Bayesian nonparametric generalized linear models

Description

Fits Dirichlet process mixtures of joint response-covariate models, where the covariates are continuous while the discrete responses are represented utilizing continuous latent variables. See ‘Details’ section for a full model description.

Usage

1
2
3
4
bnpglm(formula,family,data,offset,sampler="slice",StorageDir,
       ncomp,sweeps,burn,thin=1,seed,prec,V,Vdf,Mu.nu,Sigma.nu,
       Mu.mu,Sigma.mu,Alpha.xi,Beta.xi,Alpha.alpha,Beta.alpha,Turnc.alpha,
       Xpred,offsetPred,...)

Arguments

formula

a formula defining the response and the covariates e.g. y ~ x.

family

a description of the kernel of the response variable. Currently eight options are supported: 1. "poisson", 2. "negative binomial", 3. "generalized poisson", 4. "hyper-poisson", 5. "ctpd", 6. "com-poisson", 7. "binomial" and 8. "beta binomial". The first six kernels are used for count data analysis while the last two are used for binomial data analysis. Kernels 3.-6. allow for both over- and under-dispersion relative to the Poisson distribution. See ‘Details’ section for some of the kernel details.

data

an optional data frame, list or environment (or object coercible by ‘as.data.frame’ to a data frame) containing the variables in the model. If not found in ‘data’, the variables are taken from ‘environment(formula)’.

offset

this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be ‘NULL’ or a numeric vector of length equal to the sample size. One ‘offset’ term can be included in the formula, and if more are required, their sum should be used.

sampler

the MCMC algorithm to be utilized. The two options are sampler="slice" which implements a slice sampler (Walker, 2007; Papaspiliopoulos, 2008) and sampler="truncated" which proceeds by truncating the countable mixture at ncomp components (see argument ncomp).

StorageDir

a directory to store files with the posterior samples of models parameters and other quantities of interest. If a directory is not provided, files are created in the current directory and removed when the sampler completes.

ncomp

number of mixture components. Defines where the countable mixture of densities [in (1) below] is truncated. Even if sampler="slice" is chosen, ncomp needs to be specified as it is used in the initialization process.

sweeps

total number of posterior samples, including those discarded in burn-in period (see argument burn) and those discarded by the thinning process (see argument thin).

burn

length of burn-in period.

thin

thinning parameter.

seed

optional seed for the random generator.

prec

precision parameter. Updating the parameters of the response distribution requires a Metropolis - Hastings step, with proposal distributions centered at current values and with precision equal to this argument. It can be of length one (for "poisson" and "binomial" kernels) or of length two (for "negative binomial", "beta binomial", "generalized-poisson", "hyper-poisson" and "com-poisson" kernels) or of length three (for the "ctpd" kernel).

V

optional scale matrix V of the prior Wishart distribution assigned to precision matrix T_h. See ‘Details’ section.

Vdf

optional degrees of freedom Vdf of the prior Wishart distribution assigned to precision matrix T_h. See ‘Details’ section.

Mu.nu

optional prior mean μ_{ν} of the covariance vector ν_h. See ‘Details’ section.

Sigma.nu

optional prior covariance matrix Σ_{ν} of ν_h. See ‘Details’ section.

Mu.mu

optional prior mean μ_{μ} of the mean vector μ_h. See ‘Details’ section.

Sigma.mu

optional prior covariance matrix Σ_{μ} of μ_h. See ‘Details’ section.

Alpha.xi

an optional parameter that depends on the specified family.

  1. If family="poisson", this argument is parameter α_{ξ} of the prior of the Poisson rate: ξ \sim Gamma(α_{ξ},β_{ξ}).

  2. If family="negative binomial", this argument is a two-dimensional vector that includes parameters α_{1ξ} and α_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Gamma(α_{2ξ},β_{2ξ}), where ξ_1 and ξ_2 are the two parameters of the Negative Binomial pmf.

  3. If family="generalized-poisson", this argument is a two-dimensional vector that includes parameters α_{1ξ} and α_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim N(α_{2ξ},β_{2ξ})I[ξ_2 \in R_{ξ_2}], where ξ_1 and ξ_2 are the two parameters of the Generalized Poisson pmf. Parameter ξ_2 has to be in the range R_{ξ_2} (which is automatically done during posterior sampling).

