Marginal likelihood

So far all has made sense to me except for the below equation (eq 11 in link), the log marginal likelihood of the GP: $$ -1/2 [Y^{T} K_y^{-1}Y] -1/2 [log(|K_y|)] - N/2[log(2 \pi)]$$ The author explains that this step is necessary to optimize the hyperparameters of the kernel function. I've used some algebra and found that this is simply the log ...

The marginal likelihood in a posterior formulation, i.e P(theta|data) , as per my understanding is the probability of all data without taking the 'theta' into account. So does this mean that we are integrating out theta?The marginal likelihood is the probability of getting your observations from the functions in your GP prior (which is defined by the kernel). When you minimize the negative log marginal likelihood over $\theta$ for a given family of kernels (for example, RBF, Matern, or cubic), you're comparing all the kernels of that family (as defined by ...Bayesian statistics is an approach to data analysis based on Bayes’ theorem, where available knowledge about parameters in a statistical model is updated with the information in observed data.Marginal likelihood of implicit model. 2. Computing the Gaussian posterior from likelihood and prior. 5. Deriving Log Marginal Likelihood for Gaussian Process. Hot Network Questions Best practice for redundant conditions in if-elif-else statementsMarginal Likelihood는 두 가지 관점에서 이야기할 수 있는데, 첫 번째는 말그대로 말지널을 하여 가능도를 구한다는 개념으로 어떠한 파라미터를 지정해서 그것에 대한 가능도를 구하면서 나머지 파라미터들은 말지널 하면 된다. (말지널 한다는 것은 영어로는 ...

The marginal likelihood is used in Gómez-Rubio and Rue (Citation 2018) to compute the acceptance probability in the Metropolis-Hastings (MH) algorithm, which is a popular MCMC method. Combining INLA and MCMC allows to increase the number of models that can be fitted using R-INLA. The MCMC algorithm is simple to implement as only the ...The likelihood of each class given the evidence is known as the posterior probability in the Naive Bayes algorithm. By employing the prior probability, likelihood, and marginal likelihood in combination with Bayes' theorem, it is determined. As the anticipated class for the item, the highest posterior probability class is selected.The “Bayesian way” to compare models is to compute the marginal likelihood of each model p ( y ∣ M k), i.e. the probability of the observed data y given the M k model. This quantity, the marginal likelihood, is just the normalizing constant of Bayes’ theorem. We can see this if we write Bayes’ theorem and make explicit the fact that ...Definition. The Bayes factor is the ratio of two marginal likelihoods; that is, the likelihoods of two statistical models integrated over the prior probabilities of their parameters. [9] The posterior probability of a model M given data D is given by Bayes' theorem : The key data-dependent term represents the probability that some data are ... the variational lower bound on the marginal likelihood and that, under some mild conditions, even works in the intractable case. The method optimizes a proba-bilistic encoder (also called a recognition network) to approximate the intractable posterior distribution of the latent variables. The crucial element is a reparame-That edge or marginal would be beta distributed, but the remainder would be a (K − 1) (K-1) (K − 1)-simplex, or another Dirichlet distribution. Multinomial–Dirichlet distribution Now that we better understand the Dirichlet distribution, let’s derive the posterior, marginal likelihood, and posterior predictive distributions for a very ...Keywords: BIC, marginal likelihood, singular models, tree models, Bayesian networks, real log-canonical threshold 1. Introduction A key step in the Bayesian learning of graphical models is to compute the marginal likelihood of the data, which is the likelihood function averaged over the parameters with respect to the prior distribution.tive marginal maximum likelihood estimator using numerical quadrature. A key feature of the approach is that in the marginal distribution of the manifest vari-ables the complicated integration can be reduced, often to a single dimension. This allows a direct approach to maximizing the log-likelihood and makes the

The rise of e-commerce is spurring a decline in retailers' profit margins, according to an analysis of six key European markets and more than 250 retailers. The unstoppable ascent of e-commerce is spurring a corresponding decline in retaile...That edge or marginal would be beta distributed, but the remainder would be a (K − 1) (K-1) (K − 1)-simplex, or another Dirichlet distribution. Multinomial–Dirichlet distribution Now that we better understand the Dirichlet distribution, let’s derive the posterior, marginal likelihood, and posterior predictive distributions for a very ...In the Bayesian setting, the marginal likelihood is the key quantity for model selection purposes. Several computational methods have been proposed in the literature for the computation of the marginal likelihood. In this paper, we briefly review different estimators based on MCMC simulations. We also suggest the use of a kernel density estimation procedure, based on a clustering scheme ...Nilai likelihood yang baru adalah 0.21. (yang kita ketahui nanti, bahwa nilai ini adalah maximum likelihood) Perhatikan bahwa pada estimasi likelihood ini, parameter yang diubah adalah mean dan std, sementara berat tikus (sisi kanan) tetap ( fixed ). Jadi yang kita ubah-ubah adalah bentuk dan lokasi dari distribusi peluangnya.

