Combining probabilities with Bayes' Theorem, especially as used for conditional inference.

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How Do I choose parameters of prior on regression coefficients in a Bayesian linear model?

I'm trying to perform a linear regression in a Bayesian way. The response is normal,the prior I would like to put over Beta (vector of regression coefficients) and Sigma^2 (variance of the error ...
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When is BIC reasonable approximation to evidence?

I've recently seen a few papers in physics using the Bayesian information criterion (BIC) to evaluate models. I'm much more familiar with Bayesian evidence, $p(x|M)$. I've read in a few places, e.g. ...
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Classic medical Bayes Theorem question with a twist [duplicate]

Typical statistics question about taking a test to detect a deadly disease and what are the chances that you have the disease given a positive result. Now I ask what happens when you take the test ...
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30 views

What is the Bernoulli class conditional distribution?

What is the Bernoulli class conditional distribution? I am trying to implement a procedure for computing a naive Bayes classifier for binary features with a Bernoulli class conditional distribution. ...
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29 views

Find Bayes rule/action under given prior

I am able to solve for Bayes actions/rules with no data and am able to follow problems with simple data. However, I'm not sure how to solve a question where the data, $X$, is conditional on the state ...
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24 views

How to calculate $P(A)$ given only $P(A|B)$ , $P(A|B')$ and $P(B)$?

Assume $A$ and $B$ are two dependant events with only the following details provided $P(A|B)$, $P(A|\neg B)$ and $P(B)$ How to calculate the value of $P(A)$?
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Calculating elasticities from spatialprobit Bayesian coefficients

I have run a model in R using the spatialprobit package and would like to calculate elasticities with respect to some of my coefficients of interest. I am a bit ...
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29 views

Why is Bayes Classifier the ideal classifier?

It is considered the ideal case in which the probability structure underlying the categories is known perfectly. Why is that with Bayes classifier we achieve the best performance that can be achieved ...
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1answer
32 views

What is the difference between a Naive Bayesian Classifier and Bayesian Linear Regression?

The difference between linear regression and Bayesian regression is that: Linear regression: You are trying to minimize (y-bx)^2 Bayesian regression: You are trying to do similar given a prior on ...
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Assign the direction and sign (+,-) of an arrow using Bayes rule [duplicate]

Being two vertex: A,B and one arrow -> that can be positive or negative, (i.e, it can activate or desactivate). How I can compute the associated probability of each posible combination: A->+B (A ...
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26 views

Buckets of dice - Binomial probabilities of either event occuring

I'm doing a bit of analysis on a dice rolling mechanic for a role playing game. This is a buckets of dice system where the result is based on the count of the number of dice rolling a given target ...
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24 views

Bayes Rule clarification

On page 12 of this tutorial, it states $$P(\pi \mid \mathbf{L}; \gamma_{\pi1}, \gamma_{\pi0}) = P(\mathbf{L} \mid \pi) P(\pi \mid \gamma_{\pi1}, \gamma_{\pi0})$$ I'm having some trouble seeing why ...
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36 views

Bayesian Hierarchical Clustering with mixed data

I want to perform Bayesian hierarchical clustering in R. I have 4 variables that 3 of them are nominal and 1 is discrete. So my data are mixture of nominal and discrete data. Data are: ...
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1answer
37 views

How to calculate prior probabilities

I have two possible events $A$ and $B$ that could lead to $n$ possible consequences $X_1, X_2, \ldots , X_n$, $P(A) + P(B) = 1$, $P(X_1) + P(X_2) + \ldots + P(X_n) = 1$. I know all conditional ...
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1answer
52 views

MCMC advice: Ignoring some parameters in a MCMC scheme?

I am after some general advice regarding my MCMC scheme, which is causing me some grief. Essentially, I have a large (2N + 9 parameters) MCMC scheme which works great. However, the problem is that ...
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For each x, I observe A and know P(C). What can I say about E(A|C)?