  4. If family="hyper-poisson", this argument is a two-dimensional vector that includes parameters α_{1ξ} and α_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Gamma(α_{2ξ},β_{2ξ}), where ξ_1 and ξ_2 are the two parameters of the hyper Poisson pmf.

  5. If family="ctpd", this argument is a three-dimensional vector that includes parameters α_{1ξ}, α_{2ξ} and α_{3ξ} of the priors: ξ_i \sim Gamma(α_{iξ},β_{iξ}), i=1,2, and ξ_3 \sim N(α_{3ξ},β_{3ξ})I[ξ_3 \in R_{ξ_3}], where ξ_i, i=1,2,3, are the three parameters of the complex triparametric Pearson distribution. Parameter ξ_3 has to be in the range R_{ξ_3} (which is automatically done during posterior sampling).

  6. If family="com-poisson", this argument is a two-dimensional vector that includes parameters α_{1ξ} and α_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Gamma(α_{2ξ},β_{2ξ}), where ξ_1 and ξ_2 are the two parameters of the COM-Poisson pmf.

  7. If family="binomial", this argument is parameter α_{ξ} of the prior of the Binomial probability: ξ \sim Beta(α_{ξ},β_{ξ}).

  8. If family="beta binomial", this argument is a two-dimensional vector that includes parameters α_{1ξ} and α_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Gamma(α_{2ξ},β_{2ξ}), where ξ_1 and ξ_2 are the two parameters of the Beta Binomial pmf.

See ‘Details’ section.

Beta.xi

an optional parameter that depends on the specified family.

  1. If family="poisson", this argument is parameter β_{ξ} of the prior of the Poisson rate: ξ \sim Gamma(α_{ξ},β_{ξ}).

  2. If family="negative binomial", this argument is a two-dimensional vector that includes parameters β_{1ξ} and β_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Gamma(α_{2ξ},β_{2ξ}), where ξ_1 and ξ_2 are the two parameters of the Negative Binomial pmf.

  3. If family="generalized poisson", this argument is a two-dimensional vector that includes parameters β_{1ξ} and β_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Normal(α_{2ξ},β_{2ξ})I[ξ_2 \in R_{ξ_2}], where ξ_1 and ξ_2 are the two parameters of the Generalized Poisson pmf. Parameter ξ_2 has to be in the range R_{ξ_2} (which is automatically done during posterior sampling). Note that β_{2ξ} is a standard deviation.

  4. If family="hyper-poisson", this argument is a two-dimensional vector that includes parameters β_{1ξ} and β_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Gamma(α_{2ξ},β_{2ξ}), where ξ_1 and ξ_2 are the two parameters of the hyper Poisson pmf.

  5. If family="ctpd", this argument is a three-dimensional vector that includes parameters β_{1ξ}, β_{2ξ} and β_{3ξ} of the priors: ξ_i \sim Gamma(α_{iξ},β_{iξ}), i=1,2, and ξ_3 \sim N(α_{3ξ},β_{3ξ})I[ξ_3 \in R_{ξ_3}], where ξ_i, i=1,2,3, are the three parameters of the complex triparametric Pearson distribution. Note that β_{3ξ} is a standard deviation.

  6. If family="com-poisson", this argument is a two-dimensional vector that includes parameters β_{1ξ} and β_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Gamma(α_{2ξ},β_{2ξ}), where ξ_1 and ξ_2 are the two parameters of the COM-Poisson pmf.

  7. If family="binomial", this argument is parameter β_{ξ} of the prior of the Binomial probability: ξ \sim Beta(α_{ξ},β_{ξ}).

  8. If family="beta binomial", this argument is a two-dimensional vector that includes parameters β_{1ξ} and β_{2ξ} of the priors: ξ_1 \sim Gamma(α_{1ξ},β_{1ξ}) and ξ_2 \sim Gamma(α_{2ξ},β_{2ξ}), where ξ_1 and ξ_2 are the two parameters of the Beta Binomial pmf.

See ‘Details’ section.

Alpha.alpha

optional shape parameter α_{α} of the Gamma prior assigned to the concentration parameter α. See ‘Details’ section.

Beta.alpha

optional rate parameter β_{α} of the Gamma prior assigned to concentration parameter α. See ‘Details’ section.

Turnc.alpha

optional truncation point c_{α} of the Gamma prior assigned to concentration parameter α. See ‘Details’ section.

