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How do you handle multimodal and non-normal distributions using Bayesian methods?

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Multimodal and non-normal distributions are common in real-world data, but they pose challenges for traditional statistical methods that assume normality and unimodality. Bayesian methods, on the other hand, offer a flexible and principled way to handle these complex distributions, by using prior knowledge, likelihood functions, and posterior inference. In this article, you will learn how to use Bayesian methods to handle multimodal and non-normal distributions, and how to apply them to some examples using Python and PyMC3.

本文章的要点总结

Use mixture models:

Mixture models are excellent for handling multimodal distributions. With Python and PyMC3, you can fit these models to your data, revealing hidden subgroups.### *Opt for non-parametric models:Non-parametric models adapt to the data's shape, ideal for non-normal distributions. In PyMC3, you can use Dirichlet process mixtures to let the data dictate the distribution form.

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1 What are multimodal and non-normal distributions?

A distribution is multimodal if it has more than one peak or mode, indicating that the data has multiple subgroups or clusters. For example, the height distribution of a mixed-gender population may be bimodal, with one mode for males and one for females. A distribution is non-normal if it does not follow the familiar bell-shaped curve of the normal distribution, which is often assumed in many statistical methods. For example, the income distribution of a population may be skewed, with a long tail of high earners. Multimodal and non-normal distributions can be challenging to analyze, because they may violate the assumptions of parametric tests, linear models, or hypothesis testing.

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Nathan Brouwer, PhD

Data Scientist with 15 years of experience wrangling insights from complex datasets in R and Python
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Its always important to emphasize that the distribution of the raw data isn't what we are concerned with, but the distribution of the residuals (aka errors) *conditional* on the model.

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Alan Chernoff

Visiting Professor at TCNJ
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Another thing worth noting is that in non-normal distributions the variance (and thus standard deviation) may be misleading measures of spread. Since the variance is the mean squared distance from the mean, data points that are far from the mean have a disproportionate effect on the variance, and in a non-normal distribution, this may be a large amount of the data. In the same way the mean of a data set may not be a 'good' measure of center for a non-normal distribution, variance and standard deviation are likely no longer 'good' measures of spread, and other statistics derived from them, such as correlation, may no longer be useful either.

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2 Why use Bayesian methods?

Bayesian methods are based on the idea of updating your beliefs about a parameter or a model based on new data and prior information. Bayesian methods use probability distributions to represent uncertainty and variability, and they can handle any kind of distribution, whether multimodal, non-normal, or otherwise. Bayesian methods also allow you to compare and evaluate different models, using criteria such as Bayes factors, information criteria, or posterior predictive checks. Bayesian methods can also incorporate domain knowledge, expert opinions, or previous studies into the analysis, by using informative priors or hierarchical models.

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Aaron Schumacher

Data Scientist at Meta
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To be clear, Bayesian methods will not automatically handle anything in particular, and frequentist approaches don't necessarily have a problem with non-normal distributions. The Central Limit Theorem doesn't require that the underlying distribution be normal.

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Eric Schles

Senior Data Scientist
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Yea, I think that's fair. You can also use bootstrapping to recover a distribution of parameter values from frequentist approaches as well. (responding to @aaron's comment)

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3 How to use Bayesian methods?

Bayesian methods follow a general framework of specifying a model, computing a posterior distribution, and performing inference and diagnostics. A model consists of a prior distribution, which represents your initial beliefs about a parameter or a model, and a likelihood function, which represents how the data are generated given the parameter or the model. A posterior distribution is the updated distribution of the parameter or the model after observing the data, and it is computed using Bayes' theorem. Inference and diagnostics involve summarizing the posterior distribution, checking its validity and fit, and comparing different models.

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Alan Chernoff

Visiting Professor at TCNJ
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A common concern that new practitioners of Bayesian methodology may have is the question of how the prior distribution is selected. This is by no means a trivial question, and in my opinion is one of the more valid criticisms of Bayesian statistics. While selecting a prior distribution to update may be somewhat subjective, if you really want to attempt to be as objective as possible, uninformative prior distributions, such as Jeffrey's prior, do exist. However, it is worth noting that 'uninformative' is a bit of a misnomer here, as any distribution will contain some inherent assumptions and thus information. But when you have a good reason to inform your prior distribution selection, Bayesian updating is incredibly useful.

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4 How to use Python and PyMC3?

Python is a popular programming language for data analysis and visualization, and PyMC3 is a Python library for probabilistic programming and Bayesian modeling. PyMC3 allows you to easily define and fit Bayesian models using a declarative syntax, and it uses advanced sampling algorithms such as Markov chain Monte Carlo (MCMC) or variational inference (VI) to approximate the posterior distribution. PyMC3 also provides various tools for inference and diagnostics, such as posterior summaries, plots, statistics, and checks.

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Garry Perkins
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What is wrong with PyMC? I never even knew there was a 2, no less a three. Does this PyMC3 use Tensorflow probability? Then again, what do I know? I am not a math guy, but I would like to know what the dominant model is (and not get stuck in an offshoot with limited functionality, or weird file type issues and similar).

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5 How to handle multimodal distributions?

One way to handle multimodal distributions is to use a mixture model, which assumes that the data are generated by a combination of several sub-distributions, each with its own parameters and proportions. For example, you can use a Gaussian mixture model to fit a bimodal distribution, by specifying two normal distributions with different means and standard deviations, and a categorical distribution for the mixing proportions. You can then use PyMC3 to define and fit the mixture model, and to infer the posterior distributions of the parameters and the proportions. You can also use PyMC3 to plot the posterior predictive distribution, which shows how well the mixture model captures the multimodality of the data.

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Aaron Hendrickson

Applied Mathematician | Currently developing characterization algorithms for DSERN image sensors
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Using a mixture model in conjunction with the EM algorithm is a great option to fit multimodal data. The example towards the beginning of the article is a good one to motivate this analysis method. The heights of a population may be described by a gaussian mixture with two components (one component for male and another for female). After fitting the data with EM, it's possible to then estimate the latent variables of the model by assigning a label, i.e. 'male' or 'female', to each data point.

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6 How to handle non-normal distributions?

Another way to handle non-normal distributions is to use a non-parametric model, which does not assume a specific functional form for the distribution, but instead lets the data determine the shape and complexity of the distribution. For example, you can use a Dirichlet process mixture model to fit a skewed or heavy-tailed distribution, by specifying a base distribution (such as a normal or a Student's t distribution) and a concentration parameter (which controls the number of clusters), and letting the Dirichlet process generate the mixture components adaptively. You can then use PyMC3 to define and fit the Dirichlet process mixture model, and to infer the posterior distributions of the parameters and the clusters. You can also use PyMC3 to plot the posterior predictive distribution, which shows how well the Dirichlet process mixture model captures the non-normality of the data.

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7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How do you handle multimodal and non-normal distributions using Bayesian methods?

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3

4

5

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1 What are multimodal and non-normal distributions?

2 Why use Bayesian methods?

3 How to use Bayesian methods?

4 How to use Python and PyMC3?

5 How to handle multimodal distributions?

6 How to handle non-normal distributions?

7 Here’s what else to consider

Statistics

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How do you handle multimodal and non-normal distributions using Bayesian methods?

1

2

3

4

5

6

7

1 What are multimodal and non-normal distributions?

2 Why use Bayesian methods?

3 How to use Bayesian methods?

4 How to use Python and PyMC3?

5 How to handle multimodal distributions?

6 How to handle non-normal distributions?

7 Here’s what else to consider

Statistics

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