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Jun 22, 2021 · Estimation of GLMMs is a non-trivial task due to the fact that the likelihood (the quantity that should be maximized) cannot be written down in closed form. The current implementation of GPBoost (version 0.6.3) is based on the Laplace approximation. Model estimation in Python and R can be done as follows: Pythonmarginal likelihood /p(Y j )p( ) Bernstein - Von Mises Theorem: For a large sample, Bayes estimate is close to the MLE. The posterior distribution of the parameter around the posterior mean is also close to the distribution of the MLE around the truth, Sample from N( ^ n; Hn( ^Log marginal likelihood for Gaussian Process. Log marginal likelihood for Gaussian Process as per Rasmussen's Gaussian Processes for Machine Learning equation 2.30 is: log p ( y | X) = − 1 2 y T ( K + σ n 2 I) − 1 y − 1 2 log | K + σ n 2 I | − n 2 log 2 π. Where as Matlab's documentation on Gaussian Process formulates the relation as.In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional normal distribution to higher dimensions.One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution.You can use this marginal distribution to calculate probabilities. I really like hierarchical models because they let you express complex system in terms of more tractable components. For example, calculating the expected number of votes for candidate 1 is easy in this setting. ... Bernoulli or binomial likelihood, beta prior. Marginalize over ...

This is an up-to-date introduction to, and overview of, marginal likelihood computation for model selection and hypothesis testing. Computing normalizing constants of probability models (or ratio of constants) is a fundamental issue in many applications in statistics, applied mathematics, signal processing and machine learning. This article provides a comprehensive study of the state-of-the ...Once you have the marginal likelihood and its derivatives you can use any out-of-the-box solver such as (stochastic) Gradient descent, or conjugate gradient descent (Caution: minimize negative log marginal likelihood). Note that the marginal likelihood is not a convex function in its parameters and the solution is most likely a local minima ...3The influence of invariance on the marginal likelihood In this work, we aim to improve the generalisation ability of a function f: X!Yby constraining it to be invariant. By following the Bayesian approach and making the invariance part of the prior on f(), we can use the marginal likelihood to learn the correct invariances in a supervised ...Mar 25, 2021 · The marginal likelihood is useful for model comparison. Imagine a simple coin-flipping problem, where model M0 M 0 is that it's biased with parameter p0 = 0.3 p 0 = 0.3 and model M1 M 1 is that it's biased with an unknown parameter p1 p 1. For M0 M 0, we only integrate over the single possible value. 18 Şub 2019 ... I was checking sklearn's implementation of log marginal likelihood of a Gaussian Process (GP). The implementation is based on Algorithm 2.1 ...Marginal or conditional likelihoods can be used. These are proper likelihoods23 so all the likelihood ratio based evidential techniques can be employed. Unfortunately, marginal and conditional likelihoods are not always obtainable. Royall [2000] recommends the use of profile likelihood 24 ratio as a general solution.A marginal likelihood just has the effects of other parameters integrated out so that it is a function of just your parameter of interest. For example, suppose your likelihood function takes the form L (x,y,z). The marginal likelihood L (x) is obtained by integrating out the effect of y and z.Maximum Likelihood with Laplace Approximation. If you choose METHOD=LAPLACE with a generalized linear mixed model, PROC GLIMMIX approximates the marginal likelihood by using Laplace’s method. Twice the negative of the resulting log-likelihood approximation is the objective function that the procedure minimizes to determine parameter estimates.Marginal likelihood vs. prior predictive probability. 5. Relation between Bayesian analysis and Bayesian hierarchical analysis? 1. How do interpret a vague prior for hierarchical modeling? 4. Posterior predictive distributions and predictive intervals. 1.The likelihood function is a product of density functions for independent samples. A density function can have non-negative values. The log-likelihood is the logarithm of a likelihood function. If your likelihood function L ( x) has values in ( 0, 1) for some x, then the log-likelihood function log L ( x) will have values between ( − ∞, 0).you will notice that no value is reported for the log marginal-likelihood (LML). This is intentional. As we mentioned earlier, Bayesian multilevel models treat random effects as parameters and thus may contain many model parameters. For models with many parameters or high-dimensional models, the computation of LML can be time consuming, and its ...