Xpred

an optional design matrix the rows of which include the covariates x for which the conditional distribution of Y|x,D (where D denotes the data) is calculated. These are treated as ‘new’ covariates i.e. they do not contribute to the likelihood. The matrix shouldn't include a column of 1's.

offsetPred

the offset term associated with the new covariates Xpred. offsetPred is a vector of length equal to the rows of Xpred. If family is one of poisson or negative binomial or generalized poisson, its entries are the associated Poisson offsets. If family is one of binomial or beta binomial, its entries are the Binomial number of trials. If offsetPred is missing, it is taken to be the mean of offset, rounded to the nearest integer.

...

Other options that will be ignored.

Details

Function bnpglm returns samples from the posterior distributions of the parameters of the model:

f(y_i,x_i) = ∑_{h=1}^{∞} π_h f(y_i,x_i|θ_h), \hspace{80pt} (1)

where y_i is a univariate discrete response, x_i is a p-dimensional vector of continuous covariates, and π_h, h ≥q 1, are obtained according to Sethuraman's (1994) stick-breaking construction: π_1 = v_1, and for l ≥q 2, π_l = v_l ∏_{j=1}^{l-1} (1-v_j), where v_k are iid samples v_k \simBeta (1,α), k ≥q 1.

The discrete responses y_i are represented as discretized versions of continuous latent variables y_i^*. Observed discrete and continuous latent variables are connected by:

y_{i} = q \iff c_{i,q-1} < y^*_{i} < c_{i,q}, q=0,1,2,…,

where the cut-points are obtained as: c_{i,-1} = -∞, while for q ≥q 0, c_{i,q} = c_{q}(λ_{i}) = Φ^{-1}\{F(q;λ_i)\}. Here Φ(.) is the cumulative distribution function (cdf) of a standard normal variable and F() denotes an appropriate cdf. Further, latent variables are assumed to independently follow a N(0,1) distribution, where the mean and variance are restricted to be zero and one as they are non-identifiable by the data. Choices for F() are described next.

For counts, currently six options are supported. First, F(.;λ_i) can be specified as the cdf of a Poisson(H_i ξ_h) variable. Here λ_i=(ξ_h,H_i)^T, ξ_h denotes the Poisson rate associated with cluster h, and H_i the offset term associated with sampling unit i. Second, F(.;λ_i) can be specified as the negative binomial cdf, where λ_i= (ξ_{1h},ξ_{2h},H_i)^T. This option allows for overdispersion within each cluster relative to the Poisson distribution. Third, F(.;λ_i) can be specified as the Generalized Poisson cdf, where, again, λ_i=(ξ_{1h},ξ_{2h},H_i)^T. This option allows for both over- and under-dispersion within each cluster. The other three options, that also allow for both over- and under-dispersion relative to the Poisson distribution, are the Hyper Poisson (HP), COM-Poisson and the Complex Triparametric Pearson (CTP) kernels. The HP and COM-Poisson kernels have 2 parameters and the CTPD kernel has 3 parameters.

For Binomial data, currently two options are supported. First, F(.;λ_i) may be taken to be the cdf of a Binomial(H_i,ξ_h) variable, where ξ_h denotes the success probability of cluster h and H_i the number of trials associated with sampling unit i. Second, F(.;λ_i) may be specified to be the beta-binomial cdf, where λ=(ξ_{1h},ξ_{2h},H_i)^T.

Details on all kernels are provided in the tables below. The first table provides the probability mass functions and the mean in the presence of an offset term (which may be taken to be one). The column ‘Sample’ indicates for which parameters the routine provides posterior samples. The second table provides information on the assumed priors along with the default values of the parameters of the prior distributions and it also indicates the function arguments that allow the user to alter these. Lastly, the third tables provides some details on the less frequently used kernels.

