Bayesian hierarchical modeling

Statistical methods and models commonly involve multiple parameters that can be regarded as related or connected in such a way that the problem implies a dependence of the joint probability model for these parameters.{{sfn|Gelman|Carlin|Stern|Rubin|2004|page=117}}

Individual degrees of belief, expressed in the form of probabilities, come with uncertainty.{{cite journal |last1=Good |first1=I.J. |date=1980 |title=Some history of the hierarchical Bayesian methodology |journal=Trabajos de Estadistica y de Investigacion Operativa |volume=31 |pages=489–519 |doi=10.1007/BF02888365 |s2cid=121270218 |url=http://dialnet.unirioja.es/servlet/oaiart?codigo=2368428 }} Amidst this is the change of the degrees of belief over time. As was stated by Professor José M. Bernardo and Professor Adrian F. Smith, “The actuality of the learning process consists in the evolution of individual and subjective beliefs about the reality.” These subjective probabilities are more directly involved in the mind rather than the physical probabilities. Hence, it is with this need of updating beliefs that Bayesians have formulated an alternative statistical model which takes into account the prior occurrence of a particular event.Bernardo, Smith(1994). [https://books.google.com/books?id=11nSgIcd7xQC&dq=bernardo+degroot+lindley&pg=PA497 Bayesian Theory]. Chichester, England: John Wiley & Sons, {{ISBN|0-471-92416-4}}, p. 23

The assumed occurrence of a real-world event will typically modify preferences between certain options. This is done by modifying the degrees of belief attached, by an individual, to the events defining the options.{{sfn|Gelman|Carlin|Stern|Rubin|2004|pages=6–8}}

Suppose in a study of the effectiveness of cardiac treatments, with the patients in hospital j having survival probability $\backslash theta\_j$, the survival probability will be updated with the occurrence of y, the event in which a controversial serum is created which, as believed by some, increases survival in cardiac patients.

In order to make updated probability statements about $\backslash theta\_j$, given the occurrence of event y, we must begin with a model providing a joint probability distribution for $\backslash theta\_j$ and y. This can be written as a product of the two distributions that are often referred to as the prior distribution $P(\backslash theta)$ and the sampling distribution $P(y\backslash mid\backslash theta)$ respectively:

: $P(\backslash theta,\; y)\; =\; P(\backslash theta)P(y\backslash mid\backslash theta)$

Using the basic property of conditional probability, the posterior distribution will yield:

: $P(\backslash theta\backslash mid\; y)=\backslash frac\{P(\backslash theta,y)\}\{P(y)\}\; =\; \backslash frac\{P(y\backslash mid\backslash theta)P(\backslash theta)\}\{P(y)\}$

This equation, showing the relationship between the conditional probability and the individual events, is known as Bayes' theorem. This simple expression encapsulates the technical core of Bayesian inference which aims to deconstruct the probability, $P(\backslash theta\backslash mid\; y)$, relative to solvable subsets of its supportive evidence.{{sfn|Gelman|Carlin|Stern|Rubin|2004|pages=6–8}}

The usual starting point of a statistical analysis is the assumption that the n values $y\_1,\; y\_2,\; \backslash ldots,\; y\_n$ are exchangeable. If no information – other than data y – is available to distinguish any of the $\backslash theta\_j$’s from any others, and no ordering or grouping of the parameters can be made, one must assume symmetry of prior distribution parameters.Bernardo, Degroot, Lindley (September 1983). [https://books.google.com/books?id=myfRtgAACAAJ&q=Proceedings+of+the+Second+Valencia+International+Meeting “Proceedings of the Second Valencia International Meeting”]. [https://books.google.com/books?id=wYj-_uFLOe4C&q=Proceedings%20of%20the%20Second%20Valencia%20International%20Meeting Bayesian Statistics 2]. Amsterdam: Elsevier Science Publishers B.V, {{ISBN|0-444-87746-0}}, pp. 167–168 This symmetry is represented probabilistically by exchangeability. Generally, it is useful and appropriate to model data from an exchangeable distribution as independently and identically distributed, given some unknown parameter vector $\backslash theta$, with distribution $P(\backslash theta)$.

