# likelihood_functions: Likelihood-Based Mixture Model Statistics In lcmix: Layered and chained mixture models

## Description

See Details for a description of the individual functions.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20``` ```## S3 method for class 'mixmod' logLik(object, ...) ## S3 method for class 'mixmod' qval(x, map=FALSE, ...) ## S3 method for class 'mixmod' qfun(x, map=FALSE, ...) ## S3 method for class 'mdmixmod' qfun(x, map=FALSE, ...) ## S3 method for class 'mixmod' aic(x, ...) ## S3 method for class 'mixmod' bic(x, ...) ## S3 method for class 'mixmod' entropy(x, map=FALSE, ...) ## S3 method for class 'mixmod' iclbic(x, map=FALSE, ...) ## S3 method for class 'mdmixmod' siclbic(x, map=FALSE, ...) ```

## Arguments

 `x, object` an object of class `mixmod` or `mdmixmod`. `map` logical; if `TRUE`, the maximum a posteriori (MAP) estimates rather than the posterior probabilities will be used when estimating expectations with respect to hidden data. `...` currently unused.

## Details

`logLik` calculates L(theta|X), the log-likelihood of the estimated parameters theta with respect to the observed data X, while `qval` calculates the “Q-value”, the expectation with respect to the hidden data of the log-likelihood with respect to the complete data: Q(theta) = E[L(theta|X,Y)] for `mixmod` and Q(theta) = E[L(theta|X,Y,Y0)] for `mdmixmod`. `qfun` returns the hidden and observed portions of the Q-value separately, as elements of a vector.

`aic`, `bic`, `entropy`, `iclbic`, and `siclbic` calculate various information criteria for model selection with mixture models of class `mixmod` and `mdmixmod`. These criteria are Akaike's information criterion (AIC, Akaike, 1974), the Bayes information criterion (BIC, Schwarz, 1978), the classification entropy (Biernacki et al., 2000), the integrated complete likelihood BIC (ICL-BIC, Biernacki et al., 2000), and the simplified ICL-BIC (SICL-BIC) for objects of class `mdmixmod`, respectively. They are defined as follows:

 AIC = 2 L(theta|X) - 2 |Theta| BIC = 2 L(theta|X) - |Theta| log(N) entropy = 2 L(theta|X) - 2 Q(theta) ICL-BIC = 2 Q(theta) - |Theta| log(N) SICL-BIC = 2 E[L(theta|X,Y0)] - |Theta| log(N) (`mdmixmod` only)

where |Theta| is the size of the parameter space and N is the size of the data. Generally, the model which provides the highest value of any information criterion should be selected. Current testing indicates that ICL-BIC is preferred for `mixmod` and BIC for `mdmixmod`.

## Value

A numeric vector for `qfun`, a numeric scalar for the other functions.

## Note

Some authors define AIC, BIC, and ICL-BIC as the negative of the quantities given in Details.

Daniel Dvorkin

## References

Akaike, H. (1974) A new look at the statistical model identification, IEEE Transactions on Automatic Control 19(6), 716–723.

Biernacki, C. and Celeux, G. and Govaert, G. (2000) Assessing a mixture model for clustering with the integrated completed likelihood, IEEE Transactions on Pattern Analysis and Machine Intelligence 22(7), 719–725.

McLachlan, G.J. and Thriyambakam, K. (2008) The EM Algorithm and Extensions, John Wiley & Sons.

Schwarz, G. (1978) Estimating the dimension of a model, The Annals of Statistics 6(2), 461–464.

`mixmod` and `mdmixmod` for details of the hidden variable structure.
 ```1 2 3 4 5 6 7 8``` ```## Not run: data(CiData) fit <- mixmod(CiData\$expression, 3) bic(fit) # -95405.4 qval(fit) # -50055.35 qval(fit, map=TRUE) # -49738.53 ## End(Not run) ```