Kernel PMF Offset Mean Sample
Poisson \exp(-Hξ) (Hξ)^y /y! H H ξ ξ
Negative Binomial \frac{Γ(y+ξ_1)}{Γ(ξ_1)Γ(y+1)}(\frac{ξ_2}{H+ξ_2})^{ξ_1}(\frac{H}{H+ξ_2})^{y} H H ξ_1/ξ_2 ξ_1, ξ_2
Generalized Poisson ξ_1 \{ξ_1+(ξ_2-1)y\}^{y-1} ξ_2^{-y} \times H Hξ_1 ξ_1,ξ_2
~~ \exp\{-[ξ_1+(ξ_2-1)y]/ξ_2\}/y!
Hyper Poisson \frac{1}{_1F_1(1,ξ_2,ξ_3)} \frac{ξ_3^y}{(ξ_2)_y} H H ξ_1 = ξ_3 - ξ_1,ξ_2
~~ (ξ_2-1) \frac{_1F_1(1,ξ_2,ξ_3)-1}{_1F_1(1,ξ_2,ξ_3)}
CTP f_0 \frac{(ξ_3+ξ_4 i)_y (ξ_3-ξ_4 i)_y}{(ξ_2)_y y!} H H ξ_1 = \frac{ξ_3^2+ξ_4^2}{ξ_2-2ξ_3-1} ξ_1, ξ_2, ξ_3
COM-Poisson \frac{ξ_3^y}{Z(ξ_2,ξ_3)(y!)^{ξ_2}} H H ξ_1 = ξ_3 \frac{\partial \log(Z)}{\partial ξ_3} ξ_1,ξ_2
Binomial {N \choose y} ξ^y (1-ξ)^{N-y} N N ξ ξ
Beta Binomial {N \choose y} \frac{{Beta}{(y+ξ_1,N-y+ξ_2)}}{{Beta}{(ξ_1,ξ_2)}} N N ξ_1/(ξ_1+ξ_2) ξ_1,ξ_2
Kernel Priors Default Values
Poisson ξ \sim Gamma(α_{ξ},β_{ξ}) Alpha.xi = 1.0, Beta.xi = 0.1
Negative Binomial ξ_i \sim Gamma(α_{ξ_i},β_{ξ_i}), i=1,2 Alpha.xi = c(1.0,1.0), Beta.xi = c(0.1,0.1)
Generalized Poisson ξ_1 \sim Gamma(α_{ξ_1},β_{ξ_1})
ξ_2 \sim TN(α_{ξ_2},β_{ξ_2}) (β_{ξ_2} \equiv st.dev.) Alpha.xi = c(1.0,1.0), Beta.xi = c(0.1,1.0)
TN: truncated normal
Hyper Poisson ξ_i \sim Gamma(α_{ξ_i},β_{ξ_i}), i=1,2 Alpha.xi = c(1.0,0.5), Beta.xi = c(0.1,0.5)
CTP ξ_i \sim Gamma(α_{ξ_i},β_{ξ_i}), i=1,2
ξ_3 \sim TN(α_{ξ_3},β_{ξ_3}) (β_{ξ_3} \equiv st.dev.) Alpha.xi = c(1.0,1.0,0.0)
TN: truncated normal Beta.xi = c(0.1,0.1,100.0)
COM-Poisson ξ_i \sim Gamma(α_{ξ_i},β_{ξ_i}), i=1,2 Alpha.xi = c(1.0,0.5), Beta.xi = c(0.1,0.5)
Binomial ξ \sim Beta(α_{ξ},β_{ξ}) Alpha.xi = 1.0, Beta.xi = 1.0
Beta Binomial ξ_i \sim Gamma(α_{ξ_i},β_{ξ_i}), i=1,2 Alpha.xi = c(1.0,1.0), Beta.xi = c(0.1,0.1)
Kernel Notes
Generalized Poisson ξ_1 > 0 is the mean and ξ_2 > 1/2 is a dispersion parameter. When ξ_2 = 1,
the pmf reduces to the Poisson. Parameter values ξ_2 > 1 suggest over-
dispersion and parameter values 1/2 < ξ_2 < 1 suggest under-dispersion
relative to the Poisson.
Hyper Poisson ξ_1 > 0 is the mean and ξ_2 > 0 is a dispersion parameter. When ξ_2 = 1,
the pmf reduces to the Poisson. When ξ_2 > 1 the pmf is over-dispersed
and when ξ_2 < 1 the pmf is under-dispersed relative to the Poisson.
COM-Poisson The mean is ξ_1 (> 0) and the variance approximately ξ_1/ξ_2, so
similar comments as for the hyper Poisson hold.
CTPD Things are a bit more complex here. See Rodriguez-Avi et al. (2004) for the details.

