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

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Forward-Backward algorithm for CDFs

Suppose I have some densities $\alpha(x)$ and $\beta(x)$. I need to compute conditional density over the state space that is said to be governed by Bayes rule: $$\gamma(x) = ...
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23 views

How can we prove this equation using marginalization and conditioning? [on hold]

I want to prove $$P(A|C) = \sum_{B} P(AB|C) $$ How can we prove this using marginalization and conditioning?
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31 views

Intuition of Bayesian normalizing constant

In the commonly mentioned mammography screening problem with a screening likelihood of 80%, a prior of 10% and a false positive rate of 50%, or its variants, it is easy to explain that the conditional ...
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3answers
251 views

Monty hall problem, getting different probabilities using different formulas?

In my Monty hall problem, I am computing what is the probability that P(H=1|D=3) i.e. price is behind door 1 and the 3rd door is opened. $P(H=1|D=3) = p(H=1) * \frac{p(D=3|H=1)} {p(D=3)} = 1/3 * ...
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1answer
40 views

Hidden Markov Model and Naive Bayes similarity

I understand Naive Bayes classifier and already have made a few implementaions. What i dont understand is, considering that i have a train set with all the X observations and Y states, what stops me ...
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1answer
105 views
+50

allocating degree belief to forecast value

We have a number of providers for a forecast of wind power generation per country per date. Values are forecast up to one week ahead. Forecasts may be compared with actual values of reported wind ...
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40 views

Bayesian Approach To Combine Multiple Weighted Inputs

I'm beginning to learn about Bayesian theory but I'm stumped on the ideal approach for combining multiple weighted inputs. Here's an example to make this more concrete. Let's say that I want to ...
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Marginalizing versus optimizing the nuisance and hyper parameters

In a Bayesian setting, the classical approach when studying a given variable of interest (here $\theta_0$) consists of marginalizing the full posterior over nuisance and hyper parameters (here ...
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2answers
56 views

What am I doing wrong in this Baye's theorem equation

I have to find the probability that that a woman does not have cancer, given that her mammogram came back positive. Facts needed: 0.1% of all women have breast cancer probability of a positive ...
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17 views

Best approach for calculating confidence score (p-value) - for Bayes models

This is a generic question on the best approach for giving a p-value to a "most likely" solution given by a graphical Bayes model. I am building a graphical model in Stan, for finding the most likely ...
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Comparing results from reference coding and orthogonal coding in a linear model?

The problem: I'm trying to fit a zero-inflated negative binomial model to count data (catches of larval fish). I have three factors, and an offset variable, which is the volume of water filtered by ...
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1answer
36 views

How to do calculate both causal and diagnostic inferences simultaneosly in bayesian networks?

Consider a simple Bayesian network as given below. Question: How to find $P(S|C,W)$? It is fairly straight forward to compute the causal inference $ P(W|S) = P(W|S,R)\cdot P(R) + ...
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Questions regarding the Bayes Classifier in *Introduction to Statistical Learning*

I am having trouble grokking some very elementary material regarding Bayesian Classification in Introduction to Statistical Learning at the end of pg. 37 to the very top of pg. 39 (i.e., the section ...
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45 views

How to compute probability

I have a dataset consisting of 4000 observation from each 324 continuous features are extracted. Each observation has been labeled a class. Since each feature from that dataset is continuous, have I ...
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1answer
30 views

Calculate Gini coefficient - Python

given a distribution of $\sigma^2$ ($y \sim LogN(\mu, \sigma^2$) I want to calculate the posterior distribution of the Gini coefficient which is given by: $G = 2\Phi(\frac{\sigma}{\sqrt{2}})-1$ ...
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54 views

Kneser Ney smoothing, why the maths allows division by 0?

