Markov chain from a Dirichlet process

Generates a Markov chain whose rows are drawn from a truncated Dirichlet process with the stick-breaking (GEM) construction, as MarkovChain.dirichlet_process() of PyDTMC does.

Usage

dirichletChain(
  n,
  diffusion,
  states = NULL,
  diagonalBias = NULL,
  shiftConcentration = FALSE,
  byrow = TRUE,
  seed = NULL,
  name = "Dirichlet process chain"
)

Arguments

n: The number of states, at least 2. It can be omitted when states is given.
diffusion: The concentration parameter $\alpha > 0$ of the Dirichlet process. Small values concentrate the probability of each row on its first states; large values spread it more evenly.
states: An optional character vector of n state names. Defaults to as.character(1:n).
diagonalBias: An optional positive number $\beta$. When given, a draw from $\mathrm{Beta}(\beta, 1)$ is added to each diagonal entry before the row is renormalised, which makes the chain more likely to stay where it is; larger values give a stronger bias.
shiftConcentration: If TRUE, the columns are reversed, so that the probability concentrates on the last states instead of the first ones.
byrow: Whether the transition matrix of the result is stored by rows (the default) or by columns.
seed: An optional whole number, as in randomMarkovChain.
name: The name slot of the result.

Value

A markovchain object with n states.

Details

For each row, $b_1, \ldots, b_n$ are independent $\mathrm{Beta}(1, \alpha)$ draws and the weights are $$w_j = b_j \prod_{k < j} (1 - b_k),$$ normalised to sum to one (the truncation at $n$ states leaves out the mass $\prod_k (1 - b_k)$). PyDTMC only accepts whole values of $\alpha$ between 1 and $n$; any positive value is accepted here, since the construction is defined for every $\alpha > 0$.

References

Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica, 4(2), 639-650.

Examples

dirichletChain(5, diffusion = 2, seed = 1)
#> Dirichlet process chain 
#>  A  5 - dimensional discrete Markov Chain defined by the following states: 
#>  1, 2, 3, 4, 5 
#>  The transition matrix  (by rows)  is defined as follows: 
#>           1         2           3          4          5
#> 1 0.6297278 0.1236413 0.009434792 0.07335242 0.16384368
#> 2 0.2641065 0.1507858 0.453149400 0.01874959 0.11320873
#> 3 0.6825133 0.0346931 0.161828110 0.10215682 0.01880864
#> 4 0.1936219 0.2395930 0.123421772 0.13778836 0.30557497
#> 5 0.1578606 0.3053509 0.218171241 0.29528612 0.02333114
#> 
# a chain that tends to stay in its current state
dirichletChain(5, diffusion = 2, diagonalBias = 5, seed = 1)
#> Dirichlet process chain 
#>  A  5 - dimensional discrete Markov Chain defined by the following states: 
#>  1, 2, 3, 4, 5 
#>  The transition matrix  (by rows)  is defined as follows: 
#>            1          2           3           4          5
#> 1 0.80587070 0.06482367 0.004946549 0.038457799 0.08590129
#> 2 0.13330163 0.57137879 0.228716666 0.009463421 0.05713949
#> 3 0.37722940 0.01917509 0.536737127 0.056462719 0.01039566
#> 4 0.10173955 0.12589530 0.064852562 0.546946757 0.16056583
#> 5 0.08668598 0.16767726 0.119804333 0.162150415 0.46368201
#>