There was some confusion on the Stan list that I wanted to clear up, having to do with fitting mixture models. The lack of discrete parameters in Stan means that we cannot do model comparison as a hierarchical model with an indexical parameter at the top level. There might be ways to work around this restriction by using clever programming contrivances, but presently there is nothing as straight forward as the model specification in JAGS.
The Stan discussion thread is here. There are huge advantages in efficiency from marginalization, and in some cases it allows us to do inference in models where discrete sampling gets stuck. In other cases, discrete sampling is challenging if not impossible, as in Ising models or the combinatorially similar variable selection models.
It's come up with mixtures, the Cormack-Jolly-Seber model in ecology, and change-point models, all of which are explained in the manual. Bob also points out that what's "very easy" for one person is a "contrivance" for another. And this goes in both directions. If you're already familiar with some software, whether it's Excel or Emacs or Mathematica or whatever, and it does the job for you, then you might as well continue using it.
But if the software doesn't do everything you need, then it's time to learn something new. The point of this post is that if Stan is working for you, or it could be working for you, but you heard that "we cannot do model comparison as a hierarchical model with an indexical parameter at the top level" then, no, this is not actually something to worry about. You actually can fit these models in Stan, it's completely straightforward.
If you don't want to use Stan, that's fine too.
We wrote Stan because existing programs couldn't fit the models we want to fit. Also we are planning to write a mixture model function in Stan that does exactly what's shown in the example above but doesn't have the "lpdf" things in case that upsets people. For example combine Gibbs steps on discrete variables with Nuts on continuous?Kijiji automatic posting
Indeed, the mixture model implementation of the model above should be much faster than any discrete latent-variable implementation. So just on computational grounds alone we prefer how Stan does this.
The quality of Stan really lifts the bar in terms of academic software, and I hope will inspire us all to lift our game there! For these reasons though I would like to put my case that this type of capability would be really great to have.
FWIW I expect the sampler they propose would be a little problematic in general due to the unbounded size of the matrix inversions that would occasionally come up. I imagine this as being an extension to say rstan where fit is replaced with a function that runs a single sample and returns the full state of the chain and additional kernels can be applied that maintain ergodicity. The idea that this could be a template for future algorithm researchers is I think exciting.
I hope this development continues both within and beyond your team. Giving algorithm developers access to both a modelling language and vector calculus routines has a lot of potential!
Using the modelling language could also ease the transition from new algorithm development to actual use in the field by quite a lot.In statisticsa mixture model is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation belongs.
Formally a mixture model corresponds to the mixture distribution that represents the probability distribution of observations in the overall population. However, while problems associated with "mixture distributions" relate to deriving the properties of the overall population from those of the sub-populations, "mixture models" are used to make statistical inferences about the properties of the sub-populations given only observations on the pooled population, without sub-population identity information.
Some ways of implementing mixture models involve steps that attribute postulated sub-population-identities to individual observations or weights towards such sub-populationsin which case these can be regarded as types of unsupervised learning or clustering procedures. However, not all inference procedures involve such steps. Mixture models should not be confused with models for compositional datai. However, compositional models can be thought of as mixture models, where members of the population are sampled at random.
Conversely, mixture models can be thought of as compositional models, where the total size reading population has been normalized to 1. A typical finite-dimensional mixture model is a hierarchical model consisting of the following components:.Burp n. 27 del 08/03/2019
In addition, in a Bayesian settingthe mixture weights and parameters will themselves be random variables, and prior distributions will be placed over the variables. In such a case, the weights are typically viewed as a K -dimensional random vector drawn from a Dirichlet distribution the conjugate prior of the categorical distributionand the parameters will be distributed according to their respective conjugate priors.
This characterization uses F and H to describe arbitrary distributions over observations and parameters, respectively. Typically H will be the conjugate prior of F. The two most common choices of F are Gaussian aka " normal " for real-valued observations and categorical for discrete observations.Weather and climate unit 6th grade
Other common possibilities for the distribution of the mixture components are:. A typical non-Bayesian Gaussian mixture model looks like this:. A Bayesian version of a Gaussian mixture model is as follows:. A Bayesian Gaussian mixture model is commonly extended to fit a vector of unknown parameters denoted in boldor multivariate normal distributions. In a multivariate distribution i.
Note that this formulation yields a closed-form solution to the complete posterior distribution. Such distributions are useful for assuming patch-wise shapes of images and clusters, for example.
One Gaussian distribution of the set is fit to each patch usually of size 8x8 pixels in the image. A typical non-Bayesian mixture model with categorical observations looks like this:. A typical Bayesian mixture model with categorical observations looks like this:.
Financial returns often behave differently in normal situations and during crisis times. A mixture model  for return data seems reasonable.
Sometimes the model used is a jump-diffusion modelor as a mixture of two normal distributions. See Financial economics Challenges and criticism for further context.The case studies on this page are intended to reflect best practices in Bayesian methodology and Stan programming. To contribute a case study, please contact us through the Stan Forums.