We study a class of interacting particle systems for implementing a marginal maximum likelihood estimation (MLE) procedure to optimize over the parameters of a latent variable model. To do so, we propose a continuous-time interacting particle system which can be seen as a Langevin diffusion over an extended state space, where the number of particles acts as the inverse temperature parameter in ...

In this paper, we present a novel approach to the estimation of a density function at a specific chosen point. With this approach, we can estimate a normalizing constant, or equivalently compute a marginal likelihood, by focusing on estimating a posterior density function at a point. Relying on the Fourier integral theorem, the proposed method is capable of producing quick and accurate ...Fast Marginal Likelihood Maximisation for Sparse Bayesian Models 3 where w is the parameter vector and where ' = [`1:::`M] is the N £ M 'design' matrix whosecolumns comprise the complete set of M 'basis vectors'. The sparse Bayesian framework makes the conventional assumption that the errors are modelledThe marginal likelihood quantifies the agreement between data and prior in a geometric sense made precise in de Carvalho et al. (2019). In classical (frequentist) statistics, the concept of marginal likelihood occurs instead in the context of a joint parameter θ = ( ψ, λ), where ψ is the actual parameter of interest, and λ is a non ...When deciding whether or not a company's stock is a good addition to your portfolio, you need to analyze various aspects of the company. When deciding whether or not a company's stock is a good addition to your portfolio, you need to analyz...parameter estimation by (Restricted) Marginal Likelihood, Generalized Cross Validation and similar, or using iterated nested Laplace approximation for fully Bayesian inference.The new version also sports significantly faster likelihood calculations through streaming single-instruction-multiple-data extensions (SSE) and support of the BEAGLE library, allowing likelihood calculations to be delegated to graphics processing units (GPUs) on compatible hardware. ... Marginal model likelihoods for Bayes factor tests can be ...That edge or marginal would be beta distributed, but the remainder would be a (K − 1) (K-1) (K − 1)-simplex, or another Dirichlet distribution. Multinomial–Dirichlet distribution Now that we better understand the Dirichlet distribution, let’s derive the posterior, marginal likelihood, and posterior predictive distributions for a very ...Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site

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Maximum likelihood is nonetheless popular, because it is computationally straightforward and intuitive and because maximum likelihood estimators have desirable large-sample properties in the (largely fictitious) case in which the model has been correctly specified. ... penalization may be used for the weight-estimation process in marginal ...A maximum marginal likelihood estimation with an expectation-maximization algorithm has been developed for estimating multigroup or mixture multidimensional item response theory models using the generalized partial credit function, graded response function, and 3-parameter logistic function. The procedure includes the estimation of item ...1.7 An important concept: The marginal likelihood (integrating out a parameter) 1.8 Summary of useful R functions relating to distributions; 1.9 Summary; 1.10 Further reading; 1.11 Exercises; 2 Introduction to Bayesian data analysis. 2.1 Bayes’ rule; 2.2 Deriving the posterior using Bayes’ rule: An analytical example. 2.2.1 Choosing a ...このことから、 周辺尤度はモデル（と θ の事前分布）の良さを量るベイズ的な指標と言え、証拠（エビデンス） (Evidence)とも呼ばれます。. もし ψ を一つ選ぶとするなら p ( D N | ψ) が最大の一点を選ぶことがリーズナブルでしょう。. 周辺尤度を ψ について ...This integral happens to have a marginal likelihood in closed form, so you can evaluate how well a numeric integration technique can estimate the marginal likelihood. To understand why calculating the marginal likelihood is difficult, you could start simple, e.g. having a single observation, having a single group, having μ μ and σ2 σ 2 be ... However, existing REML or marginal likelihood (ML) based methods for semiparametric generalized linear models (GLMs) use iterative REML or ML estimation of the smoothing parameters of working linear approximations to the GLM. Such indirect schemes need not converge and fail to do so in a non-negligible proportion of practical analyses.We propose an efficient method for estimating the marginal likelihood for models where the likelihood is intractable, but can be estimated unbiasedly. It is based on first running a sampling method such as MCMC to obtain samples for the model parameters, and then using these samples to construct the proposal density in an importance sampling ...The marginal likelihood is the average likelihood across the prior space. It is used, for example, for Bayesian model selection and model averaging. It is defined as . ML = \int L(Θ) p(Θ) dΘ. Given that MLs are calculated for each model, you can get posterior weights (for model selection and/or model averaging) on the model byI'm trying to optimize the marginal likelihood to estimate parameters for a Gaussian process regression. So i defined the marginal log likelihood this way: def marglike(par,X,Y): l,sigma_n = par n ...2. To put simply, likelihood is "the likelihood of θ θ having generated D D " and posterior is essentially "the likelihood of θ θ having generated D D " further multiplied by the prior distribution of θ θ. If the prior distribution is flat (or non-informative), likelihood is exactly the same as posterior. Share.higher dates increase the likelihood that you will have one or two distress incidents as opposed to none. We see the same thing in group 3, but the effects are even larger. ... Appendix A: Adjusted Predictions and Marginal Effects for Multinomial Logit Models . We can use the exact same commands that we used for ologit (substituting mlogit for ….

