Questions tagged [map-estimation]

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81 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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15 views

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 $x \geq 0$...
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32 views

MAP Estimation of Covariance Matrix of a Multivariate Normal Distribution

I have a general prior multivariate normal distribution and I want to update it with new samples which are more local for my case. I want to do it with MAP estimation. With MAP estimation it is ...
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16 views

Practical Estimation of Maximum a Posteriori Values

Suppose I have samples from a posterior density function, generated using MCMC methods. I wish to summarize the posterior marginally for any given parameter in the model (of which there may be many) - ...
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1answer
20 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 ...
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93 views

Bias correcting penalized maximum likelihood / maximum a posteriori estimates

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 $t$, $\...
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31 views

Naive Bayes - why MAP?

According to Wikipedia on (Classic) Naive Bayes The naive Bayes classifier combines this model with a decision rule. One common rule is to pick the hypothesis that is most probable; this is known ...
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23 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(\...
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1answer
25 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. ...
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34 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 ...
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12 views

MAP estimate and Maximization step

I have a very simple question. My reference textbook is the Murphy, "machine learning, a probabilistic perspective". Let's imagine we are trying to fit a GMM $\gamma$ with MAP. We know that the ...
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3answers
191 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 $\...
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13 views

Distance correlation and a corresponding mapping

I have two long vectors, say X and Y (of equal length). I computed the Distance Correlation as implemented in Scipy and I got a ...
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81 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 ...
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26 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 ...
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1answer
49 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,\...
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1answer
96 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+...
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60 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 ...
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50 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(\...
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1answer
50 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 ...
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3answers
425 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) = \...
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1answer
593 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' ...
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1answer
392 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(\...
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2answers
639 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 ...
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1answer
150 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(\...
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1answer
101 views

The explanation for the need to compute (rather then optimize) the posterior of latent variables

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 ...
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208 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 ...