Questions tagged [map-estimation]

Estimation by maximizing the posterior density function

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Is there a reason to use variational inference for point estimates?

I have seen Bayesian hierarchical models, particularly in computational biology, that use variational inference, but do not use the uncertainty provided by a variational solution. For example, MOFA is ...
26 views

MCMC with using MAP as starting value

let $X$ be a random variable from my target distribution $\pi(x)$, which I know up to a normalizing constant, and I want to calculate $Ef(X)$ for some know function $f$. The dimensions of $X$ are ...
64 views

Why should MAP be invariant under reparameterization?

I learned why MAP suffers from being reparametrization invariance while MLE not from this answer, but I don't know why reparametrization invariance even matters? What is the non-linear mapping ...
56 views

Why is MAP and ML widely accepted? [closed]

(ML as in Maximum Likelihood and MAP as in Maximum A-posteriori) I'm going trough a course book on my own, and without really having peers to talk to I'm turning to stack exchange with these rather ...
47 views

How to choose estimates after Bayesian regression?

In a Bayesian logistic regression with two predictor variables $x_{1}$ and $x_{2}$, I did MCMC (2000 samples) to estimate posterior distribution. Now it's done, how can I choose the final estimates ...
210 views

Differences between MLE and MAP estimators

Generally speaking, what are the differences between an MLE and a MAP estimator? If I wanted to improve the performance of a model, how would these differences come into play? Are there specific ...
14 views

Quantifying uncertainty in MAP nonlinear regression

I am interested in clinical pharmacokinetics, where we have a given medicine's population model, with its parameters (mean and standard deviation), and our goal is take one or two blood samples from a ...
26 views

Bayesian Inference Intuition: Beta and Binomial vs Gamma and Poisson

When the data is assumed to be binomial distributed, and the prior probability is assumed to be a beta distribution, the posterior follows the distribution $Beta (\alpha - 1 + k , \beta - 1 + n- k$). ...
293 views

MLE and MAP with Naive Bayes

From what I understand, Naive Bayes classifies by doing: $$y \leftarrow argmax_{y_k}P(Y=y_k)\prod_{i}P(X_i|Y=y_k)$$ There are two things there we need to know: $P(Y=y_k)$ and all the $P(X_i|Y=y_k)$ ...
19 views

Bayesian inference MAP question

Hello I am stuck with this problem and was wondering if someone can solve it for me. Thanks
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Deriving the posterior distribution over the model parameters: are the model parameters and data independent?

We are told (in Section 9.2.3, Deisenroth et al.: Mathematics for Machine Learning) that we can compute the posterior over a model's parameters $\boldsymbol\theta$ (here in the context of linear ...
81 views

The prior in MAP and Bayesian interference

We can use a Normal distribution as a prior when handling a Normal distribution as likelihood in Bayesian inference However if we want to do MAP given a Bernoulli as likelihood can we use Normal ...
49 views

Does it make sense to condition on fixed values of some parameters before doing MCMC on the other parameters?

I have a Bayesian model with a large number of parameters (around 50), and as usual my goal is to infer the posterior distribution for the parameters, with MCMC. However, I am only interested in the ...
232 views

Shouldn't log likelihood always be normalized by data size in bayesian estimation?

This is very interesting problem. I wonder if the whole bayesian statistics is neglecting it or if I am super confused. I will illustrate it on a bayesian Maximum a Posteriori (MAP) estimation and ...
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Finding the MAP for a function whose conditioning depends on an exponential integral