Further, joint vectors (y_i^{*},x_{i}) are modeled utilizing Gaussian distributions. Then, with θ_h denoting model parameters associated with the hth cluster, the joint density f(y_{i},x_{i}|θ_h) takes the form

f(y_{i},x_{i}|θ_h) = \int_{c_{i,y_i-1}}^{c_{i,y_i}} N_{p+1}(y_{i}^{*},x_{i}|μ_{h},C_h) dy_{i}^{*},

where μ_h and C_h denote the mean vector and covariance matrix, respectively.

The joint distribution of the latent variable y_i^{*} and the covariates x_{i} is

(y_{i}^{*},x_{i}^T)^T|θ_h \sim N_{p+1}≤ft( \begin{array}{ll} ≤ft( \begin{array}{l} 0 \\ μ_h \\ \end{array} \right), & C_h=≤ft[ \begin{array}{ll} 1 & ν_h^T \\ ν_h & Σ_h \\ \end{array} \right] \end{array}\right),

where ν_h denotes the vector of covariances cov(y_{i}^{*},x_{i}|θ_h). Sampling from the posterior of constrained covariance matrix C_h is done using methods similar to those of McCulloch et al. (2000). Specifically, the conditional x_{i}|y_{i}^{*} \sim N_{p}(μ_h+y_{i}^{*}ν_h, B_h = Σ_h - ν_h ν_h^T) simplifies matters as there are no constraints on matrix B_h (other than positive definiteness). Given priors for B_h and ν_h, it is easy to sample from their posteriors, and thus obtain samples from the posterior of Σ_h=B_h+ν_h ν_h^T.

Specification of the prior distributions:

  1. Define T_h=B_h^{-1} = (Σ_{h} - ν_h ν_h^T)^{-1}, h ≥q 1. We specify that a priori T_h \sim Wishart_{p}(V,Vdf), where V is a p \times p scale matrix and Vdf is a scalar degrees of freedom parameter. Default values are: V = I_{p}/p and Vdf=p, however, these can be changed using arguments V and Vdf.

  2. The assumed prior for ν_h is N_p(μ_{ν},Σ_{ν}), h ≥q 1, with default values μ_{ν}=0 and Σ_{ν} = I_{p}. Arguments Mu.nu and Sigma.nu allow the user to change the default values.

  3. A priori μ_{h} \sim N_p(μ_{μ},Σ_{μ}), h ≥q 1. Here the default values are μ_{μ} = \bar{x} where \bar{x} denotes the sample mean of the covariates, and Σ_{μ} = D where D denotes a diagonal matrix with diagonal elements equal to the square of the observed range of the covariates. Arguments Mu.mu and Sigma.mu allow the user to change the default values.

  4. For count data, with family="poisson", a priori we take ξ_{h} \sim Gamma(α_{ξ},β_{ξ}), h ≥q 1. The default values are α_{ξ}=1.0,β_{ξ}=0.1, that define a Gamma distribution with mean α_{ξ}/β_{ξ}=10 and variance α_{ξ}/β_{ξ}^2=100. Defaults can be altered using arguments Alpha.xi and Beta.xi.

    For count data with family="negative binomial" a priori we take ξ_{jh} \sim Gamma(α_{jξ},β_{jξ}), j=1,2, h ≥q 1. The default values are α_{jξ}=1.0,β_{jξ}=0.1, j=1,2. Default values for \{α_{jξ}: j=1,2\} can be altered using argument Alpha.xi, and default values for \{β_{jξ}: j=1,2\} can be altered using argument Beta.xi.

    For count data with family="generalized poisson", a priori we take ξ_{1h} \sim Gamma(α_{1ξ},β_{1ξ}), and ξ_{2h} \sim Normal(α_{2ξ},β_{2ξ})I[ξ_{2h} \in R_{ξ_{2}}]. The default values are α_{jξ}=1.0, j=1,2 and β_{1ξ}=0.1, β_{2ξ}=1.0. Default values for \{α_{jξ}: j=1,2\} can be altered using argument Alpha.xi, and default values for \{β_{jξ}: j=1,2\} can be altered using argument Beta.xi.

    For count data with family="hyper-poisson" a priori we take ξ_{jh} \sim Gamma(α_{jξ},β_{jξ}), j=1,2, h ≥q 1. The default values are α_{1ξ}=1.0, α_{2ξ}=0.5 and β_{1ξ}=0.1, β_{2ξ}=0.5. Default values for \{α_{jξ}: j=1,2\} can be altered using argument Alpha.xi, and default values for \{β_{jξ}: j=1,2\} can be altered using argument Beta.xi.

