User-defined constrained parameterizations#967
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bob-carpenter
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bob-carpenter
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I renamed and tagged "WIP" because I want to go over and add a little more text background and also citations to the original threads that introduced these ideas. |
bob-carpenter
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@WardBrian. This is now code complete, spell checked, and read over by me. Sorry, but I sort of got carried away, and as a result, it's a lot. |
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Fun read. A bunch of comments below
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It might make sense to reassess https://mc-stan.org/docs/stan-users-guide/reparameterization.html#changes-of-variables and see if that should just be blended into this new chapter (or at least cross-referenced)
| real log_sigma; | ||
| } | ||
| transformed parameters { | ||
| real<lower=0> sigma = exp(log_sigma); |
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It might be worth explicitly (re)-stating that the lower= syntax only performs checks in blocks other than parameters
| if (cols(x) != N + binom(N, 2)) { | ||
| reject("cols(x) = ", cols(x), | ||
| "; but N + binom(N, 2) = ", N + binom(N, 2)); | ||
| } |
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binom is not a function; it's choose (which is correctly used in the parameters block below)
| functions { | ||
| matrix toeplitz_matrix(vector x, int M, int N) { | ||
| if (num_elements(x) != M + N - 1) { | ||
| reject("x must have M + N - 1 elements", |
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I think these rejects should be fatal_errors, since the sizes will never change there is no recovery possible
| ```stan | ||
| functions { | ||
| matrix symmetric_matrix(vector x, int N) { | ||
| matrix[N, N] y = lower_triangular_matrix(x, N); |
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| matrix[N, N] y = lower_triangular_matrix(x, N); | |
| matrix[N, N] y = lower_triangular_matrix(x, N); // defined as above |
Or any other indication that this is not a built-in and therefore this example is not fully self-contained
| ``` | ||
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| The transform itself, which takes a vector $v$ of size $\binom{N}{2}$, | ||
| creates a skew-symmetric matrix (the `skew_symmetric` function is defined |
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| creates a skew-symmetric matrix (the `skew_symmetric` function is defined | |
| creates a skew-symmetric matrix (the `skew_symmetric_matrix` function is defined |
| * @return The `N` x `N` rotation matrix for which `v` is the basis | ||
| */ | ||
| matrix special_orthogonal_jacobian(vector v, int N) { | ||
| matrix[N, N] S = skew_symmetric(v, N); |
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| matrix[N, N] S = skew_symmetric(v, N); | |
| matrix[N, N] S = skew_symmetric_matrix(v, N); |
| { | ||
| tuple(matrix[D, D], matrix[D, D]) phi_Gamma | ||
| = stationary_var1_jacobian(raw, D); | ||
| phi = phi_Gamma.1; | ||
| Gamma = phi_Gamma.2; | ||
| } |
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We have tuple unpacking
| { | |
| tuple(matrix[D, D], matrix[D, D]) phi_Gamma | |
| = stationary_var1_jacobian(raw, D); | |
| phi = phi_Gamma.1; | |
| Gamma = phi_Gamma.2; | |
| } | |
| (phi, Gamma) = stationary_var1_jacobian(raw, D); |
| covariance `Gamma`. These need to be pulled out of the tuple. This | ||
| is one function returning a tuple because there is shared work. The | ||
| extra braces mean that `phi_Gamma` is no longer a top-level | ||
| `transformed parameters` variable and thus will not be saved in the | ||
| output (it would be redundant). The model could have been written | ||
| directly in terms of `phi_Gamma.1` and `phi_Gamma.2`, but it's easier | ||
| to follow as written. Note that `phi` is used in its transposed form |
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Which makes this explanation redundant
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| ## Matrices matching known margins | ||
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| Suppose $X$ is an $M \times N$ matrix $X$, the marginals of which are |
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| Suppose $X$ is an $M \times N$ matrix $X$, the marginals of which are | |
| Suppose $X$ is an $M \times N$ matrix, the marginals of which are |
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This is a whole new chapter on user-defined constrained parameters with some challenging examples. I've tested that all the code actually works.
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