All Questions
Tagged with density-function bayesian
44 questions
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How to solve this question about the beta distribution in a Bayesian analysis? [closed]
This question appeared in Prof. Babak Shahbaba's book (Biostatistics With R: An Introduction to Statistics Through Biological Data) in the questions of its chapter 13.
Q4. Suppose that we are ...
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4
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5
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281
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What is $p(y|x)$ given $X=Y+Z$, $Z$ is a standard normal, and $Y$ is a random variable
We have $X=Y+Z$ where $Z$ is a standard normal and $Y$ is a random variable with $p(y)$ as its density. $Y$ and $Z$ are independent.
The conditional probability $p(x|y)$ is obvious to be $\mathcal N_x(...
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2
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516
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Creating half normal probability distribution
I have come across a problem where a half normal distribution is based on a single number, namely the sum of all costs. The exact definition of the number is not important. The important think is that ...
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How to validate rejection sampling?
What is a principled approach to validating samples generated from rejection sampling actually follow the target function?
I am looking for some thing more than a simple histogram + density plot.
...
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38
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I need help understanding why this integral is the probability for winning by switching in the Monty Hall problem
I need help understanding this probability from the Monty Hall problem. Why does this integral give the probability of winning by switching if the Car is behind 1, Monty shows goat behind 3 and Player ...
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How can I derive the distribution of the L2,1 norm if the ditribution of L1 norm is given?
I understand that the L1 norm promotes sparsity and is a Laplace prior in the LASSO regression framework. I am interested in how this prior changes when we apply L2,1 regularisation instead? Is it ...
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22
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Esitmate of minimal of a function changed after transforming the variable
I want to perform MCMC or HMC for solving minimization problem of a function $f(x)$, then define the corresponding density $$g(x) = \exp\left(-f(x)\right)$$
Because the function of the future apply is ...
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How is the q(z) function added at the end of this Bayesian formula?
At the bottom of this Bayesian formula why is a q(z) is brought into numerator and denominator positions?
Is this within the rules of Algebra? Could anything be placed in the numerator and denominator?...
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2
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181
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How to understand the posterior distribution is the same as likelihood function
So I read this post Why is the posterior distribution the same as the likelihood function when uniform prior distribution is used in Bayesian Analysis, and learned that when we have a uniform prior, ...
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93
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Best probability density function to use for the prior of a variance parameter in Bayesian inference
This answer provided some good general advice, but in my specific case I want to create a model of my prior beliefs about the variance of a normally-distributed random variable:
$$x \sim \mathcal{N}(0,...
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Why is the beta distribution so flat when a, b=1?
If the beta distribution is a prior of a Bernoulli distribution (i.e. a rate of success for a binary outcome), then it is completely counterintuitive to me that the beta distribution should be ...
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344
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Gradient of Log Normalizing Constant - Does it have a name and do we know of any properties?
Suppose $p(x) = \frac{\tilde{p}(x)}{Z}$ is a density function. I was wondering if the gradient of the log of the normalizing constant has a name and if we know any properties of it (e.g. maybe some ...
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2
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346
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Maximum Value of Kernel Function in ABC
Are there cases where a kernel function, must have 1 as the maximum value ??
The definition of a Kernel can be found in the following link,
https://en.wikipedia.org/wiki/Kernel_(statistics)#In_non-...
- 2,286
-1
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1
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How do I read this posterior distribution? [closed]
This might be a very strange question, but I am having a bit of trouble. Here is a posterior distribution.
If I am reading this passage out loud, when I get near the end to π(δ1), do I read it as pi ...
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2
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2k
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Why can I use a PDF when computing bayes rule?
My understanding is that PDFs are 0-valued at all individual points, and only when we integrate over a specific region do we get a non-zero value. However, my professor keeps using PDFs when ...
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3
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689
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Which pdf to choose for the prior of an angle?
I have a system in which one uncertain variable is a direction in two dimensions. If I want to define a prior for this, is there an elegant way to reflect the fact that the parameter space dimension ...
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1
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272
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Sampling posterior distribution of a function
I have the following problem: let's say I have a function $y=f(x)$. Let $f$ be defined for all $x$ but it it might not be invertible. Further assume $x \sim p(x)$ with some probability density $p(x)$.
...
- 103
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2
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178
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Given distribution of $X$ and $X|Y=y$, is it possible to find distribution of $Y$?
What the title says!
My intuition is NO since in Bayesian statistics we typically specify the prior and likelihood, and from those two we can compute the posterior and so on. We can interpret $Y$ = ...
- 61
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2
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872
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What's the big deal with normalization constants in Bayesian inference? [duplicate]
I read this sentence in a book:
"... therefore this method is particularly useful for Bayesian inference since it doesn't require a normalization constant"
The method is a computational algorithm ...
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2
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837
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Differences between a frequentist and a Bayesian density prediction
What are some essential differences between a frequentist density forecast/prediction and a Bayesian posterior for an outcome of a random variable?
Of course, there will be differences in how they ...
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Sampling from joint distribution by writing its density as a product of conditional densities
In Gelman et al. "Bayesian Data Analysis Ed3" the authors often do the following (e.g. on pg. 65): Given two parameters $\mu$ and $\sigma^2$ and data y joint posterior density $p(\mu,\sigma^2)$ is ...
