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

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51 views

Can I use Bayes Theorem to find a conditional distribution rather than conditional probability?

I might be going about this the wrong way, but I'm trying to develop an understanding of a particular conditional value, say $P(CustomerBuysFries | CustomerBuysHamburger) = P(F|H)$. Ultimately, I want ...
3
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3answers
66 views

Bayesian logit model - intuitive explanation?

I must confess that I previously haven't heard of that term in any of my classes, undergrad or grad. What does it mean for a logistic regression to be Bayesian? I'm looking for an explanation with a ...
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1answer
30 views

how to minimize the probability of error in a Bayes decision rule

According to the Bayes decision rule for a 2 class classification problem: $d(x) = w_1 : P(w_1 |x) ≥ P(w_2|x) $ And $P(error|x) = min[P(w_1 |x), P(w_2|x)]$ where $P(w_i |x) = p(x|w_i) * P(w_i)$ ...
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44 views

why use diagonal $\Sigma$ when working with Bayes decision theory?

My prof. said in the class that for Bayes decision rule, the likelihood is Gaussian and in practice, we will almost always work with a diagonal $\Sigma$. Why is that? I know that a diagonal $\Sigma$ ...
1
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1answer
32 views

pattern classification when the prior probabilities are not equal

In the case of 2 class classification, the decision boundary occurs when 2 discriminant functions are equal: $$ g_1(x) = g_2(x) $$ $$ g_i(x) = p(x|w_i)P(w_i) $$ $$ p(x|w_i) = ...
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12 views

Varying transition probabilities by position

I'm still very new to Bayesian Tables, Hidden Markov Models and the likes, but have an otherwise solid computational and linguistics background. I've been diving into NLTK (Natural Language Toolkit) ...
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0answers
28 views

Need for iid in MLE

I am studying about parametric estimation in supervised learning using maximum likelihood estimation. Here is what I learned: Separate our training data according to class; i.e., we have c data sets ...
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5 views

Bayesian approach for comparing the predictability of different datasets for another

Suppose I have three datasets A, B and C with not necessarily the same amount of data. Now, I want to know whether dataset A or dataset B is better in predicting C. I thought of using a Bayesian ...
3
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1answer
33 views

Why do we want low autocorrelation for MCMC convergence?

Usually, autocorrelation is one diagnostical tool for judging the convergence of a MCMC trail. Low autocorrelation is desired as this would mean that the parameter space is well explored. I have a ...
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56 views

How do I solve a under-determined quadratic multi-variate system?

I performed some simulations with known values of input variables $X_1$, $X_2$ and $X_3$, to find output response $Y$. The variables are distributed as following: $$ X_1 = N(\mu_1, \sigma_1) $$ $$ ...
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30 views

Confusion about joint and conditional probability

A person has 16 headaches (H) in 30 days, means P(H) = 16/30 = 0.53 The person tracked his Stress Level and Lack of Sleep for those 30 days. It is assumed that High Stress (HS) and Lack of Sleep ...
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28 views

Conditional probability with multiple dependent conditions

A set of items is said to be incorrect if at least one item in the set is incorrect. Initially, all items have equal probability of being correct/incorrect (P = 0.5). A series of infinite events ...
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12 views

Find a likelihood function from varying data for Bayes Theorem

I'm not sure how to model a likelihood function for the following problem: Assume I've got a sensor producing the following raw values (normalized to an interval $\pm$1): In reality there are more ...
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23 views

A linear model with prior information

Suppose I have this experimental data: I have measurements of drug response from patients (let's say its blood pressure). Specifically, I have measurements after being treated with drug A (30 ...
3
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1answer
82 views

Bayes theorem: normalisation denominator and likelihood

I have been racking my brains trying to understand Bayes theorem. So, the way I have understood is that the likelihood is the probability of observing the particular outcome given a set of parameter ...
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7answers
1k views

What do/did you do to remember Bayes' rule?

