pointed to C as the following: A --> C <-- B Also Suppose that each node takes a peobability between (0,1) so that the ...
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8 views

Interpreting data from a Gaussian Mixture Model using Gibbs sampling

I have data from a population with suspected subtypes within it. I have used a Gibbs sampler with different numbers of potential subtypes to produce Markov chains and posterior distributions. I am ...
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50 views

how to calculate expected utility for probability decision problem?

consider a decision problem: classifying $x$ as belonging to one of two classes $C_1, C_2$. there are prior probabilities for each class, $p(C_1), p(C_2)$ and likelihood probabilities for data given ...
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35 views

What assumptions did I make when I strengthened my independence criterion across a new random variable?

I have an algorithm which tries to calculate some $\operatorname{Pr(X | Y_1 Y_2 \dots )}$ (where juxtaposition means event intersection, "given $Y_1$ and $Y_2$ and ... have happened".) We have some ...
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66 views

Bayesian Approach: Is a die from a 3-D printer fair?

In a recent post "Fair die or not from 3-D printer"on this site @Eumel reported making a die on a 3-D printer, providing data on the faces seen in 150 rolls, and wondered about "the chances that the ...
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19 views

Bayes classification

What are the synonyms for 1) Bayes classifier 2) Bayes decision rule 3) Bayes decision function for uniform distribution I found many terms in literature and got confused because they look similar (...
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57 views

Simple example of “Maximum A Posteriori”

I've been immersing myself into Bayesian statistics in school and I'm having a very difficult time grasping argmax and ...
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51 views

Conditional Probability calcualtion

In the following BBN network, 1)what is meant by P(Martin Late|train strike,Norman Late)? Does this mean probability of martin Late given that Train Strike And ...
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48 views

Change of Variable technique for two variables?

If, $\theta_1 = ln \frac p{1-p}$ $\theta_2 = ln \frac q{1-q}$ $\theta_2|\theta_1 \sim N(\theta_1, \sigma^2)$ which means $f(\theta_1,\theta_2) \propto e^{\frac{-(\theta_1-\theta_2)^2}{2\sigma^...
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45 views

When do we expand the numerator of the Bayes' Theorem

I am trying to understand why the proposed solution below to the following question is wrong:- A box contains three cards: a card that is black on both sides, one that is white on both sides and a ...
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8 views

Bayesian serial link d-separation?

I don't get how I prove d-separation for a serial link: $$ (A)\rightarrow(B)\rightarrow(C) $$ I am trying to prove that if $B$ is known with certainty (hard evidence), then the probability of $C$ ...
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8 views

Bayes risk with loss function that penalizes all errors equally

Loss function $L(\alpha(x),y = 1$ if $a(x) = y$, else 0. If $y\in \{-1,1\}$, then $\sum_y L(\alpha(x),y)p(y|x) = -p(y \neq \alpha(x) |x)$. (taken from http://www.stat.ucla.edu/~yuille/courses/...
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22 views

Determining posterior gaussian distribution having marginalised over hyperparameters.

When applying gaussian process machine learning to regression problems where we want to determine the value a function $f$ takes at a new input point $x_{n+1}$, given observations of function values ...
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46 views

Integration in solving coin toss problem via Bayesian appoach

The following is taken from here: You have a coin that when flipped ends up head with probability $p$ and ends up tail with probability $1−p$. (The value of p is unknown.) Trying to ...
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17 views

Combining two Gaussian posterior distributions from different data to refine estimated distribution.

