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Bayesian analysis of finite mixture models of distributions from exponential families

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Abstract

This paper deals with the Bayesian analysis of finite mixture models with a fixed number of component distributions from natural exponential families with quadratic variance function (NEF-QVF). A unified Bayesian framework addressing the two main difficulties in this context is presented, i.e., the prior distribution choice and the parameter unidentifiability problem. In order to deal with the first issue, conjugate prior distributions are used. An algorithm to calculate the parameters in the prior distribution to obtain the least informative one into the class of conjugate distributions is developed. Regarding the second issue, a general algorithm to solve the label-switching problem is presented. These techniques are easily applied in practice as it is shown with an illustrative example.

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Acknowledgments

Comments from David Ríos-Insua are gratefully acknowledged. This research was partially supported by Ministerio de Educatión y Ciencia, Spain (Project TSI2004-06801-C04-03).

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Authors and Affiliations

  1. Department of Mathematics, University of Extremadura, Avda. de la Universidad s/n, 10071, Cáceres, Spain

    M. J. Rufo, J. Martín & C. J. Pérez

Authors
  1. M. J. Rufo
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  2. J. Martín
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  3. C. J. Pérez
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Corresponding author

Correspondence to C. J. Pérez.

Appendix: Theoretical results

Appendix: Theoretical results

Lemma 1 is required for Proposition 1. In Algorithm 3, step 1 follows from Proposition 2 and step 2 follows from Proposition 1.

Lemma 1 The following expression holds:

$$\int \mathrm{NQ}\left(\mu_{l}, V\left(\mu_{l}\right)\right) \log \left(\mathrm{NQ}\left(\mu_{i}, V\left(\mu_{i}\right)\right)\right) \mathrm{d} x=\theta_{i}\left(\mu_{i}\right) \mu_{l}-M\left(\theta_{i}\left(\mu_{i}\right)\right).$$

Proof

$$\begin{array}{l}{\int \mathrm{NQ}\left(\mu_{l}, V\left(\mu_{l}\right)\right) \log \left(\mathrm{NQ}\left(\mu_{i}, V\left(\mu_{i}\right)\right)\right) \mathrm{d} x=E\left(X_{l} \theta_{i}\left(\mu_{i}\right)-M\left(\theta_{i}\left(\mu_{i}\right)\right)\right)} \\ {\quad=E\left(X_{l} \theta_{i}\left(\mu_{i}\right)\right)-E\left(M\left(\theta_{i}\left(\mu_{i}\right)\right)\right)=\theta_{i}\left(\mu_{i}\right) E\left(X_{l}\right)-M\left(\theta_{i}\left(\mu_{i}\right)\right)} \\ {\quad=\theta_{i}\left(\mu_{i}\right) \mu_{l}-M\left(\theta_{i}\left(\mu_{i}\right)\right) \quad \text { where } \quad X_{l} \sim N Q\left(\mu_{l}, V\left(\mu_{l}\right)\right)}.\end{array}$$

Proposition 1 The following expression holds:

$$\begin{array}{r}{D[v(\boldsymbol{\phi}) \| \widehat{\boldsymbol{\phi}}]=C-\sum\limits_{j=1}^{k}\left\{\omega_{v(j)} \log \ \widehat{\omega}_{j}+\left(1-\omega_{v(j)}\right) \log \left(1-\widehat{\omega}_{j}\right)\right.} \\ {+\omega_{v(j)} \widehat{\theta}_{j}\left(\widehat{\mu}_{j}\right) \mu_{v(j)}-\omega_{v(j)} M\left(\widehat{\theta}_{j}\left(\widehat{\mu}_{j}\right)\right) \}}\end{array}$$

where C is a function that does not depend on the permutation v or on \(\widehat{\boldsymbol{\phi}}=(\widehat{\boldsymbol{\omega}}, \widehat{\boldsymbol{\mu}})\).