I think a good way to remember the formula is to think of the formula like this: The probability that some event A has a particular outcome given an independent event B's outcome = the probability of ...
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40 views

P(U | C) = ? where U = (A, B) where A and B are independent events

I want to calculate $P(C | A, B)$ or $P(C | U)$. I know $P(C | U) = \frac{P(U | C) \times P(C)}{P(U)}$. Since $A$ and $B$ are independent, $P(U) = P(A)\times P(B)$ But how to calculate $P(U | C)$? ...
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50 views

E[x|y] where y=cdf(x) and y is random

I have a grid approximation of a cdf, $F_x$. The cdf has support for $x>=0$ From there, calculating the $E[x]$ is straight forward with some std numerical integration techniques. In my case, ...
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14 views

What methods can be used for solving Bayes related problems? [closed]

Here are methods using formula and table. Here are methods using tables and probability tree. Here is another interesting graphical method. Are there any another useful techniques that can be used to ...
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53 views

Significance of explanatory variables in Bayesian models

I was wondering if there is a general way to handle parameters of which posterior distributions include zero. Should one remove these parameters and refit the model? E.g. You fit a regression model ...
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0answers
17 views

Combining probabilities/information from different sources

Lets say I have three independent sources and each of them make predictions for the weather tomorrow. The first one says that the probability of rain tomorrow is 0, then the second one says that the ...
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23 views

Set prior for logistic regression in R when using unequal group sizes (29 versus 48 cases)

I have 29 cases for negative outcome (0) and 48 cases for positive outcome (1). I fit my data with logistic regression model ...
2
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1answer
77 views

Can $P(A \cap B)$ be computed as $P(A)*P(B|A)$ and vice versa in any example?

Following Bayes example is taken from here A math teacher gave her class two tests. 25% of the class passed both tests and 42% of the class passed the first test. What percent of those who ...
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3answers
97 views

Are there alternatives to the Bayesian update rule?

Are there any other methods to update my belief in a hypothesis aside from the Bayesian update rule?
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1answer
31 views

Prior predictive density given by $f(y) = {f(y\mid \lambda) g(\lambda)}\big/{g(\lambda | y)}$?

(I guess stats.SE is the right place for this) I'm reading Albert's book "Bayesian computation with R". To get theprior predictive density, he extensively uses this formula $$f(y) = \frac{f(y\mid ...
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38 views

Why is Bayes theorem more popular than the normal definition of P(A|B)? [duplicate]

As everyone knows, the conditional probability of A given B is $P(A|B) = \frac{P(A\cap B)}{P(B)}$, and Bayes' theorem is derived from that equation to $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$. I'm pretty ...
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0answers
31 views

A way to make Bayes' rule common sense to me? [closed]

Although I understand Bayes' rule/theorem I always forget its intuition. I solved a lot of exercises to practice it. I remember the equation, but I find it hard to remember the intuition itself. I ...
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1answer
61 views

Linear discriminant analysis (Fisher) = Bayes?

I'd like to ask a question, I am reading book right now about mail filtering, both methods: naïve Bayes and Fisher are there very similar in implementation. I am also writing a paper on Bayesian spam ...
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1answer
446 views

Please help me understand this bayes probabilities chart

I am trying to understand a research paper. It contains a chart of prior and posterior probabilities (Bayes Probabilities) of an individual stock. However, I need guidance in interpreting what the ...
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13 views

How to choose between two models on the basis of the normalised posterior distributions?

Suppose you are given two normalised posterior densities $\pi_1(\theta|y)$ and $\pi_2(\theta|y)$, based on the data $y$, and arising from model 1 and model 2, respectively. You are asked to find ...
0
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1answer
25 views

Find corresponding linear discriminant function in a two-class, three-dimensional classification

I am new to Patter Recognition and I am kind of stuck at a homework assignment. Any help regarding the issue will be appreciated. Thank you very much. In a two-class, three-dimensional ...
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26 views

bayes estimate possion distribution function

Let {X\, ...,Xn) be random sample from random variable which has Poisson distribution with parameter A. Assume that the prior distribution A for is Gamma(1, 1) and that you have observed sample of ...
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1answer
91 views

Interpreting prior and posterior

I am bit puzzled on how we can interpret the posterior. Assume a coin which is 0.1 probable to be unfair. So our prior probability on the coin being unfair is 0.1, and being fair is 0.9. Also by ...
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0answers
27 views

What are some advanced algorithms in bayesian networks? [closed]

What are some advanced algorithms in bayesian networks? I am familiar with the conventional algorithms of network construction and inference in bayesian networks. What are some algorithms that provide ...
2
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3answers
55 views