    For count data with family="ctpd", a priori we take ξ_{1h} \sim Gamma(α_{1ξ},β_{1ξ}), ξ_{2h} \sim Gamma(α_{2ξ},β_{2ξ}) and ξ_{3h} \sim Normal(α_{3ξ},β_{3ξ})I[ξ_{3h} \in R_{ξ_{3}}]. The default values are α_{1ξ}=1.0, α_{2ξ}=1.0, α_{3ξ}=0.0 and β_{1ξ}=0.1, β_{2ξ}=0.1, β_{3ξ}=100.0. Default values for \{α_{jξ}: j=1,2\} can be altered using argument Alpha.xi, and default values for \{β_{jξ}: j=1,2\} can be altered using argument Beta.xi.

    For count data with family="com-poisson" a priori we take ξ_{jh} \sim Gamma(α_{jξ},β_{jξ}), j=1,2, h ≥q 1. The default values are α_{1ξ}=1.0, α_{2ξ}=0.5 and β_{1ξ}=0.1, β_{2ξ}=0.5. Default values for \{α_{jξ}: j=1,2\} can be altered using argument Alpha.xi, and default values for \{β_{jξ}: j=1,2\} can be altered using argument Beta.xi.

    For binomial data, with family="binomial", a priori we take ξ_{h} \sim Beta(α_{ξ},β_{ξ}), h ≥q 1. The default values are α_{ξ}=1.0,β_{ξ}=1.0, that define a uniform distribution. Defaults can be altered using arguments Alpha.xi and Beta.xi.

    For binomial data with family="beta binomial", a priori we take ξ_{jh} \sim Gamma(α_{jξ},β_{jξ}), j=1,2, h ≥q 1. The default values are α_{jξ}=1.0,β_{jξ}=0.1. Default values for \{α_{jξ}: j=1,2\} can be altered using argument Alpha.xi, and default values for \{β_{jξ}: j=1,2\} can be altered using argument Beta.xi.

  5. The concentration parameter α is assigned a Gamma(α_{α},β_{α}) prior over the range (c_{α},∞), that is, f(α) \propto α^{α_{α}-1} \exp\{-α β_{α}\} I[α > c_{α}], where I[.] is the indicator function. The default values are α_{α}=2.0, β_{α}=4.0, and c_{α}=0.25. Users can alter the default using using arguments Alpha.alpha, Beta.alpha and Turnc.alpha.

Value

Function bnpglm returns the following:

call

the matched call.

seed

the seed that was used (in case replication of the results is needed).

meanReg

if Xpred is specified, the function returns the posterior mean of the expectation of the response given each new covariate x.

modeReg

if Xpred is specified, the function returns the posterior mean of the conditional mode of the response given each new covariate x.

Q05Reg

if Xpred is specified, the function returns the posterior mean of the conditional 5% quantile of the response given each new covariate x.

Q10Reg

if Xpred is specified, the function returns the posterior mean of the conditional 10% quantile of the response given each new covariate x.

Q15Reg

if Xpred is specified, the function returns the posterior mean of the conditional 15% quantile of the response given each new covariate x.

Q20Reg

if Xpred is specified, the function returns the posterior mean of the conditional 20% quantile of the response given each new covariate x.

Q25Reg

if Xpred is specified, the function returns the posterior mean of the conditional 25% quantile of the response given each new covariate x.

Q50Reg

if Xpred is specified, the function returns the posterior mean of the conditional 50% quantile of the response given each new covariate x.

Q75Reg

if Xpred is specified, the function returns the posterior mean of the conditional 75% quantile of the response given each new covariate x.

Q80Reg

if Xpred is specified, the function returns the posterior mean of the conditional 80% quantile of the response given each new covariate x.

Q85Reg

if Xpred is specified, the function returns the posterior mean of the conditional 85% quantile of the response given each new covariate x.

Q90Reg

if Xpred is specified, the function returns the posterior mean of the conditional 90% quantile of the response given each new covariate x.

Q95Reg

if Xpred is specified, the function returns the posterior mean of the conditional 95% quantile of the response given each new covariate x.

denReg

if Xpred is specified, the function returns the posterior mean conditional density of the response given each new covariate x. Results are presented in a matrix the rows of which correspond to the different xs.

denVar

if Xpred is specified, the function returns the posterior variance of the conditional density of the response given each new covariate x. Results are presented in a matrix the rows of which correspond to the different xs.