If we apply Bayesian inference to try and determine the distribution of a multivariate Gaussian $\textbf{x}$, and we have two predictions $$ \textbf{x}\sim N(\textbf{a}_1,\Sigma _1)~~ and ~~ \textbf{...
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32 views

How to work out $P(B\mid\neg A)$ using Bayes' formula

I am trying to work out the probability of something using Bayes' theorem: $$P(A \mid B) =\frac{P(B\mid A)P(A)}{P(B\mid A)P(A) + P(B\mid \neg A)P(\neg A)}$$ So in the question I know what $P(B\mid A)...
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32 views

Correcting multivariate distribution by additional info about its marginal

Assume that I have a posterior distribution $p(\theta_1, \theta_2|X)$ and I obtain an additional information in the form of a marginal density $q(\theta_1|Y)$ that is of the same type as $p(\theta_1|X)...
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Bayesian model to estimate the parameter of a Bernoulli law

Suppose we have iid boolean variables $X_1,...,X_T = X_{1:T}$ and the associated deterministic parameters $k_1,...,k_T=k_{1:T}$ and $c_1,...,c_T=c_{1:T}$, where for each $t \in \mathbb{N}$, $k_{t} \in ...
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Explain how the following expression was derived?

Can someone explain how the author gets to the expression after the words "This leads to:"
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27 views

Show covering number $N(\epsilon,\mathcal{P},h) < \infty$ for all $\epsilon >0$

Let $\mathcal{P} = \{P_{\theta}: \theta \in \Theta\}$ be a dominated model of distributions on $[0,1]$. For the parameter space $\Theta$ we have $$\Theta := \{\theta: [0,1] \rightarrow \mathbb{R} \...
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34 views

Conditional probability connecting three terms (chain rule)

How can I express $\Pr(a \mid c)$ in terms of $\Pr(a \mid b)$, $\Pr(b \mid c)$, and $\Pr(c)$? Is it possible? I'm thinking the chain rule might have something to do with it, but I'm having trouble ...
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29 views

Bayes Network 2 parents one child

I have the following Bayes network: S R \ / H I know that: $$ P(s) = .7$$$$ P(r) = .01$$$$ P(h|s,r) = 1$$$$ P(h|!s,r) = .9$$$$ P(h|s,!r) = .7$$$$ P(...
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26 views

A baisian estimation problem - How to formulate a Baisian estimation function for a given problem.

Two envelopes are given. Envelope 1 contains $x$ dollars and envelope 2 contains $2x$ dollars. We opened one of them and found in it $100$$. Now we have the option to change envelopes or not. ...
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175 views

Prove the estimator $\hat{B}$ of ridge regression = mean of the posterior distribution under a Gaussian prior

I want to prove that the estimator of ridge regression is the mean of the posterior distribution under Gaussian prior. $$y \sim N(X\beta,\sigma^2I),\quad \text{prior }\beta \sim N(0,\gamma^2 I).$$ $...
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22 views

Posterior for Beta Binomial Distribution with Repeated Observations

I'm working on a question with simultaneous learning about an underlying population and individual members of the population. The basic setup is: Let $N_g$ be the size of a population. At any point ...
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37 views

Can the parameter of prior probability depends on data?

In Bayseian approach https://en.wikipedia.org/wiki/Prior_probability we often use prior probability. Can we have a prior probability distribution with parameters and while estimating the posterior ...
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A Bayesian exercise

I have encountered the following problem in a book I am reading: Suppose you are offered to participate in the following game: Two fair dies are thrown untill '1' will apear (in one of them at ...
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30 views

How to find the posterior distribution

So suppose I have a coin that has a probability $\mu$ of landing on heads, and $1-\mu$ of landing on tails. I am giving the prior distribution $\mu$ ~ Uniform[0,1], and my realization $D_1 = \{H,T\}$....
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24 views

Assumptions leading to the mutual independence of random variables

I know that $P(AB) = P(A)P(B) \land P(BC) = P(B)P(C) \land P(AC) = P(A)P(C)$ does not imply $P(ABC) = P(A)P(B)P(C)$. But does $P(ABC) = P(A)P(BC) = P(B)P(AC) = P(C)P(AB)$ imply $P(ABC) = P(A)P(B)P(C)$...
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102 views