We require. This report shows you how to author a Jupyter Notebook for your Stan model and data so that anyone with a modern web browser and a Google account can run your analysis with Google Colaboratory free cloud servers.
View HTML. In this document, we discuss the implementation of Bayesian model-based inference for causal effects in Stan. We start by providing an introduction to the Bayesian inferential framework by analyzing a simulated dataset generated under unconfounded treatment assignment. Then we analyze an example dataset obtained from a completely randomized experiment focusing on the specification of the joint distribution of the potential outcomes.
This case study shows how we can apply Bayesian inference to Hidden Markov Models HMMs using Stan to extract useful information from basketball player tracking data. Specifically we show how to tag drive events and how to determine defensive assignment. Before diving into basketball data we show how to fit an HMM in Stan using a simple example. In this case study, we use Stan to build a series of models to estimate the probability of a successful putt using data from professional golfers.
We fit and check the fit of a series of models, demonstrating the benefits of modeling based on substantive rather than purely statistical principles. We successfully fit to a small dataset and then have to expand the model to fit a new, larger dataset.Idle skilling ascension
We use weakly informative priors and a model-misfit error term to enable the fit. The dIRT model is fit using Stan version 2. In this tutorial, we illustrate how to fit a multilevel linear model within a full Bayesian framework using rstanarm. This tutorial is aimed primarily at educational researchers who have used lme4 in R to fit models to their data and who may be interested in learning how to fit Bayesian multilevel models.
However, for readers who have not used lme4 before, we briefly review the use of the package for fitting multilevel models. Lotka and Volterra formulated parameteric differential equations that characterize the oscillating populations of predators and prey. A statistical model to account for measurement error and unexplained variation uses the deterministic solutions to the Lotka-Volterra equations as expected population sizes.
Stan is used to encode the statistical model and perform full Bayesian inference to solve the inverse problem of inferring parameters from noisy data. Posterior predictive checks for replicated data show the model fits this data well. Full Bayesian inference may be used to estimate future or past populations.Michael Betancourt recently wrote a nice case study describing the problems often encountered with gaussian mixture models, specifically the estimation of parameters of a mixture model and identifiability i.
For this I will use R, but of course Stan is also available in wrappers of python, ruby and others. Firstly lets get the required libraries:. Then we need to generate some toy data. Working in a 4 dimensional parameter space, I want to create 3 gaussian mixtures at different locations:. To run the model in R only takes 1 line too. Here I use iteration steps, of which are warmup for adaptation of the NUTS sampler parameters.
As you can see we get very good Rhat values and effective samples, also the timescale is reasonable. We recover the input parameters really well too. Marginalised 1D posteriors of the 3 gaussian mixture means. The red dotted line is the truth. Firstly lets get the required libraries: library MASS require rstan Then we need to generate some toy data. Share this: Twitter Facebook. Like this: Like Loading Related posts. Post to Cancel.
Unfortunately when those data data generating processes are degenerate the resulting mixture model suffers from inherent combinatorial non-identifiabilities that frustrate accurate computation. Consequently, in order to utilize mixture models reliably in practice we need strong and principled prior information to ameliorate these frustrations.
In this case study I will first introduce how mixture models are implemented in Bayesian inference. I will then discuss the non-identifiability inherent to that construction as well as how the non-identifiability can be tempered with principled prior information.
Lastly I will demonstrate how these issues manifest in a simple example, with a final tangent to consider an additional pathology that can arise in Bayesian mixture models. To implement such a model we need to construct the corresponding likelihood and then subsequent posterior distribution.
MCMC for Hierarchical Mixture Models
By combining assignments with a set of data generating processes we admit an extremely expressive class of models that encompass many different inferential and decision problems. Similarly, if both the measurements and the assignments are given then inference over the mixture model admits classification of future measurements.
Finally, semi-supervised learning corresponds to inference over a mixture model where only some of the assignments are known. In practice discrete assignments are difficult to fit accurately and efficiently, but we can facilitate inference by marginalizing the assignments out of the model entirely. Marginalizing out the discrete assignments yields a likelihood that depends on only continuous parameters, making it amenable to state-of-the-art tools like Stan.
Moreover, modeling the latent mixture probabilities instead of the discrete assignments admits more precise inferences as a consequence of the Rao-Blackwell theorem. From any perspective the marginalized mixture likelihood is the ideal basis for inference. Consequently we will consider only a single measurement in the proceeding section, returning to multiple measurements in the example.
This introduces a subtle challenge because if the measurement cannot discriminate between the components then it cannot discriminate between the component parameters. Because of this labeling degeneracy the posterior distribution will be non-identified. In particular, it will manifest multimodality, with one mode for each of the possible labelings. Hence even for a relatively small number of components the posterior distribution will have too many modes for any statistical algorithm to accurately quantify unless the modes collapse into each other.
For example, if we applied Markov chain Monte Carlo then any chain would be able to explore one of the modes but it would not be able to transition between the modes, at least not within in any finite running time. Even if we had a statistical algorithm that could transition between the degenerate modes and explore the entire mixture posterior, typically there will be too many modes to complete that exploration in any reasonable time.