For each subject x in a population, I observe x's age, A(x). I can calculate the probability that x has some property of interest c, $P[C(x) = 1]$, where C(x) is a binary variable indicating ...
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314 views

Bayes Theorem with multiple conditions

I don't understand how this equation was derived. $P(I|M_{1}\cap M_{2}) \leq \frac{P(I)}{P(I')}\cdot \frac{P(M_{1}|I)P(M_{2}|I)}{P(M_{1}|I')P(M_{2}|I')}$ This equation was from the paper "Trial by ...
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Strange election results and probability of election fraud

Suppose an election is held for the leadership position in a major political party. Four candidates are running. After the election, the following results are announced: ...
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Making inferences from Bayes networks and CPTs

So I'm practicing working with Bayes Networks and conditional probability tables and I feel like some of my numbers simply don't make sense. Here's the situation: I have a bag of three different ...
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76 views

Practical Bayes Theorem - how to use it?

I am testing a variant of LDA (Latent Dirichlet Allocation) algorithm on some text data. Given some documents $d$ where each document is expressed as a distribution over words $w$, the algorithm ...
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172 views

Interpretation of Bayes Theorem applied to positive mammography results

I'm trying to wrap my head around the result of Bayes Theorem applied to the classic mammogram example, with the twist of the mammogram being perfect. That is, Incidence of cancer: $.01$ ...
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67 views

Different Confidence vs. Credible Interval (Continuous case, noninformative prior) [duplicate]

Okay, so, credible intervals aren't the same as confidence intervals. We all know that. In fact, they're only guaranteed to be the same when they're about a location or a scale parametre with a ...
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Inferring intermediate probabilities in a real-time system if the ending distribution is known?

For example, Let's say I have a bar of 4 LED lights that light sequentially, as below: [1] [2] [3] [4] Meaning that if [3] is lit, both [1] and [2] are also lit, etc. During some period of time - ...
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Calculating Bayes Factor from a correlation coefficient

I'm wondering whether anyone knows whether it is possible to directly calculate a Bayes Factor (comparing null model of zero correlation to non-zero correlation) given just a correlation coefficient ...
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114 views

What can we conclude from a Bayesian credible interval?

I learned that the credible interval does not have the frequentist property, but recently I read the following statements that derived from the credible interval/region: Point (0,0) is on the ...
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68 views

Bayes theorem in odds form - incorrect in Tetlock's 'Superforecasting' book?

Page 170 in Philip Tetlock's et al. Superforecasting book shows Bayes' theorem in odds form as: $$\frac{P (H|D)}{P (\neg H|D)} = P (D|H) P (D|\neg H) \frac{ P (H)}{P (\neg H)}$$ Posterior Odds = ...
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why use Mean of posterior distribution instead of probability?

I'm reading the Think Bayes by Allen B. Downey, and on this example I don't understand well the purpose of Mean in the chapter 3.2 The locomotive problem. ...
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30 views

Prior pdf decay in Recursive Bayesian Estimation

I'm doing Recursive Bayesian Estimation numerically. I have a state vector, x, that I'm trying to estimate by regularly taking noisy measurements, z. I use Posterior = Likelihood x Prior / ...
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30 views

How to compute the likelihood distribution?

I am not an expert in statistics, but I need the bayesian inference to resolve a problem in artificial intelligence. The problem is not about estimation, no. The problem is about computing the ...
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1answer
34 views

Determine probability values for use in Bayes rule

In most examples I find, the various probabilities are just there and it is a simple question of applying Bayes rule to those numbers. However, I'm having a hard time finding details about how to ...
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23 views

Which is a better method to combine probability in Naive Bayes?

The Wikipedia article "Naive Bayes Spam Filtering" https://en.wikipedia.org/wiki/Naive_Bayes_spam_filtering#Mixed_methods) mentions two methods of combining probability that addresses underflow ...
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Simple and direct application of Bayes Theorem

Suppose that $\theta \in \Theta=\{0,1\}$ such that $P(\theta = 0) = 0.1$. Let $X$ be a r.v such that, given $\theta=0$, $X \sim N(50,1)$ and given $\theta = 1$, $X \sim N(52,1)$. Show that the ...
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81 views

Case-Control Likelihood Function Logistic Regression

I'm reading Applied Logistic Regression (2nd Edition) by Hosmer and Lemeshow and I can't follow the derivation given in section 6.3 Case Control Studies, equtions 6.2, 6.3 and 6.4: It is not ...
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Empirical Bayes