For a fixed number n, the set $y\_1,\; y\_2,\; \backslash ldots,\; y\_n$ is exchangeable if the joint probability $P(y\_1,\; y\_2,\; \backslash ldots,\; y\_n)$ is invariant under permutations of the indices. That is, for every permutation $\backslash pi$ or $(\backslash pi\_1,\; \backslash pi\_2,\; \backslash ldots,\; \backslash pi\_n)$ of (1, 2, …, n), $P(y\_1,\; y\_2,\; \backslash ldots,\; y\_n)=\; P(y\_\{\backslash pi\_1\},\; y\_\{\backslash pi\_2\},\; \backslash ldots,\; y\_\{\backslash pi\_n\}).${{sfn|Gelman|Carlin|Stern|Rubin|2004|pages=121–125}}

The following is an exchangeable, but not independent and identical (iid), example:

Consider an urn with a red ball and a blue ball inside, with probability $\backslash frac\{1\}\{2\}$ of drawing either. Balls are drawn without replacement, i.e. after one ball is drawn from the n balls, there will be n − 1 remaining balls left for the next draw.

: $\backslash text\{Let\; \}$

Y_i =

\begin{cases}

1, & \text{if the }i\text{th ball is red},\\

0, & \text{otherwise}.

\end{cases}

The probability of selecting a red ball in the first draw and a blue ball in the second draw is equal to the probability of selecting a blue ball on the first draw and a red on the second, both of which are 1/2:

: $[P(y\_1\; =\; 1,\; y\_2\; =0)\; =\; P(y\_1=0,y\_2=1)=\; \backslash frac\{1\}\{2\}]$)

This makes $y\_1$ and $y\_2$ exchangeable.

But the probability of selecting a red ball on the second draw given that the red ball has already been selected in the first is 0. This is not equal to the probability that the red ball is selected in the second draw, which is 1/2:

: $[P(y\_2=1\backslash mid\; y\_1=1)=0\; \backslash ne\; P(y\_2=1)=\; \backslash frac\{1\}\{2\}]$).

Thus, $y\_1$ and $y\_2$ are not independent.

If $x\_1,\; \backslash ldots,\; x\_n$ are independent and identically distributed, then they are exchangeable, but the converse is not necessarily true.Diaconis, Freedman (1980). [http://projecteuclid.org/download/pdf_1/euclid.aop/1176994663 “Finite exchangeable sequences”]. Annals of Probability, pp. 745–747

Infinite exchangeability is the property that every finite subset of an infinite sequence $y\_1$, $y\_2,\; \backslash ldots$ is exchangeable. For any n, the sequence $y\_1,\; y\_2,\; \backslash ldots,\; y\_n$ is exchangeable.

Bayesian hierarchical modeling makes use of two important concepts in deriving the posterior distribution, namely:

1. Hyperparameters: parameters of the prior distribution
2. Hyperpriors: distributions of Hyperparameters

Suppose a random variable Y follows a normal distribution with parameter $\backslash theta$ as the mean and 1 as the variance, that is $Y\backslash mid\; \backslash theta\; \backslash sim\; N(\backslash theta,1)$. The tilde relation $\backslash sim$ can be read as "has the distribution of" or "is distributed as". Suppose also that the parameter $\backslash theta$ has a distribution given by a normal distribution with mean $\backslash mu$ and variance 1, i.e. $\backslash theta\backslash mid\backslash mu\; \backslash sim\; N(\backslash mu,1)$. Furthermore, $\backslash mu$ follows another distribution given, for example, by the standard normal distribution, $\backslash text\{N\}(0,1)$. The parameter $\backslash mu$ is called the hyperparameter, while its distribution given by $\backslash text\{N\}(0,1)$ is an example of a hyperprior distribution. The notation of the distribution of Y changes as another parameter is added, i.e. $Y\; \backslash mid\; \backslash theta,\backslash mu\; \backslash sim\; N(\backslash theta,1)$. If there is another stage, say, $\backslash mu$ following another normal distribution with a mean of $\backslash beta$ and a variance of $\backslash epsilon$, then $\backslash mu\; \backslash sim\; N(\backslash beta,\backslash epsilon)$, $\backslash mbox\; \{\; \}$$\backslash beta$ and $\backslash epsilon$ can also be called hyperparameters with hyperprior distributions.{{sfn|Gelman|Carlin|Stern|Rubin|2004|page=117}}