Let $X$ be such that $X \sim exp( \lambda = 1)$ and let $Y$ be such that $Y \sim U[0,x]$, where $x$ is the realization of $X$. Given that information I know that: $f_{X}(x) = e^{-x}$ for $x \geq 0$...
Suppose an estimator $\hat\theta_T$ is defined as the value of $\theta$ maximizing: $$\sum_{t=1}^T{l(y_t|\theta)}+\mu_T g(\theta),$$ where $l(y_t|\theta)$ is the log-likelihood of observation $t$, $\... 1answer 22 views MAP, MLE and parametrised data It is often said that maximum likelihood is used to obtain estimates of distrubtion's parameters. However, what is unclear is whether it will produce consistent estimate parameters other than those of ... 0answers 25 views Logistic regression fitting methods clarification Each book I read propose a different fitting method for Logistic Regression. The general idea is to maximize this expression. $$Pr\left(\beta|y,X,M\right) = \frac{Pr\left(y|\beta,X,M\right) Pr(\... 1answer 37 views Assign an error to the parameters of MAP estimate Through a MCMC Gibbs sampler I obtain M chains of the parameters vector \mathbf{\theta}, meaning that each component of \mathbf{\theta} is the value of one parameter at a given iteration. ... 0answers 57 views Discriminative Models with Class Priors In discriminative models, we model p(Y|X) directly while in generative models we model p(X|Y)p(Y) where X is the input and Y is the output variable. I am confused when the parameters and ... 3answers 220 views Bayesian parameter estimation with proportion data I am trying to do a Bayesian analysis using a model that comes from the literature in non-Bayesian form: y = \Phi\Bigg(\frac{1}{\alpha} * log(A/\beta)\Bigg). Because the model uses the function \... 0answers 160 views Can you find the posterior mode of an unknown distribution without MCMC? I was wondering if you wanted to compute the MAP estimate of an unknown posterior distribution, is there a non-sampling based method that would suffice? As in, if you don’t need to know anything more ... 1answer 196 views What are the possible estimates of the parameters of the multinomial distribution? The expected value of the parameters of the multinomial distribution (taking into account the Dirichlet prior D(\alpha) and the posterior Dirichlet-Multinomial) is: \pi_i = α_i+ x_i / \sum_{j} α_j+... 1answer 108 views Finding MAP estimate I think after all the reading I've done I still don't fully understand MAP estimation. I came across a problem that's leaving me dumbfounded. Suppose A ~ N(0,\sigma^2_1) and \epsilon ~ N(0,\... 1answer 116 views Example of maximum a posteriori that does not match the mean of a marginalized posterior Given a N-parameter likelihood and prior, I can obtain the marginalized posterior for each parameter through Bayesian MCMC. I can also estimate the maximum a posteriori (MAP) of the N-parameter ... 0answers 210 views confusion related to maximum a posteriori estimation [duplicate] I was reading this article in wikipedia related to MAP http://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation. However, I had this confusion when it says MAP estimation is a limit of Bayes ... 3answers 717 views MAP is a solution to L(\theta) = \mathcal{I}[\theta \ne \theta^{*}] I have come across these slides (slide # 16 & #17) in one of the online courses. The instructor was trying to explain how Maximum Posterior Estimate(MAP) is actually the solution L(\theta) = \... 1answer 54 views Confused about maximum a posteriori estimation [closed] I am new to Bayesian statistics, and I just came across MAP. When our prior is a continuous distribution (pdf) on \theta how can we calculate g(\theta) in the numerator? Edit: I assumed g(\... 1answer 71 views Multidimensional Bayes point estimates Consider the posterior distribution p(\theta|x). We aim to find a "good" estimate of the random variable \theta. The Bayes risk associated with the loss function L(\hat{\theta}, \theta) is ... 1answer 967 views How can (L1 / L2) regularization be equivalent to using a prior when priors can't be changed? I understand the argument for how training with an L1/L2 regularizer is the same thing as finding the MAP estimate when the prior is Gaussian/Laplace. But there's a crucial difference. In Bayes' ... 1answer 608 views MAP estimation for multiple parameters Consider N observed data points x_i (i=1,..,N), and a likelihood that depends on p parameters: f(x_i|\theta_n) (n=1,..p). From Bayes' theorem$$p(\theta_n|x_i) = \frac{f(x_i|\theta_n)g(\... 2answers 1k views What is an example of a transformation on a posterior distribution such that the MAP estimate will be non-invariant? Suppose that we have a posterior distribution$p(\theta\mid y)$and we wish to define a transformation on$\theta$such that$\phi = f(\theta)$. I know that generally such transformations will not ... 1answer 266 views For a posterior$p(\theta |y)$, if I specify a one-to-one transformation$\phi = g(\theta)$, how can I apply the transformation? [duplicate] Suppose I have a posterior distribution,$p(\theta \mid y)$, where$y$was my data and$\theta$is a random variable with some prior distribution. If I specify a one-to-one transformation$\phi = g(\...
The most common usage of the variational inference looks like to be in computing the marginal distribution $P(X)$ in the denominator of the Bayes formula when computing the posterior probability of ...