Proof

$$\begin{array}{*{20}{c}} {D\left[ {v(\phi )\left\| {\hat \phi } \right.} \right]} \\ { = \sum\limits_{j = 1}^k {\left[ {{w_{v(j)}}\text{NQ}\left( {{\mu _{v(j)}},V\left( {{\mu _{v(j)}}} \right)} \right)\left\| {{{\hat w}_j}\text{NQ}\left( {{{\hat \mu }_j},V\left( {{{\hat \mu }_j}} \right)} \right)} \right.} \right]} } \\ { = \sum\limits_{j = 1}^k {\left\{ {\left( {{w_{v(j)}}\log \frac{{{w_{v(j)}}}}{{{{\hat w}_j}}}} \right) + \left( {\left( {1 - {w_{v(j)}}} \right)\log \frac{{1 - {w_{v(j)}}}}{{1 - {{\hat w}_j}}}} \right) + \left( {{w_{v(j)}}\int {\text{NQ}\left( {{\mu _{v(j)}},V\left( {{\mu _{v(j)}}} \right)} \right)\log \frac{{\text{NQ}\left( {{\mu _{v(j)}},V\left( {{\mu _{v(j)}}} \right)} \right)}}{{\text{NQ}\left( {{{\hat \mu }_j},V\left( {{{\hat \mu }_j}} \right)} \right)}}dx} } \right)} \right\}} } \\ { = \sum\limits_{j = 1}^k {\left\{ {\left( {{w_{v(j)}}\log {w_{v(j)}}} \right) + \left( {\left( {1 - {w_{v(j)}}} \right)\log \left( {1 - {w_{v(j)}}} \right)} \right) + \left( {{w_{v(j)}}\int {\text{NQ}\left( {{\mu _{v(j)}},V\left( {{\mu _{v(j)}}} \right)} \right)\log \text{NQ}\left( {{\mu _{v(j)}},V\left( {{\mu _{v(j)}}} \right)} \right)dx} } \right)} \right\} - \sum\limits_{j = 1}^k {\left\{ {\left( {{w_{v(j)}}\log {{\hat w}_j}} \right) + \left( {\left( {1 - {w_{v(j)}}} \right)\log \left( {1 - {{\hat w}_j}} \right)} \right) + \left( {{w_{v(j)}}\int {\text{NQ}\left( {{\mu _{v(j)}},V\left( {{\mu _{v(j)}}} \right)} \right)\log \text{NQ}\left( {{{\hat \mu }_j},V\left( {{{\hat \mu }_j}} \right)} \right)dx} } \right)} \right\}} } } \\ { = \sum\limits_{j = 1}^k {\left\{ {\left( {{w_j}\log {w_j}} \right) + \left( {\left( {1 - w} \right)\log \left( {1 - w} \right)} \right) + \left( {{w_j}\int {\text{NQ}\left( {{\mu _j},V\left( {{\mu _j}} \right)} \right)\log \text{NQ}\left( {{\mu _j},V\left( {{\mu _j}} \right)} \right)dx} } \right)} \right\} - \sum\limits_{j = 1}^k {\left\{ {\left( {{w_{v(j)}}\log {{\hat w}_j}} \right) + \left( {\left( {1 - = {w_{v(j)}}} \right)\log \left( {1 - {{\hat w}_j}} \right)} \right) + \left( {{w_{v(j)}}\int {\text{NQ}\left( {{\mu _{v(j)}},V\left( {{\mu _{v(j)}}} \right)} \right)\log \text{NQ}\left( {{{\hat \mu }_j},V\left( {{{\hat \mu }_j}} \right)} \right)dx} } \right)} \right\}} } } \end{array}$$

[By Lemma 1]

$$\begin{array}{*{20}{c}} { = C - \sum\limits_{j = 1}^k {\left\{ {{w_{v(j)}}\log {{\hat w}_j} + \left( {1 - {w_{v(j)}}} \right)\log \left( {1 - {{\hat w}_j}} \right)} \right.} } \\ {\left. {\;\;\;\;\;\; + {w_{v(j)}}{{\hat \theta }_j}\left( {{{\hat \mu }_j}} \right){\mu _{v(j)}}M\left( {{{\hat \theta }_j}\left( {{{\hat \mu }_j}} \right)} \right)} \right\}.} \end{array}$$