Bayesian probability with negative conditions

I'm trying to construct a probability model which analyzes signals if someone is in the neighbourhood. There are let's say 20 machines in the neighbourhood (of the wifi router) producing a wifi signal,...
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35 views

Perfect Bayesian Equilibria of the following game

Consider the following game between a monopolist firm and a consumer. Consumer's income is $1$, and he needs to allocate it between period 1 and period 2 consumption to maximize his utility $u(c_1,c_2)...
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31 views

Find parameters of the posterior Gaussian distribution

The question is to find $p(x|y)$ given that $p(x) \sim \mathcal{N}(\mu, \Sigma)$ and $p(y|x) \sim \mathcal{N}(Ax, \Gamma)$. I do realize that I may just obtain a posterior through application of ...
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24 views

What are the different ways to do a rating in a web application?

I tried to do an example using Bayesian Average in order to find the rating for 3 hotels. Following is my example, Hotel A 3 Votes/ 2 Star/ 1 Star,Rating- 3 Star Hotel B 1 Vote, Rating -5 Star ...
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18 views

Bayesian Analysis and Lindley's paradox?

So I have this problem to solve. Can anyone give me some hints on how to get started ? I have an understanding of conjugate priors and Driac function but have no idea how to apply it here.
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29 views

Bayes network predecessor relation

I have the following Bayes network. I know when "SHOES WET" becomes true, the probability of "GROUND WET" will change. But why will the probability of "RAINING" also change? And how can I ...
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116 views

Use Bayes's Theorem to Predict Success

I have a group of $n$ events. The successes don't all come in at once, and and I want to try to predict the actual success rate $s$. The number of successes showing in the system at any given time can ...
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31 views

Find the probability of binary bit when sent to channel

I have a question about Bayesian rule. My question is For a certain binary communication channel, the prob. that any bit sent is a $0$ is $0.49$. An error occurs with probability $0.08$ given that a $...
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58 views

Bayesian Inference and Disease Testing

I've been working my way through an introduction to Bayesian Inference in a Statistical Physics textbook (Tobochnik and Gould, 2010 - available online, excellent book). I've run across a problem that ...
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Bayesian Gaussian Mixture model

I am trying to fit basic Gaussian mixture with a Bayesian model. My likelihood function is Gaussian, with std=1, and the only parameter is the mean, chosen from $\{0,1,\dots,14,15\}$ and my prior is ...
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28 views

a new symbol for conditional independence?

I am reading Prof. Daphne Koller of Stanford University's book Probabilistic Graphical Models and Techniques where she uses a symbol |= like this: We say that an event α is conditionally ...
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Formulating simple games as Bayesian problems

Say you are in a game where every win doubles your money and every lose halves. You can walk away any time with the money you have by giving up? The rules of game is every step a problem is posed ...
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Bayesian inference on Poisson rate parameter

Given a Poisson arrival process of rate $\lambda$, and an observation of $k$ arrivals during an (arbitrary externally imposed) window of duration $t$, and prior p.d.f. $p(\lambda)$, is the following (...
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Choosing a non-convex global optimization algorithm based on the number of permitted steps

Can anyone comment on the most suitable approach for the following optimization problem: We are given finite bounds for a set of $n$ real-valued parameters of an unknown deterministic function. The ...
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1answer
37 views

Bayes Theorem hypothetical situation

Let's say I have a friend named Dave. There was a murder committed next door. Dave is most likely the killer. P(dave committed murder)=0.99 However, the probability that Dave would leave a blond ...
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66 views

.Decision Theory and Bayesian Inference

An organization uses a spam filtering software to block email messages that may potentially be spam messages. The spam filter can be set to one of two security modes: High-Security-Mode (H) or Low-...
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Simple derivation of empirical Bayes

I'm trying to derive the Bayes risk shown below in the first picture. From the definition of Bayes risk, in the next picture Here is my derivation of the Bayes Risk: $r(\tau,\delta^{\tau}(X))=...