Let $y\_j$ be an observation and $\backslash theta\_j$ a parameter governing the data generating process for $y\_j$. Assume further that the parameters $\backslash theta\_1,\; \backslash theta\_2,\; \backslash ldots,\; \backslash theta\_j$ are generated exchangeably from a common population, with distribution governed by a hyperparameter $\backslash phi$.

The Bayesian hierarchical model contains the following stages:

: $\backslash text\{Stage\; I:\; \}\; y\_j\backslash mid\backslash theta\_j,\backslash phi\; \backslash sim\; P(y\_j\backslash mid\backslash theta\_j,\backslash phi)$

: $\backslash text\{Stage\; II:\; \}\; \backslash theta\_j\backslash mid\backslash phi\; \backslash sim\; P(\backslash theta\_j\backslash mid\backslash phi)$

: $\backslash text\{Stage\; III:\; \}\; \backslash phi\; \backslash sim\; P(\backslash phi)$

The likelihood, as seen in stage I is $P(y\_j\backslash mid\backslash theta\_j,\backslash phi)$, with $P(\backslash theta\_j,\backslash phi)$ as its prior distribution. Note that the likelihood depends on $\backslash phi$ only through $\backslash theta\_j$.

The prior distribution from stage I can be broken down into:

: $P(\backslash theta\_j,\backslash phi)\; =\; P(\backslash theta\_j\backslash mid\backslash phi)P(\backslash phi)$ [from the definition of conditional probability]

With $\backslash phi$ as its hyperparameter with hyperprior distribution, $P(\backslash phi)$.

Thus, the posterior distribution is proportional to:

: $P(\backslash phi,\backslash theta\_j\backslash mid\; y)\; \backslash propto\; P(y\_j\; \backslash mid\backslash theta\_j,\backslash phi)\; P(\backslash theta\_j,\backslash phi)$ [using Bayes' Theorem]

: $P(\backslash phi,\backslash theta\_j\backslash mid\; y)\; \backslash propto\; P(y\_j\backslash mid\backslash theta\_j\; )\; P(\backslash theta\_j\; \backslash mid\backslash phi\; )\; P(\backslash phi)$Bernardo, Degroot, Lindley (September 1983). [https://books.google.com/books?id=wYj-_uFLOe4C&q=Proceedings%20of%20the%20Second%20Valencia%20International%20Meeting “Proceedings of the Second Valencia International Meeting”]. [https://books.google.com/books?id=wYj-_uFLOe4C&q=Proceedings%20of%20the%20Second%20Valencia%20International%20Meeting Bayesian Statistics 2]. Amsterdam: Elsevier Science Publishers B.V, {{ISBN|0-444-87746-0}}, pp. 371–372

As an example, a teacher wants to estimate how well a student did on the SAT. The teacher uses the current grade point average (GPA) of the student for an estimate. Their current GPA, denoted by $Y$, has a likelihood given by some probability function with parameter $\backslash theta$, i.e. $Y\backslash mid\backslash theta\; \backslash sim\; P(Y\backslash mid\backslash theta)$. This parameter $\backslash theta$ is the SAT score of the student. The SAT score is viewed as a sample coming from a common population distribution indexed by another parameter $\backslash phi$, which is the high school grade of the student (freshman, sophomore, junior or senior).{{sfn|Gelman|Carlin|Stern|Rubin|2004|pages=120–121}} That is, $\backslash theta\backslash mid\backslash phi\; \backslash sim\; P(\backslash theta\backslash mid\backslash phi)$. Moreover, the hyperparameter $\backslash phi$ follows its own distribution given by $P(\backslash phi)$, a hyperprior.