Proposition 2 The minimum over \(\widehat{\boldsymbol{\phi}}=(\widehat{\boldsymbol{\omega}}, \widehat{\boldsymbol{\mu}})\) of \(D=\sum\nolimits_{t=1}^{N} D\left[v_{t}\left(\boldsymbol{\phi}^{(t)}\right) \| \widehat{\boldsymbol{\phi}}\right]\) is achieved at:

$$\widehat{\omega}_{j}=\frac{1}{N} \sum\limits_{t=1}^{N} \omega_{v_{t}(j)}^{(t)} \quad and \quad \widehat{\mu}_{j}=\frac{\sum\nolimits_{t=1}^{N} \omega_{v_{t}(j)}^{(t)} \mu_{v_{t}(j)}^{(t)}}{\sum\nolimits_{t=1}^{N} \omega_{v_{t}(j)}^{(t)}}.$$

Proof By Proposition 1, the problem can be divided into the following steps:

  1. 1.

    Choose \(\widehat{\omega}_{j}(j=1, \ldots, k)\) to maximize

    $$\sum\limits_{t=1}^{N} \omega_{v_{t}(j)}^{(t)} \log\ \widehat{\omega}_{j}+\left(1-\omega_{v_{t}(j)}^{(t)}\right) \log \left(1-\widehat{\omega}_{j}\right).$$
    (5)
  2. 2.

    Choose \(\widehat{\mu}_{j}(j=1, \ldots, k)\) to minimize

    $$\sum\limits_{t=1}^{N}-\omega_{v_{t}(j)}^{(t)} \widehat{\theta}_{j}\left(\widehat{\mu}_{j}\right) \mu_{v_{t}(j)}^{(t)}+\omega_{v_{t}(j)}^{(t)} M\left(\widehat{\theta_{j}}\left(\widehat{\mu}_{j}\right)\right).$$
    (6)

Dividing (5) by N, it follows:

$$\frac{\sum\nolimits_{t=1}^{N} \omega_{v_{t}(j)}^{(t)}}{N} \log\ \widehat{\omega}_{j}+\left(1-\frac{\sum\nolimits_{t=1}^{N} \omega_{v_{t}(j)}^{(t)}}{N}\right) \log \left(1-\widehat{\omega}_{j}\right).$$

Deriving and equaling to zero, it follows:

$$\widehat{\omega}_{j}=\frac{\sum\nolimits_{t=1}^{N} \omega_{v_{t}(j)}^{(t)}}{N}.$$

To solve (6), it is calculated:

$$\frac{\partial\left[\widehat{\theta}_{j}\left(\widehat{\mu}_{j}\right)\left(-\sum\nolimits_{t=1}^{N} \omega_{v_{t}(j)}^{(t)} \mu_{v_{t}(j)}^{(t)}\right)+M\left(\widehat{\theta}_{j}\left(\widehat{\mu}_{j}\right)\right) \sum\nolimits_{t=1}^{N} \omega_{v_{t}(j)}^{(t)}\right]}{\partial \widehat{\mu}_{j}},$$

and making the derivative equal to zero, it follows:

$$\widehat{\mu}_{j}=\frac{\sum\nolimits_{t=1}^{N} \omega_{v_{t}(j)}^{(t)} \mu_{v_{t}(j)}^{(t)}}{\sum\nolimits_{t=1}^{N} \omega_{v_{t}(j)}^{(t)}}.$$

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Rufo, M.J., Martín, J. & Pérez, C.J. Bayesian analysis of finite mixture models of distributions from exponential families. Computational Statistics 21, 621–637 (2006). https://doi.org/10.1007/s00180-006-0018-8

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