the marginal likelihood (2) for each model k separately, and then if desired use this infor mation to form Bayes factors (Chib, 1995; Chib and Jeliazkov, 2001). Neal (2001) combined aspects of simulated annealing and importance sampling to provide a method of gatheringWe select the value of G based on the maximum value of the corresponding marginal likelihood value. Footnote 4 Note that the value of G can also be selected by using the well known Bayesian information criterion (BIC), However, BIC is just an asymptotic version of the marginal likelihood and Bayes factors when the sample size …Definition. The Bayes factor is the ratio of two marginal likelihoods; that is, the likelihoods of two statistical models integrated over the prior probabilities of their parameters. [9] The posterior probability of a model M given data D is given by Bayes' theorem : The key data-dependent term represents the probability that some data are ...Fast marginal likelihood estimation of penalties for group-adaptive elastic net Mirrelijn M. van Nee∗ 1, Tim van de Brug , and Mark A. van de Wiel1,2 1Epidemiology and Data Science, Amsterdam University Medical Centers, The Netherlands 2MRC Biostatistics Unit, Cambridge University, UK Abstract Nowadays, clinical research routinely uses omics data, such as gene expression, forThe marginal likelihood is the probability of getting your observations from the functions in your GP prior (which is defined by the kernel). When you minimize the negative log marginal likelihood over $\theta$ for a given family of kernels (for example, RBF, Matern, or cubic), you're comparing all the kernels of that family (as defined by ...The proposed method is developed in the context of MCMC chains produced by the Metropolis-Hastings algorithm, whose building blocks are used both for sampling and marginal likelihood estimation, thus economizing on prerun tuning effort and programming. This article provides a framework for estimating the marginal likelihood for the purpose of Bayesian model comparisons. The approach extends ...Mar 25, 2021 · The marginal likelihood is useful for model comparison. Imagine a simple coin-flipping problem, where model M0 M 0 is that it's biased with parameter p0 = 0.3 p 0 = 0.3 and model M1 M 1 is that it's biased with an unknown parameter p1 p 1. For M0 M 0, we only integrate over the single possible value. However, existing REML or marginal likelihood (ML) based methods for semiparametric generalized linear models (GLMs) use iterative REML or ML estimation of the smoothing parameters of working linear approximations to the GLM. Such indirect schemes need not converge and fail to do so in a non-negligible proportion of practical analyses.Table 2.7 displays a summary of the DIC, WAIC, CPO (i.e., minus the sum of the log-values of CPO) and the marginal likelihood computed for the model fit to the North Carolina SIDS data. All criteria (but the marginal likelihood) slightly favor the most complex model with iid random effects. Note that because this difference is small, we may ...2. Pairwise Marginal Likelihood The proposed pairwise marginal likelihood (PML) belongs to the broad class of pseudo-likelihoods, first proposed by Besag (1975) and also termed composite likelihood by Lindsay (1988). The motivation behind this class is to replace the likelihood by a func-tion that is easier to evaluate, and hence to maximize. Marginal likelihood, M jM j M N + 2 I) noise Understanding the marginal likelihood (1). Models Consider 3 models M1, M2 and M3. Given our data: We want to compute the marginal likelihood for each model. We want to obtain the predictive distribution for each model. 2 0 −2 −6 −4 −2 0 2 4 6 2 0 −2 −6 −4 −2 0 2 , Whether you’re a small business owner or you have some things from around the house you want to get rid of, you’re likely looking to reach a wider number of people and increase the likelihood that you’ll find new customers or connect with t..., This is similar to a different question I asked (The PDF of the Data Given (Marginal Likelihood) the Likelihood and the Prior of a Normal Distribution with Prior on the Mean) yet with totally different model (This is about the conjugate prior Gamma Gamma model and the other question about the Normal Normal conjugate prior model). I am using ..., the marginal likelihood, but is presented as an example of using the Laplace