These relationships can be used to calculate the likelihood of a specific SAT score relative to a particular GPA:

: $P(\backslash theta,\backslash phi\backslash mid\; Y)\; \backslash propto\; P(Y\backslash mid\backslash theta,\backslash phi)P(\backslash theta,\backslash phi)$

: $P(\backslash theta,\backslash phi\backslash mid\; Y)\; \backslash propto\; P(Y\backslash mid\backslash theta)P(\backslash theta\backslash mid\backslash phi)P(\backslash phi)$

All information in the problem will be used to solve for the posterior distribution. Instead of solving only using the prior distribution and the likelihood function, using hyperpriors allows a more nuanced distinction of relationships between given variables.Box G. E. P., Tiao G. C. (1965). [http://projecteuclid.org/download/pdf_1/euclid.aoms/1177699906 "Multiparameter problem from a bayesian point of view"]. [http://projecteuclid.org/all/euclid.aoms Multiparameter Problems From A Bayesian Point of View Volume 36 Number 5]. New York City: John Wiley & Sons, {{ISBN|0-471-57428-7}}

In general, the joint posterior distribution of interest in 2-stage hierarchical models is:

: $P(\backslash theta,\backslash phi\backslash mid\; Y)\; =\; \{P(Y\backslash mid\backslash theta,\backslash phi)\; P(\backslash theta,\backslash phi)\; \backslash over\; P(Y)\}\; =\; \{P(Y\backslash mid\backslash theta)P(\backslash theta\backslash mid\backslash phi)P(\backslash phi)\; \backslash over\; P(Y)\}$

: $P(\backslash theta,\backslash phi\backslash mid\; Y)\; \backslash propto\; P(Y\backslash mid\backslash theta)P(\backslash theta\backslash mid\backslash phi)P(\backslash phi)$

For 3-stage hierarchical models, the posterior distribution is given by:

: $P(\backslash theta,\backslash phi,\; X\backslash mid\; Y)\; =\; \{P(Y\backslash mid\backslash theta)P(\backslash theta\backslash mid\backslash phi)P(\backslash phi\backslash mid\; X)P(X)\; \backslash over\; P(Y)\}$

: $P(\backslash theta,\backslash phi,\; X\backslash mid\; Y)\; \backslash propto\; P(Y\backslash mid\backslash theta)P(\backslash theta\backslash mid\backslash phi)P(\backslash phi\backslash mid\; X)P(X)$

File:Bayesian research cycle.png

A three stage version of Bayesian hierarchical modeling could be used to calculate probability at 1) an individual level, 2) at the level of population and 3) the prior, which is an assumed probability distribution that takes place before evidence is initially acquired:

Stage 1: Individual-Level Model

$\{y\}\_\{ij\}\; =\; f(t\_\{ij\};\backslash theta\_\{1i\},\backslash theta\_\{2i\},\backslash ldots,\backslash theta\_\{li\},\backslash ldots,\backslash theta\_\{Ki\}\; )\; +\; \backslash epsilon\_\{ij\},\backslash quad\; \backslash epsilon\_\{ij\}\; \backslash sim\; N(0,\; \backslash sigma^2),\; \backslash quad\; i\; =1,\backslash ldots,\; N,\; \backslash ,\; j\; =\; 1,\backslash ldots,\; M\_i.$

Stage 2: Population Model

$\backslash theta\_\{li\}=\; \backslash alpha\_l\; +\; \backslash sum\_\{b=1\}^\{P\}\backslash beta\_\{lb\}x\_\{ib\}\; +\; \backslash eta\_\{li\},\; \backslash quad\; \backslash eta\_\{li\}\; \backslash sim\; N(0,\; \backslash omega\_l^2),\; \backslash quad\; i\; =1,\backslash ldots,\; N,\; \backslash ,\; l=1,\backslash ldots,\; K.$

Stage 3: Prior

$\backslash sigma^2\; \backslash sim\; \backslash pi(\backslash sigma^2),\backslash quad\; \backslash alpha\_l\; \backslash sim\; \backslash pi(\backslash alpha\_l),\; \backslash quad\; (\backslash beta\_\{l1\},\backslash ldots,\backslash beta\_\{lb\},\backslash ldots,\backslash beta\_\{lP\})\; \backslash sim\; \backslash pi(\backslash beta\_\{l1\},\backslash ldots,\backslash beta\_\{lb\},\backslash ldots,\backslash beta\_\{lP\}),\; \backslash quad\; \backslash omega\_l^2\; \backslash sim\; \backslash pi(\backslash omega\_l^2),\; \backslash quad\; l=1,\backslash ldots,\; K.$

