# A log of a posteriori that the covariance matrix is invariant under permutation More precisely, it is the logarithm of an unnormalized posterior probability. It is the goal function for optimization algorithms in the `find_MAP()` function. The `perm_proposal` that maximizes this function is the Maximum A Posteriori (MAP) Estimator. ```r log_posteriori_of_gipsmult(g) ``` ## Arguments - `g`: An object of a `gipsmult` class. ## Returns Returns a value of the logarithm of an unnormalized A Posteriori. ## Details It is calculated using [formulas (33) and (27) from references](https://arxiv.org/abs/2004.03503). If `Inf` or `NaN` is reached, it produces a warning. ## Examples ```r # In the space with p = 2, there is only 2 permutations: perm1 <- permutations::as.cycle("(1)(2)") perm2 <- permutations::as.cycle("(1,2)") S1 <- matrix(c(1, 0.5, 0.5, 2), nrow = 2, byrow = TRUE) S2 <- matrix(c(2, 1, 3, 7), nrow = 2, byrow = TRUE) g1 <- gipsmult(list(S1, S2), c(100, 100), perm = perm1) g2 <- gipsmult(list(S1, S2), c(100, 100), perm = perm2) log_posteriori_of_gipsmult(g1) # -354.4394, this is the MAP Estimator log_posteriori_of_gipsmult(g2) # -380.0079 exp(log_posteriori_of_gipsmult(g1) - log_posteriori_of_gipsmult(g2)) # 127131902082 # g1 is 127131902082 times more likely than g2. # This is the expected outcome because S1[1,1] and S2[1,1] # differ significantly from S1[2,2] and S2[2,2] respectively. # ======================================================================== S3 <- matrix(c(1, 0.5, 0.5, 1.1), nrow = 2, byrow = TRUE) S4 <- matrix(c(2, 1, 3, 2.137), nrow = 2, byrow = TRUE) g1 <- gipsmult(list(S3, S4), c(100, 100), perm = perm1) g2 <- gipsmult(list(S3, S4), c(100, 100), perm = perm2) log_posteriori_of_gipsmult(g1) # -148.6485 log_posteriori_of_gipsmult(g2) # -145.3019, this is the MAP Estimator exp(log_posteriori_of_gipsmult(g2) - log_posteriori_of_gipsmult(g1)) # 28.406 # g2 is 28.406 times more likely than g1. # This is the expected outcome because S1[1,1] and S2[1,1] # are very close to S1[2,2] and S2[2,2] respectively. ``` ## References Piotr Graczyk, Hideyuki Ishi, Bartosz Kołodziejek, Hélène Massam. "Model selection in the space of Gaussian models invariant by symmetry." The Annals of Statistics, 50(3) 1747-1774 June 2022. [arXiv link](https://arxiv.org/abs/2004.03503); tools:::Rd_expr_doi("10.1214/22-AOS2174") ## See Also * `find_MAP()` - The function that optimizes the `log_posteriori_of_gips` function. * `gips::compare_posteriories_of_perms()` - Uses `log_posteriori_of_gips()` to compare a posteriori of two permutations.

log_posteriori_of_gipsmult function