Further, function bnpglm creates files where the posterior samples are written. These files are (with all file names preceded by ‘BNSP.’):

alpha.txt

this file contains samples from the posterior of the concentration parameters α. The file is arranged in (sweeps-burn)/thin lines and one column, each line including one posterior sample.

compAlloc.txt

this file contains the allocations or configurations obtained at each iteration of the sampler. It consists of (sweeps-burn)/thin lines, that represent the posterior samples, and n columns, that represent the sampling units. Entries in this file range from 0 to ncomp-1.

MeanReg.txt

this file contains the conditional means of the response y given covariates x obtained at each iteration of the sampler. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the means are obtained.

muh.txt

this file contains samples from the posteriors of the p-dimensional mean vectors μ_h, h=1,2,…,ncomp. The file is arranged in ((sweeps-burn)/thin)*ncomp lines and p columns. In more detail, each sweep creates ncomp lines representing samples μ_h^{(sw)}, h=1,…,ncomp, where superscript sw represents a particular sweep. The elements of μ_h^{(sw)} are written in the columns of the file.

nmembers.txt

this file contains (sweeps-burn)/thin lines and ncomp columns, where the lines represent posterior samples while the columns represent the components or clusters. The entries represent the number of sampling units allocated to the components.

nuh.txt

this file contains samples from the posteriors of the p-dimensional covariance vectors ν_h, h=1,2,…,ncomp. The file is arranged in ((sweeps-burn)/thin)*ncomp lines and p columns. In more detail, each sweep creates ncomp lines representing samples ν_h^{(sw)}, h=1,…,ncomp, where superscript sw represents a particular sweep. The elements of ν_h^{(sw)} are written in the columns of the file.

Q05Reg.txt

this file contains the 5% conditional quantile of the response y given covariates x obtained at each iteration of the sampler. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the quantiles are obtained.

Q10Reg.txt

this file contains the 10% conditional quantile of the response y given covariates x obtained at each iteration of the sampler. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the quantiles are obtained.

Q15Reg.txt

this file contains the 15% conditional quantile of the response y given covariates x obtained at each iteration of the sampler. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the quantiles are obtained.

Q20Reg.txt

this file contains the 20% conditional quantile of the response y given covariates x obtained at each iteration of the sampler. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the quantiles are obtained.

Q25Reg.txt

this file contains the 25% conditional quantile of the response y given covariates x obtained at each iteration of the sampler. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the quantiles are obtained.

Q50Reg.txt

this file contains the 50% conditional quantile of the response y given covariates x obtained at each iteration of the sampler. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the quantiles are obtained.

Q75Reg.txt

this file contains the 75% conditional quantile of the response y given covariates x obtained at each iteration of the sampler. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the quantiles are obtained.

Q80Reg.txt

this file contains the 80% conditional quantile of the response y given covariates x obtained at each iteration of the sampler. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the quantiles are obtained.

Q85Reg.txt

this file contains the 85% conditional quantile of the response y given covariates x obtained at each iteration of the sampler. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the quantiles are obtained.

Q90Reg.txt

this file contains the 90% conditional quantile of the response y given covariates x obtained at each iteration of the sampler. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the quantiles are obtained.

Q95Reg.txt

this file contains the 95% conditional quantile of the response y given covariates x obtained at each iteration of the sampler. The rows represent the (sweeps-burn)/thin posterior samples. The columns represent the various covariate values x for which the quantiles are obtained.

Sigmah.txt

this file contains samples from the posteriors of the p \times p covariance matrices Σ_h, h=1,2,…,ncomp. The file is arranged in ((sweeps-burn)/thin)*ncomp lines and p^2 columns. In more detail, each sweep creates ncomp lines representing samples Σ_h^{(sw)}, h=1,…,ncomp, where superscript sw represents a particular sweep. The elements of Σ_h^{(sw)} are written in the columns of the file: the entries in the first p columns of the file are those in the first column (or row) of Σ_h^{(sw)}, while the entries in the last p columns of the file are those in the last column (or row) of Σ_h^{(sw)}.