Here, $y\_\{ij\}$ denotes the continuous response of the $i$-th subject at the time point $t\_\{ij\}$, and $x\_\{ib\}$ is the $b$-th covariate of the $i$-th subject. Parameters involved in the model are written in Greek letters. The variable $f(t\; ;\; \backslash theta\_\{1\},\backslash ldots,\backslash theta\_\{K\})$ is a known function parameterized by the $K$-dimensional vector $(\backslash theta\_\{1\},\backslash ldots,\backslash theta\_\{K\})$.

Typically, $f$ is a `nonlinear' function and describes the temporal trajectory of individuals. In the model, $\backslash epsilon\_\{ij\}$ and $\backslash eta\_\{li\}$ describe within-individual variability and between-individual variability, respectively. If the prior is not considered, the relationship reduces to a frequentist nonlinear mixed-effect model.

A central task in the application of the Bayesian nonlinear mixed-effect models is to evaluate posterior density:

$\backslash pi(\backslash \{\backslash theta\_\{li\}\backslash \}\_\{i=1,l=1\}^\{N,K\},\backslash sigma^2,\; \backslash \{\backslash alpha\_l\backslash \}\_\{l=1\}^K,\; \backslash \{\backslash beta\_\{lb\}\backslash \}\_\{l=1,b=1\}^\{K,P\},\backslash \{\backslash omega\_l\backslash \}\_\{l=1\}^K\; |\; \backslash \{y\_\{ij\}\backslash \}\_\{i=1,j=1\}^\{N,M\_i\})$

$\backslash propto\; \backslash pi(\backslash \{y\_\{ij\}\backslash \}\_\{i=1,j=1\}^\{N,M\_i\},\; \backslash \{\backslash theta\_\{li\}\backslash \}\_\{i=1,l=1\}^\{N,K\},\backslash sigma^2,\; \backslash \{\backslash alpha\_l\backslash \}\_\{l=1\}^K,\; \backslash \{\backslash beta\_\{lb\}\backslash \}\_\{l=1,b=1\}^\{K,P\},\backslash \{\backslash omega\_l\backslash \}\_\{l=1\}^K)$

$=\; \backslash underbrace\{\backslash pi(\backslash \{y\_\{ij\}\backslash \}\_\{i=1,j=1\}^\{N,M\_i\}\; |\backslash \{\backslash theta\_\{li\}\backslash \}\_\{i=1,l=1\}^\{N,K\},\backslash sigma^2)\}\_\{Stage\; 1:\; Individual-Level\; Model\}$

\times

\underbrace{\pi(\{\theta_{li}\}_{i=1,l=1}^{N,K}|\{\alpha_l\}_{l=1}^K, \{\beta_{lb}\}_{l=1,b=1}^{K,P},\{\omega_l\}_{l=1}^K)}_{Stage 2: Population Model}

\times

\underbrace{p(\sigma^2, \{\alpha_l\}_{l=1}^K, \{\beta_{lb}\}_{l=1,b=1}^{K,P},\{\omega_l\}_{l=1}^K)}_{Stage 3: Prior}

The panel on the right displays Bayesian research cycle using Bayesian nonlinear mixed-effects model. A research cycle using the Bayesian nonlinear mixed-effects model comprises two steps: (a) standard research cycle and (b) Bayesian-specific workflow.

A standard research cycle involves 1) literature review, 2) defining a problem and 3) specifying the research question and hypothesis. Bayesian-specific workflow stratifies this approach to include three sub-steps: (b)–(i) formalizing prior distributions based on background knowledge and prior elicitation; (b)–(ii) determining the likelihood function based on a nonlinear function $f$; and (b)–(iii) making a posterior inference. The resulting posterior inference can be used to start a new research cycle.

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Category:Bayesian networks