approximation. Lecture 16 3 Figure 1: The standard random effects graphical model 5 Full Bayes versus empirical Bayes Using the standard model from Figure 1, we are now interested in the inference for some function of θ. For, lated likelihood and composite marginal likelihood estimation approaches in the context of the multivariate ordered response model. In W. H. Greene and ..., The marginal likelihood is an integral over the unnormalised posterior distribution, and the question is how it will be affected by reshaping the log likelihood landscape. The novelty of our paper is that it has investigated this question empirically, on a range of benchmark problems, and assesses the accuracy of model selection in comparison ..., Jan 1, 2013 · This marginal likelihood, sometimes also called the evidence, is the normalisation constant required to have the likelihood times the prior PDF (when normalised called the posterior PDF) integrate to unity when integrating over all parameters. The calculation of this value can be notoriously difficult using standard techniques. , In a Bayesian framework, the marginal likelihood is how data update our prior beliefs about models, which gives us an intuitive measure of comparing model fit that is grounded in probability theory. Given the rapid increase in the number and complexity of phylogenetic models, methods for approximating marginal likelihoods are increasingly ..., Jul 16, 2020 · Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. , equivalent to the marginal likelihood for for Je reys prior p() /j j (d+1)=2 on . Result 2.2. Let y ijx i ind˘N(x> i ;˙ 2), i= 1;2;:::;n, where each x i 2Rq is a vector of covariates, is an associated vector of mean parameters of interest and ˙2 is a nuisance variance parameter. Then the pro le likelihood for is equivalent to the marginal ..., %0 Conference Proceedings %T Marginal Likelihood Training of BiLSTM-CRF for Biomedical Named Entity Recognition from Disjoint Label Sets %A Greenberg, Nathan %A Bansal, Trapit %A Verga, Patrick %A McCallum, Andrew %S Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing %D 2018 %8 oct nov %I Association for Computational Linguistics %C Brussels, Belgium %F ..., 12 May 2011 ... marginal) likelihood as opposed to the profile likelihood. The problem of uncertain back- ground in a Poisson counting experiment is ..., Evidence is also called the marginal likelihood and it acts like a normalizing constant and is independent of disease status (the evidence is the same whether calculating posterior for having the disease or not having the disease given a test result). We have already explained the likelihood in detail above., Sep 12, 2014 · Marginal-likelihood scores estimated for each species delimitation can vary depending on the estimator used to calculate them. The SS and PS methods gave strong support for the recognition of the E samples as a distinct species (classifications 3, 4, and 5, see figure 3 ). , Marginal Maximum Likelihood Estimation of Linear Models Description. Implements a survey-weighted marginal maximum estimation, a type of regression where the outcome is a latent trait (such as student ability). Instead of using an estimate, the likelihood function marginalizes student ability. Includes a variety of variance estimation strategies., Mar 5, 2023 · Gaussian Mixture Models Deep Latent Gaussian Models Variational Inference Maximum Marginal Likelihood Learning. Latent Variable Models is a very useful tool in our generative models toolbox. We will compare and give examples of shallow and deep latent variable models, and take a look at how to approximate marginal likelihood using …, denominator has the form of a likelihood term times a prior term, which is identical to what we have already seen in the marginal likelihood case and can be solved using the standard Laplace approximation. However, the numerator has an extra term. One way to solve this would be to fold in G(λ) into h(λ) and use the, However, the marginal likelihood was an unconditional expectation and the weights of the parameter values came from the prior distribution, whereas the posterior predictive distribution is a conditional expectation (conditioned on the observed data \(\mathbf{Y} = \mathbf{y}\)) and weights for the parameter values come from the posterior ..., Since the log-marginal likelihood comes from a MVN, then wouldn't $\hat \mu$ just be the Maximum Likelihood Estimate of the Multivariate Gaussian given as \begin{equation} \bar y = \frac{1}{n}\sum_{i=1}^n y_i \tag{6} \label{mean_mvn} \end{equation} as derived in another CrossValidated answer. Then the GP constant mean vector would just be $1 ..., Marginal tax rate is the rate you pay on any additional income at a certain point. It's what federal tax brackets show. Your average tax rate refers to the rate you pay in total on all of your taxable income. It's less than or equal to your..., Efc ient Marginal Likelihood Optimization in Blind Deconv olution Anat Levin 1, Yair Weiss 2, Fredo Durand 3, William T. Freeman 3 1 Weizmann Institute of Science, 2 Hebrew University, 3 MIT CSAIL Abstract In blind deconvolution one aims to estimate from an in-put blurred image y a sharp image x and an unknown blur kernel k ., Partial deivatives log marginal likelihood w.r.t. hyperparameters where the 2 terms have different signs and the y targets vector is transposed just the first time. Share, The Gaussian process marginal likelihood Log marginal likelihood has a closed form logp(yjx,M i) =-1 2 y>[K+˙2 nI]-1y-1 2 logjK+˙2 Ij-n 2 log(2ˇ) and is the combination of adata fitterm andcomplexity penalty. Occam's Razor is automatic. Carl Edward Rasmussen GP Marginal Likelihood and Hyperparameters October 13th, 2016 3 / 7, Mar 25, 2021 · The marginal likelihood is useful for model comparison. Imagine a simple coin-flipping problem, where model M0 M 0 is that it's biased with parameter p0 = 0.3 p 0 = 0.3 and model M1 M 1 is that it's biased with an unknown parameter p1 p 1. For M0 M 0, we only integrate over the single possible value. , The problem is in your usage of θ θ. Each of the Poisson distributions have a different mean. θi = niλ 100. θ i = n i λ 100. The prior is placed on not θi θ i but on the common parameter λ λ. Thus, when you write down the Likelihood you need to write it in terms of λ λ. Likelihood ∝ ∏i=1m θyi i e−θi = ∏i=m (niλ 100)yi e ..., Partial deivatives log marginal likelihood w.r.t. hyperparameters where the 2 terms have different signs and the y targets vector is transposed just the first time. Share, the full likelihood is a special case of composite likelihood; however, composite likelihood will not usually be a genuine likelihood function, that is, it may not be proportional to the density function of any random vector. The most commonly used versions of composite likelihood are composite marginal likelihood and composite conditional ..., The marginal likelihood for this curve was obtained by replacing the marginal density of the data under the alternative hypothesis with its expected value at the true value of μ. Display full size As in the case of one-sided tests, the alternative hypotheses used to define the ILRs in the Bayesian test can be revised to account for sampling ..., marginal likelihood over tokenisations. We compare different estimators for the marginal likelihood based on sampling, and show that it is feasible to estimate the marginal likeli-hood with a manageable number of samples. We then evaluate pretrained English and Ger-man language models on both the one-best-tokenisation and marginal perplexities, and, On the face of it, the crossfire on Lebanon's border with Israel appears marginal, dwarfed by the scale and intensity of the Hamas-Israel war further south. The fighting has stayed within a ..., Since the log-marginal likelihood comes from a MVN, then wouldn't $\hat \mu$ just be the Maximum Likelihood Estimate of the Multivariate Gaussian given as \begin{equation} \bar y = \frac{1}{n}\sum_{i=1}^n y_i \tag{6} \label{mean_mvn} \end{equation} as derived in another CrossValidated answer. Then the GP constant mean vector would just be $1 ..., Then we obtain a likelihood ratio test, with the ratio 0.9, slightly favoring the binomial model. Actually this marginal likelihood ratio is constant y/n, independent of the posterior distribution of . If , then we get a Bayes factor 1000 favoring the binomial model. Except it is wrong., Maximum Likelihood Estimation Generalized M Estimation. Specifying Estimator Criterion in (2) Least Squares Maximum Likelihood Robust (Contamination-resistant) Bayes (assume β. j. are r.v.’s with known prior distribution) Accommodating incomplete/missing data Case Analyses for (4) Checking Assumptions. Residual analysis. Model errors E. i ...