SigmahI.txt

this file contains samples from the posteriors of the p \times p precision matrices Σ_h^{-1}, h=1,2,…,ncomp. The file is arranged in ((sweeps-burn)/thin)*ncomp lines and p^2 columns. In more detail, each sweep creates ncomp lines representing samples (Σ_h^{-1})^{(sw)}, h=1,…,ncomp, where superscript sw represents a particular sweep. The elements of (Σ_h^{-1})^{(sw)} are written in the columns of the file: the entries in the first p columns of the file are those in the first column (or row) of (Σ_h^{-1})^{(sw)}, while the entries in the last p columns of the file are those in the last column (or row) of (Σ_h^{-1})^{(sw)}.

Th.txt

this file contains samples from the posteriors of the p \times p precision matrices T_h, h=1,2,…,ncomp. The file is arranged in ((sweeps-burn)/thin)*ncomp lines and p^2 columns. In more detail, each sweep creates ncomp lines representing samples T_h^{(sw)}, h=1,…,ncomp, where superscript sw represents a particular sweep. The elements of T_h^{(sw)} are written in the columns of the file: the entries in the first p columns of the file are those in the first column (or row) of T_h^{(sw)}, while the entries in the last p columns of the file are those in the last column (or row) of T_h^{(sw)}.

xih.txt

this file contains samples from the posteriors of parameters ξ_h, h=1,2,…,ncomp. The file is arranged in ((sweeps-burn)/thin)*ncomp lines and one or two columns, depending on the number of parameters in the selected F(.;λ). Sweeps write in the file ncomp lines representing samples ξ_h^{(sw)}, h=1,…,ncomp, where superscript sw represents a particular sweep.

Updated.txt

this file contains (sweeps-burn)/thin lines with the number of components updated at each iteration of the sampler.

Author(s)

Georgios Papageorgiou gpapageo@gmail.com

References

Consul, P. C. & Famoye, G. C. (1992). Generalized Poisson regression model. Communications in Statistics - Theory and Methods, 1992, 89-109.

McCulloch, R. E., Polson, N. G., & Rossi, P. E. (2000). A Bayesian analysis of the multinomial probit model with fully identified parameters. Journal of Econometrics, 99(1), 173-193.

Papageorgiou, G., Richardson, S. and Best, N. (2014). Bayesian nonparametric models for spatially indexed data of mixed type.

Papaspiliopoulos, O. (2008). A note on posterior sampling from Dirichlet mixture models. Technical report, University of Warwick.

Rodriguez-Avi, J., Conde-Sanchez, A., Saez-Castillo, A. J., & Olmo-Jimenez, M. J. (2004). A triparametric discrete distribution with complex parameters. Statistical Papers, 45(1), 81-95.

Saez-Castillo, A. & Conde-Sanchez, A. (2013). A hyper-poisson regression model for overdispersed and underdispersed count data. Computational Statistics & Data Analysis, 61, 148-157.

Sellers, K. F. & Shmueli, G. (2010). A flexible regression model for count data. Annals of Applied Statistics, 4(2), 943-961.

Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica, 4, 639-650.

Shmueli, G., Minka, T. P., Kadane, J. B., Borle, S., & Boatwright, P. (2005). A useful distribution for fitting discrete data: revival of the conwaymaxwellpoisson distribution. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54(1), 127-142.

Walker, S. G. (2007). Sampling the Dirichlet mixture model with slices. Communications in Statistics Simulation and Computation, 36(1), 45-54.

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
# Bayesian nonparametric GLM with Binomial response Y and one predictor X
data(simD)
pred<-seq(with(simD,min(X))+0.1,with(simD,max(X))-0.1,length.out=30)
npred<-length(pred)
# fit1 and fit2 define the same model but with different numbers of
# components and posterior samples. They both use a slice sampler
# and parameter prec=200 achieves optimal acceptance rate, about 22%.
fit1 <- bnpglm(cbind(Y,(E-Y))~X, family="binomial", data=simD, ncomp=30, sweeps=150,
               burn=100, sampler="slice", prec1=c(200), Xpred=pred, offsetPred=rep(30,npred))
fit2 <- bnpglm(cbind(Y,(E-Y))~X, family="binomial", data=simD, ncomp=50, sweeps=5000,
               burn=1000, sampler="slice", prec=c(200), Xpred=pred, offsetPred=rep(30,npred))
plot(with(simD,X),with(simD,Y)/with(simD,E))
lines(pred,fit2$medianReg,col=3,lwd=2)