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Probability of first time to an event
We have a stream of events over time. Suppose that $f_t$ is the probability density that an event happens at time $t$. For example, $f_t$ can be the probability density that any bus arrives at time $t$...
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3
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209
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what does p( y | μ,σ²) really mean?
Just started to study Bayesian Statistics. I am very confused the concept of having a conditional probability on a distribution. Specifically:
I understand what p( A | B ) where A="I am sick" and ...
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Deriving Posterior Binomial Density from Uniform Prior
I'm trying to derive the posterior density of the probability parameter of a binomial random variable, given one realization of the random variable and a uniform prior density on the probability ...
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68
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Conditional expectation of the probability that a prior parameter is greater than some value
I am working in a Bayesian setting where I have a prior $p \sim \text{Beta}(\alpha, \beta)$. For reasons that don't really matter, I'm later defining a new parameter, call it $C$, which is the ...
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Expectation and variance of the posterior distribution example: seeking elaboration on normalising constant
I have the following example:
Assume that we have an observation $Y$ from a Binomial distribution with parameter $n = 20$ and success probability $p: [Y \sim \mathrm{Bin}(20, p)]$.
Further assume ...
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329
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Posterior Predictive Density of Linear Regression
I'm trying to derive the Posterior Predictive Density of a Linear Regression Model with a diffuse, uninformative prior such that we have:
$y_{i} = x_{i}'\beta + \varepsilon_{i}$ with $\varepsilon_{i}...
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1
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2
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546
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Estimating non-centrality parameter from some obtained sample of t variates
Suppose I have a sample of 10 $t$ variates which I think has come from a non-central $t-distribution$.
I was wondering how I can estimate the non-centrality parameter $(ncp)$ of the mother non-...
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Finding the probabilities from density distribution to use in Bayesian formula
I have some search results, which have been validated based on a certain criterion and each hit has a probability of being correct assigned to it.
The search results look like this:
Var1---...
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0
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159
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Poisson posterior PDF
I want to understand this concept a little better, I'll give a reduced example of the problem I'm having and would appreciate a more intuitive answer (my statistics background is largely self taught ...
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1
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375
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How to proper evaluate the PDF of a Beta Distribution?
On page 40 of "Think Bayes - Bayesian Statistics Made Simple", Allen evaluates the PDF of the Beta distribution as
...
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1
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147
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What's an intuitive explanation for why MAP is variant under parameterization?
I understand why MAP is variant under parameterization mathematically, but I don't really understand it intuitively.
To help me out, my professor gave me an example where reparameterizing MAP "...
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1
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354
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How to compute the CDF of this random variable?
I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. Specifically, one player has the opportunity to choose any value $\eta$ from ...
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1
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274
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Is an improper prior/posterior equivalent to an undefined PDF?
A "proper" prior or posterior distribution is defined as a distribution for which the PDF integrates to 1 (or in practice, if we're working with a known distribution, one for which the PDF without ...
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Fastest way to solve Bayes estimator problem
The below problem is from an old PhD qualifying exam in our department. My own solution below is time-consuming and quite possibly wrong. It also relies on recognizing a less common distribution, so I ...
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Is there a Bayesian approach to density estimation
I am interested to estimate the density of a continuous random variable $X$. One way of doing this that I learnt is the use of Kernel Density Estimation.
But now I am interested in a Bayesian ...
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1
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676
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Why likelihood is not always a density function? [duplicate]
I try to self-learn Bayesian machine learning (mostly by studying Bishop and Kevin Murphy's books).
While working with formulas I was puzzled by the quote that "Note that the likelihood function is ...
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394
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Find posterior distribution
Let $X_{1},..,X_{n}$ be a sample from a poisson$({\lambda})$ distribution. Let the prior be ${\pi}({\lambda})=1/{\sqrt{\lambda}}$. Find the posterior distribution.
My work: We have $f(x|{\lambda})=\...
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Derive the conditional pdf of data on prior parameters
In Bayesian statistics I see this derivation often.
Given the likelihood function $f(X|\theta)$ and the prior $f( \theta |a, b)$, the author will derive $f(X|a,b)$. The steps in between are ...
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165
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Can posterior distribution for a continuous variable be greater than one?
I already asked this question here, but I am not sure where would be better to ask it? This might sound a dumb question but I am really confused about it. According to Bayes' rule we do have the ...
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Connection between PDFs/PMFs and Bayes Theorem
UPDATE Original question was confused and poorly worded. I thought about it more and don't think I have a question any longer. After thinking a bit more I came up with:
For a distribution, such as ...
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PDF Manipulation for Bayesian analysis
This post pertains to Bayesian pdf manipulation.
Firstly, assuming a prior probability specified as Gamma distribution such that $\alpha = \mu_{0}^{2}/\sigma_{0}^{2}$ and $\beta = \mu_{0}/\sigma_{0}^{...
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Is Perkins et al.'s "skill score" an application of Bayes' theorem?
Perkins et al. (2007) introduce a "skill score" for measuring climate model output against observations. The score basically consists of measuring the overlap between probability density functions of ...
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Quantile intervals vs. highest posterior density intervals
I am reading a bit about Bayesian analysis, but I cannot understand the difference between the classic quantile-based intervals and the Highest Posterior Density Intervals. What is the difference ...
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