I'm studying Natural Language Processing and the various smoothing approaches. I'm finding a little hard to understand how to handle unknown words with the Kneser-Ney smoothing. In particular I'm ...
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1answer
18 views

Confusion about the concept of Bayesian Networks

I read a lot about Bayesian networks, including focused literature. However, what I have not yet understood properly is this: What is the use case of Bayesian networks that contain of more than 2 ...
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30 views

Calculating likelihood in R

I want to "by hand" calculate a Bayes factor in a simple case. I'm sure I'm off somewhere in my calculation of likelihood of data under H1. I think I don't "scale" my likelihood under H1 correctly. ...
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How to store a large graph? [closed]

When I try to develop a spelling collector, I realize if I can include the word background (etc:before and after word) into the bayes calculation, may be could increase this spelling collector ...
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1answer
18 views

Bayes decision rule and thresholding

The best possible classification is for a set of samples drawn from any probability distribution is given by the Bayes decision rule. For any distribution, the rule is given by $f(x) = 1 ~if ...
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1answer
41 views

Bayesian inference when observed variable contains uncertainty

I have a very simple graphical model to describe the relationship between two categorical variables $c \in \{0,1\}$ and $l \in \{A,B,C\}$: $$c \rightarrow l$$ I know all the conditional ...
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1answer
32 views

Bayes theorem with unknown model

I have a book on applied statistics that uses a result I don't understand. The example in the book begins with Bayes' theorem applied to hypothesis $H$ and data $D$: $$P(H|D) = \frac{P(D|H) ...
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118 views

Can I make a decision using a Bayes factor?

Bayes factors denote how well a certain model is supported. Say that I am running a controlled experiment and I have two models: the null model and the alternative model. If I have a high Bayes ...
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23 views

Simple Marginalization Explanation Please

I am having difficulty understanding this marginalization. Let's say you have this relationship $p(a,b,c) = p(a)p(c|a)p(b|c)$ From that you are trying to get $p(a,b)$ $p(a,b) = p(a) \sum\limits_{c} ...
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51 views

Prior Parameters in Bayesian Hierarchical Linear model

I'm trying to fit a linear model to describe student performance in 2 different schools. My response variable is $$Yij= X{ij}*\beta+Z_{ij}*\gamma_j + \epsilon_{ij}$$ . $$i = 1,...,n $$ $$j = 1,2 ...
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Problem using continuous bayes theorem on multivariate normal distribution

Given three correlated normally distributed variables $ x$, $ y$ and $z$ such that: $ p\left(\begin{bmatrix} x \\ y\\ z\end{bmatrix}\right) \sim \mathcal{N}\left(\begin{bmatrix} 0 \\ 0\\ ...
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Bayesian Inference reference to self-study

I need a reference request for bayesian inference for self-study, I'm look for something containing: 1-Fundamentals about prior and posterior distribution, conflict between prior and posterior 2- ...
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55 views

Bayes' theorem problem

Three companies supply same type of products. Company A supplies 30 % more products than company B. Company C supplies the same as A. Fraction of defective products produced by company A is 4 %, by ...
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31 views

How to evaluate probabilistic classification of multi-class problem

I have 3 classes $c = \{A,B,C\}$ and use a bayesian formulation to achieve a probabilistic outcome $p(c \mid x)$ based on some observation $x$. I know there is some bias in there which I haven't ...
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50 views

Bayes theorem applied correctly on client churn?

Here is a table selected and grouped from table where i store information about client - if he churned(TRUE - he churned, FALSE he stayed) and how many refund he got. CNT counts number of rows per ...
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JACKMAN: ordered logit model, how do I get proper thresholds?

I'm working on a logistic regression model, using as a reference this Jackman lesson http://jackman.stanford.edu/classes/BASS/ch8.pdf . I implemented his jags function on my data (student scores ...
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Probability of non recurrence given a particular time of non-reccuring

I've been having a cardiac problem that may (or may not) be linked to alcohol consumption. If its happening about once a week on "social use", how long do I have to be completely alcohol free and ...
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Can frequentists use Bayes theorem?

I understand that Bayesians interpret probability as belief, which can be updated with evidence. We happen to compute the posterior with Bayes theorem. What is the difference between Bayes Theorem ...
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Calculating conditional probability through time

Suppose I can obtain non-parameteric estimates of a probability distribution at different horizons, e.g. $\rho(x, \tau)_{t=0}$ is the probability density seen at time $t=0$ for the variable to have ...
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30 views

Posterior from a Poisson likelihood and prior

I have the following Poisson mass function: $$p(y| \theta) = \frac{\theta^y e^{\theta}}{y!} $$ Which has a corresponding likelihood for n independent realizations of y as follows: ...
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50 views

Question on how to apply Bayes Theorem realistically

So I'm new to Bayes Theorem and am trying to understand it. For simplicity I'll refer to the usual example of testing for cancer. ...
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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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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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36 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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32 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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1answer
81 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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41 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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41 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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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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59 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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41 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 ...