This question originates from page 4 of Efron's book (Large Scale Inference, Empirical Bayes methods for estimation, Testing and Prediction) Let $z_i|\mu\sim N(\mu,1)$ and $\mu\sim N(B,A)$ for ...
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Bayes Factor and Hypothesis testing

This is apparently true a story. In Year 1998, 25 studies were funded to find a link between microwaves and brain tumor. There was no basis to reject the null hypothesis, and xx millions were spent to ...
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Using Bayes' rule to jointly update two parameters from a single signal

For my masters' thesis, I am playing around with the following model: A player undertakes an action that is a success with probability $a$, and a failure with probability $1 - a$. On failure, the ...
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183 views

Help with a proof of Bayes classifier optimality

I have a class assignment to provide a proof that Bayes classifier for the two label version is optimal in that it's error rate is always ${\le}$ any other classifier. I've never worked through a ...
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73 views

To show that Bayes classifier has best error rate

Show that the bayes classifier will achieve the best error rate, defined as: $$ E(f) = \int \int \mathbb{I}(y = f(x)) \cdot p(x, y) dxdy $$ where f(x) is the classifier, and p(x, y) is the intrinsic ...
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Prior on effect of treatment using CausalImpact library in R

I'm using the package causalImpact in R to estimate the causal effect of an intervention in a time series. However, I have strong prior information that the effect can't be negative. How can I encode ...
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49 views

updating a Bayes factor

A Bayes factor is defined in Bayesian testing of hypothesis and Bayesian model selection by the ratio of two marginal likelihoods: given an iid sample $(x_1,\ldots,x_n)$ and respective sampling ...
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1answer
75 views

Posterior distribution in Bayesian linear regression - why not include $p(X | \beta, \sigma^2)$?

Given parameter/s $\theta$, data $X$ and prior on the parameter/s $p(\theta)$, Bayes' theorem allows us to estimate the posterior distribution $p(\theta | X)$: $p(\theta | X) = \frac{p(\theta) p(X | ...
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29 views

Finding the posterior distribution, several priors

Bayes' rule $$ \pi(\theta|y)\propto\pi(y|\theta)\pi(\theta) $$ gives the posterior distribution. However for the first time I have encountered a problem where I have two parameters (because the ...
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85 views

How to find p(A|C) from known values for p(A|B) and p(C|B)?

Suppose, in some problem with discrete variables, I have found numerical answers for p(A|B) and p(C|B), and have already analytically derived the numerical answers for all their respective associated ...
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44 views

Role of the Objective Prior in the Unification of Bayesian and Frequentists

On a recent post asking about self-identified Bayesians (most comprehensive list: ISBA) there were great answers, yet little was mentioned about the drive towards unification. Except from Andrew ...
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Variance of Bayesian posterior

Setup Let $f(\theta)$ be a prior on $\Theta$, and $X_1,\dots,X_n$ are iid according to $P_\theta$. From Bayes' rule, we derive the posterior as $$ g(\theta\mid ...
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57 views

Confusion regarding correct conditional probability expression

Let T be an event whose occurrence is to be investigated and the probability with which a witness(W) tells the truth is $p$. Let E be the event of witness testifying that the event T has occurred, ...
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What are frequentist approaches to logistic regression besides MLE?

Related to these two questions: Bayesian logit model - intuitive explanation?, How can Bayesian inference improve upon logistic regression in incorporating psychometric data? From what I know and ...
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weighted bayes theorem

I am using a simple implementation of Bayes theorem to find the discrete probability distribution of proportion of wins. ...
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How can Bayesian inference improve upon logistic regression in incorporating psychometric data?

In order to estimate probability of default, banks and other financial institutions have often used logistic regression based on data involving credit history. Suppose an entrepreneur or borrower ...
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Finding the Complicated Posterior Probability Distribution of $θ$

Suppose, we are given a likelihood function, $f(x|θ)$ corresponding to a shifted-exponential distribution and the prior distribution on the parameter $θ$ is a standard Cauchy distribution. Now I am ...