Bayesian Risk and Subjectivity

I am studying the differences in bayesian and frequentist approaches to point estimation. I understand that there are objective and subjective approaches to Bayesian and some people don't like the ...
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1answer
80 views

Help setting up pymc to solve this problem relating to distribution of colors in M&M's

My overall goal is to work through the "Bayesian Methods for Hackers" book. So far I understand how to do simple things with pymc (like determining the parameters for a linear model and for a ...
2
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0answers
27 views

Bayes Rule with 1 Signal but 2 Unknowns

This is a question I originally posted in the math.stackexchange site, but didn't get much of an answer. Suppose I have an unknown variable $X_i = \alpha_i + \beta_i$ where $\alpha$ is one of 2 ...
0
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3answers
68 views

Bayes probability confidence

I use Bayes theorem to estimate the impact of a sales person on customer's decision to buy a product. $ P(buy|salesperson) = \frac{P(salesperson|buy) P(buy)}{P(salesperson)} $ Naturally, some ...
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1answer
54 views

“Multiple definitions of node p[1]” Error using WinBUGS [closed]

I have written the following code in WinBUGS and every time I try to compile the data after loading in the data I get the same error which is "Multiple definitions of node ...
7
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1answer
112 views

What would be an example of when L2 is a good loss function for computing a posterior loss?

L2 loss, together with L0 and L1 loss, are three a very common "default" loss functions used when summarising a posterior by the minimum posterior expected loss. One reason for this is perhaps that ...
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1answer
73 views

Posterior predictive test quantities

I've been trying to figure out problem 6.2 from Gelman's book, second edition, page 192 on Bayesian data analysis. Can anyone help? a) Set up predictive test quantities to check the following ...
0
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1answer
52 views

How to simplify $P (X|Y, Z)$ when $X$ is independent of $Y$ but not $Z$? [duplicate]

How do I simplify $P (X|Y, Z)$ if I know that $X$ is independent of $Y$ but not $Z$.
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49 views

Bayes Net: how to calculate joint distribution?

I originally posted this question to Computer Science Stack Exchange, but then I was told that CrossValidated site existed. I've been reading many questions, but none of them seem to answer my doubts. ...
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0answers
21 views

Am I fine in implementing Naive Bayes Classifier?

I have implemented one Sentiment Analysis using Naive Bayes Classifier. To do this I have taken the following steps. First I checked the problem, the nature of data (continuous or discrete) and ...
6
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7answers
201 views

Why is $p(A) \times p(B|A) = p(B) \times p(A|B)$?

Is there an easy way or simple example to get why it must be that $$ p(A) \times p(B|A) = p(B) \times p(A|B) $$? I get that $p(A) \times p(B|A)$ can be seen as probability of $B$ occurring, weighted ...
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1answer
47 views

Why is it called “mode” in MAP estimation?

When estimating parameters with MAP, why is it written that we are estimating the "mode"? I thought it would be the mean of the posterior distribution?
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1answer
47 views

Conditional probability density function

Let $\theta$ be the parameter of the probability density function $f(x)$. If it is mentioned that $f(x|\theta)$ be the conditional probability density function, then what does $f(x|\theta)$ mean? ...
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50 views

How to compute the probability of $P(a|b,e)$ given $P(a|b), P(a|e)$ and $A$ independent of $B$?

The following is given: $$ \begin{align} P(b) &= P(e) = 0.1\\ P(a|e) &= 0.2\\ P(a|b) &= 0.95\\ P(a|\neg b, \neg e) &= 0\\ B &{\perp\!\!\!\perp} E \end{align} $$ Is there any way ...
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21 views

Describing a distribution of probabilities/percentages

Is there a better way to describe this probability? \begin{equation} Pr(Y_{sij}|R_n) = \frac{Y_{sij|R_n} \sum_{ij}Y_{sij}}{\sum_{ij|R_n} Y_{ij}\sum_{sij}Y_{sij}} \end{equation} The $R_n$ describes ...
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2answers
35 views

Conditions under which “explaining away” does vs. does not occur

My understanding of "explaining away" is as follows. If there is an effect, C, that can result from two independent causes, A and B, then observing only C makes A more likely than observing both C and ...