HDPHMMpoisson {MCMCpack} | R Documentation

## Markov Chain Monte Carlo for sticky HDP-HMM with a Poisson outcome distribution

### Description

This function generates a sample from the posterior distribution of a (sticky) HDP-HMM with a Poisson outcome distribution (Fox et al, 2011). The user supplies data and priors, and a sample from the posterior distribution is returned as an mcmc object, which can be subsequently analyzed with functions provided in the coda package.

### Usage

    HDPHMMpoisson(
      formula,
      data = parent.frame(),
      K = 10,
      b0 = 0,
      B0 = 1,
      a.alpha = 1,
      b.alpha = 0.1,
      a.gamma = 1,
      b.gamma = 0.1,
      a.theta = 50,
      b.theta = 5,
      burnin = 1000,
      mcmc = 1000,
      thin = 1,
      verbose = 0,
      seed = NA,
      beta.start = NA,
      P.start = NA,
      gamma.start = 0.5,
      theta.start = 0.98,
      ak.start = 100,
      ...
    )

### Arguments

`formula` | Model formula.
`data` | Data frame.
`K` | The number of regimes under consideration. This should be larger than the hypothesized number of regimes in the data. Note that the sampler will likely visit fewer than `K` regimes.
`b0` | The prior mean of β. This can either be a scalar or a column vector with dimension equal to the number of betas. If this takes a scalar value, then that value will serve as the prior mean for all of the betas.
`B0` | The prior precision of β. This can either be a scalar or a square matrix with dimensions equal to the number of betas. If this takes a scalar value, then that value times an identity matrix serves as the prior precision of beta. Default value of 0 is equivalent to an improper uniform prior for beta.
`a.alpha, b.alpha` | Shape and scale parameters for the Gamma distribution on α + κ.
`a.gamma, b.gamma` | Shape and scale parameters for the Gamma distribution on γ.
`a.theta, b.theta` | Paramaters for the Beta prior on θ, which captures the strength of the self-transition bias.
`burnin` | The number of burn-in iterations for the sampler.
`mcmc` | The number of Metropolis iterations for the sampler.
`thin` | The thinning interval used in the simulation. The number of mcmc iterations must be divisible by this value.
`verbose` | A switch which determines whether or not the progress of the sampler is printed to the screen. If `verbose` is greater than 0 the iteration number, the current beta vector, and the Metropolis acceptance rate are printed to the screen every `verbose`th iteration.
`seed` | The seed for the random number generator. If NA, the Mersenne Twister generator is used with default seed 12345; if an integer is passed it is used to seed the Mersenne twister. The user can also pass a list of length two to use the L'Ecuyer random number generator, which is suitable for parallel computation. The first element of the list is the L'Ecuyer seed, which is a vector of length six or NA (if NA a default seed of `rep(12345,6)` is used). The second element of list is a positive substream number. See the MCMCpack specification for more details.
`beta.start` | The starting value for the β vector. This can either be a scalar or a column vector with dimension equal to the number of betas. If this takes a scalar value, then that value will serve as the starting value for all of the betas. The default value of NA will use the maximum likelihood estimate of β as the starting value for all regimes.
`P.start` | Initial transition matrix between regimes. Should be a `K` by `K` matrix. If not provided, the default value will be place `theta.start` along the diagonal and the rest of the mass even distributed within rows.
`theta.start, ak.start, gamma.start` | Scalar starting values for the θ, α + κ, and γ parameters.
`...` | further arguments to be passed.

### Details

`HDPHMMpoisson` simulates from the posterior distribution of a sticky HDP-HMM with a Poisson outcome distribution, allowing for multiple, arbitrary changepoints in the model. The details of the model are discussed in Blackwell (2017). The implementation here is based on a weak-limit approximation, where there is a large, though finite number of regimes that can be switched between. Unlike other changepoint models in `MCMCpack`, the HDP-HMM approach allows for the state sequence to return to previous visited states.

The model takes the following form, where we show the fixed-limit version:

y_t \sim \mathcal{P}oisson(μ_t)

μ_t = x_t ' β_m,\;\; m = 1, …, M

Where M is an upper bound on the number of states and β_m are parameters when a state is m at t.

The transition probabilities between states are assumed to follow a heirarchical Dirichlet process:

π_m \sim \mathcal{D}irichlet(αδ_1, …, αδ_j + κ, …, αδ_M)

δ \sim \mathcal{D}irichlet(γ/M, …, γ/M)

The κ value here is the sticky parameter that encourages self-transitions. The sampler follows Fox et al (2011) and parameterizes these priors with α + κ and θ = κ/(α + κ), with the latter representing the degree of self-transition bias. Gamma priors are assumed for (α + κ) and γ.

We assume Gaussian distribution for prior of β:

β_m \sim \mathcal{N}(b_0,B_0^{-1}),\;\; m = 1, …, M

The model is simulated via blocked Gibbs conditonal on the states. The β being simulated via the auxiliary mixture sampling method of Fuerhwirth-Schanetter et al. (2009). The states are updated as in Fox et al (2011), supplemental materials.

### Value

An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.

### References

Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. “MCMCpack: Markov Chain Monte Carlo in R.”, Journal of Statistical Software. 42(9): 1-21. https://www.jstatsoft.org/v42/i09/.

Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.lsa.umich.edu.

Sylvia Fruehwirth-Schnatter, Rudolf Fruehwirth, Leonhard Held, and Havard Rue. 2009. “Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data”, Statistics and Computing 19(4): 479-492. <doi:10.1007/s11222-008-9109-4>

Matthew Blackwell. 2017. “Game Changers: Detecting Shifts in Overdispersed Count Data,” Political Analysis 26(2), 230-239. <doi:10.1017/pan.2017.42>

Emily B. Fox, Erik B. Sudderth, Michael I. Jordan, and Alan S. Willsky. 2011.. “A sticky HDP-HMM with application to speaker diarization.” The Annals of Applied Statistics, 5(2A), 1020-1056. <doi:10.1214/10-AOAS395>

### See Also

`MCMCpoissonChange`, `HDPHMMnegbin`

### Examples

     ## Not run:
       n <- 150
       reg <- 3
       true.s <- gl(reg, n/reg, n)
       b1.true <- c(1, -2, 2)
       x1 <- runif(n, 0, 2)
       mu <- exp(1 + x1 * b1.true[true.s])
       y <- rpois(n, mu)

       posterior <- HDPHMMpoisson(y ~ x1, K = 10, verbose = 1000,
                              a.theta = 100, b.theta = 1,
                              b0 = rep(0, 2), B0 = (1/9) * diag(2),
                              seed = list(NA, 2),
                              theta.start = 0.95, gamma.start = 10,
                              ak.start = 10)

       plotHDPChangepoint(posterior, ylab="Density", start=1)

    ## End(Not run)

[Package MCMCpack version 1.5-0 Index]