Consequently if we want to accurately fit these models in practice then we need to break the labeling degeneracy and remove the extreme multimodality altogether. Exactly how we break the labeling degeneracy depends on what prior information we can exploit. In particular, our strategy will be different depending on whether our prior information is exchangeable or not. Because the posterior distribution inherits the permutation-invariance of the mixture likelihood only if the priors are exchangeable, one way to immediately obstruct the labeling degeneracy of the mixture posterior is to employ non-exchangeable priors.
This approach is especially useful when each component of the likelihood is meant to be responsible for a specific purpose, for example when each component models a known subpopulations with distinct behaviors about which we have prior information. If this principled prior information is strong enough then the prior can suppress all but the one labeling consistent with these responsibilities, ensuring a unimodal mixture posterior distribution.Clustering: Gaussian Mixture Models (12c)
When our prior information is exchangeable there is nothing preventing the mixture posterior from becoming multimodal and impractical to fit. When our inferences are also exchangeable, however, we can exploit the symmetry of the labeling degeneracies to simplify the computational problem dramatically.
Each labeling is characterized by the unique assignment of indices to the components in our mixture.Checks: 7 0.
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Facilities to help determine the appropriate number of components are also provided. Two-component Gaussian mixture model: data points, and equi-probability surfaces of the model. A Gaussian mixture model is a probabilistic model that assumes all the data points are generated from a mixture of a finite number of Gaussian distributions with unknown parameters.
One can think of mixture models as generalizing k-means clustering to incorporate information about the covariance structure of the data as well as the centers of the latent Gaussians. Scikit-learn implements different classes to estimate Gaussian mixture models, that correspond to different estimation strategies, detailed below. The GaussianMixture object implements the expectation-maximization EM algorithm for fitting mixture-of-Gaussian models.
It can also draw confidence ellipsoids for multivariate models, and compute the Bayesian Information Criterion to assess the number of clusters in the data. A GaussianMixture. Given test data, it can assign to each sample the Gaussian it mostly probably belong to using the GaussianMixture. The GaussianMixture comes with different options to constrain the covariance of the difference classes estimated: spherical, diagonal, tied or full covariance.
See GMM covariances for an example of using the Gaussian mixture as clustering on the iris dataset. See Density Estimation for a Gaussian mixture for an example on plotting the density estimation.
As this algorithm maximizes only the likelihood, it will not bias the means towards zero, or bias the cluster sizes to have specific structures that might or might not apply. When one has insufficiently many points per mixture, estimating the covariance matrices becomes difficult, and the algorithm is known to diverge and find solutions with infinite likelihood unless one regularizes the covariances artificially.
This algorithm will always use all the components it has access to, needing held-out data or information theoretical criteria to decide how many components to use in the absence of external cues.
The BIC criterion can be used to select the number of components in a Gaussian Mixture in an efficient way. In theory, it recovers the true number of components only in the asymptotic regime i. Note that using a Variational Bayesian Gaussian mixture avoids the specification of the number of components for a Gaussian mixture model. See Gaussian Mixture Model Selection for an example of model selection performed with classical Gaussian mixture.
Expectation-maximization is a well-founded statistical algorithm to get around this problem by an iterative process.
First one assumes random components randomly centered on data points, learned from k-means, or even just normally distributed around the origin and computes for each point a probability of being generated by each component of the model. Then, one tweaks the parameters to maximize the likelihood of the data given those assignments. Repeating this process is guaranteed to always converge to a local optimum.
The BayesianGaussianMixture object implements a variant of the Gaussian mixture model with variational inference algorithms. Variational inference is an extension of expectation-maximization that maximizes a lower bound on model evidence including priors instead of data likelihood. The principle behind variational methods is the same as expectation-maximization that is both are iterative algorithms that alternate between finding the probabilities for each point to be generated by each mixture and fitting the mixture to these assigned pointsbut variational methods add regularization by integrating information from prior distributions.
This avoids the singularities often found in expectation-maximization solutions but introduces some subtle biases to the model. Inference is often notably slower, but not usually as much so as to render usage unpractical. Specifying a low value for the concentration prior will make the model put most of the weight on few components set the remaining components weights very close to zero.
High values of the concentration prior will allow a larger number of components to be active in the mixture. The parameters implementation of the BayesianGaussianMixture class proposes two types of prior for the weights distribution: a finite mixture model with Dirichlet distribution and an infinite mixture model with the Dirichlet Process.
In practice Dirichlet Process inference algorithm is approximated and uses a truncated distribution with a fixed maximum number of components called the Stick-breaking representation. The number of components actually used almost always depends on the data. The examples below compare Gaussian mixture models with a fixed number of components, to the variational Gaussian mixture models with a Dirichlet process prior.
Here, a classical Gaussian mixture is fitted with 5 components on a dataset composed of 2 clusters. We can see that the variational Gaussian mixture with a Dirichlet process prior is able to limit itself to only 2 components whereas the Gaussian mixture fits the data with a fixed number of components that has to be set a priori by the user.
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