# Math¶

This submodule contains various mathematical functions. Most of them are imported directly from theano.tensor (see there for more details). Doing any kind of math with PyMC3 random variables, or defining custom likelihoods or priors requires you to use these theano expressions rather than NumPy or Python code.

 dot(a, b) Computes the dot product of two variables. constant(x[, name, ndim, dtype]) Return a symbolic Constant with value x. flatten(x[, ndim, outdim]) Reshapes the variable x by keeping the first outdim-1 dimension size(s) of x the same, and making the last dimension size of x equal to the multiplication of its remaining dimension size(s). zeros_like(model[, dtype, opt]) equivalent of numpy.zeros_like Parameters ———- model : tensor dtype : data-type, optional opt : If True, we will return a constant instead of a graph when possible. ones_like(model[, dtype, opt]) equivalent of numpy.ones_like Parameters ———- model : tensor dtype : data-type, optional opt : If True, we will return a constant instead of a graph when possible. stack(*tensors, **kwargs) Stack tensors in sequence on given axis (default is 0). concatenate(tensor_list[, axis]) Alias for join(axis, *tensor_list). sum(input[, axis, dtype, keepdims, acc_dtype]) Computes the sum along the given axis(es) of a tensor input. prod(input[, axis, dtype, keepdims, …]) Computes the product along the given axis(es) of a tensor input. lt a < b gt a > b le a <= b ge a >= b eq a == b neq a != b switch if cond then ift else iff clip Clip x to be between min and max. where if cond then ift else iff and_ bitwise a & b or_ bitwise a | b abs_ |a| exp e^a log base e logarithm of a cos cosine of a sin sine of a tan tangent of a cosh hyperbolic cosine of a sinh hyperbolic sine of a tanh hyperbolic tangent of a sqr square of a sqrt square root of a erf error function erfinv inverse error function dot(a, b) Computes the dot product of two variables. maximum elemwise maximum. minimum elemwise minimum. sgn sign of a ceil ceiling of a floor floor of a det Matrix determinant. matrix_inverse Computes the inverse of a matrix $$A$$. extract_diag Return specified diagonals. matrix_dot(*args) Shorthand for product between several dots. trace(X) Returns the sum of diagonal elements of matrix X. sigmoid Generalizes a scalar op to tensors. logsumexp(x[, axis]) invlogit(x[, eps]) The inverse of the logit function, 1 / (1 + exp(-x)). logit(p)
class pymc3.math.BatchedDiag

Fast BatchedDiag allocation

make_node(diag)

Create a “apply” nodes for the inputs in that order.

perform(node, ins, outs, params=None)

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters: node : Apply instance Contains the symbolic inputs and outputs. inputs : list Sequence of inputs (immutable). output_storage : list List of mutable 1-element lists (do not change the length of these lists) MethodNotDefined The subclass does not override this method.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

class pymc3.math.BlockDiagonalMatrix(sparse=False, format='csr')
make_node(*matrices)

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, output_storage, params=None)

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters: node : Apply instance Contains the symbolic inputs and outputs. inputs : list Sequence of inputs (immutable). output_storage : list List of mutable 1-element lists (do not change the length of these lists) MethodNotDefined The subclass does not override this method.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

class pymc3.math.LogDet

Compute the logarithm of the absolute determinant of a square matrix M, log(abs(det(M))) on the CPU. Avoids det(M) overflow/ underflow.

Notes

Once PR #3959 (https://github.com/Theano/Theano/pull/3959/) by harpone is merged, this must be removed.

make_node(x)

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs, params=None)

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters: node : Apply instance Contains the symbolic inputs and outputs. inputs : list Sequence of inputs (immutable). output_storage : list List of mutable 1-element lists (do not change the length of these lists) MethodNotDefined The subclass does not override this method.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a Numpy ndarray, with the right number of dimensions, and the correct dtype. Its shape and stride pattern, can be arbitrary. It not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

pymc3.math.block_diagonal(matrices, sparse=False, format='csr')

See scipy.sparse.block_diag or scipy.linalg.block_diag for reference

Parameters: matrices : tensors format : str (default ‘csr’) must be one of: ‘csr’, ‘csc’ sparse : bool (default False) if True return sparse format matrix
pymc3.math.cartesian(*arrays)

Makes the Cartesian product of arrays.

Parameters: arrays: 1D array-like 1D arrays where earlier arrays loop more slowly than later ones
pymc3.math.expand_packed_triangular(n, packed, lower=True, diagonal_only=False)

Convert a packed triangular matrix into a two dimensional array.

Triangular matrices can be stored with better space efficiancy by storing the non-zero values in a one-dimensional array. We number the elements by row like this (for lower or upper triangular matrices):

[[0 - - -] [[0 1 2 3]
[1 2 - -] [- 4 5 6] [3 4 5 -] [- - 7 8] [6 7 8 9]] [- - - 9]
Parameters: n : int The number of rows of the triangular matrix. packed : theano.vector The matrix in packed format. lower : bool, default=True If true, assume that the matrix is lower triangular. diagonal_only : bool If true, return only the diagonal of the matrix.
pymc3.math.invlogit(x, eps=2.220446049250313e-16)

The inverse of the logit function, 1 / (1 + exp(-x)).

pymc3.math.kron_diag(*diags)

Returns diagonal of a kronecker product.

Parameters: diags: 1D arrays The diagonals of matrices that are to be Kroneckered
pymc3.math.kron_dot(krons, m)

Apply op to krons and m in a way that reproduces op(kronecker(*krons), m)

Parameters: krons: list of square 2D array-like objects D square matrices [A_1, A_2, …, A_D] to be Kronecker’ed: A = A_1 otimes A_2 otimes … otimes A_D Product of column dimensions must be N m : NxM array or 1D array (treated as Nx1) Object that krons act upon
pymc3.math.kron_matrix_op(krons, m, op)

Apply op to krons and m in a way that reproduces op(kronecker(*krons), m)

Parameters: krons: list of square 2D array-like objects D square matrices [A_1, A_2, …, A_D] to be Kronecker’ed: A = A_1 otimes A_2 otimes … otimes A_D Product of column dimensions must be N m : NxM array or 1D array (treated as Nx1) Object that krons act upon
pymc3.math.kron_solve_lower(krons, m)

Apply op to krons and m in a way that reproduces op(kronecker(*krons), m)

Parameters: krons: list of square 2D array-like objects D square matrices [A_1, A_2, …, A_D] to be Kronecker’ed: A = A_1 otimes A_2 otimes … otimes A_D Product of column dimensions must be N m : NxM array or 1D array (treated as Nx1) Object that krons act upon
pymc3.math.kron_solve_upper(krons, m)

Apply op to krons and m in a way that reproduces op(kronecker(*krons), m)

Parameters: krons: list of square 2D array-like objects D square matrices [A_1, A_2, …, A_D] to be Kronecker’ed: A = A_1 otimes A_2 otimes … otimes A_D Product of column dimensions must be N m : NxM array or 1D array (treated as Nx1) Object that krons act upon
pymc3.math.kronecker(*Ks)
Return the Kronecker product of arguments:
$$K_1 \otimes K_2 \otimes ... \otimes K_D$$
Parameters: Ks: 2D array-like
pymc3.math.log1mexp(x)

Return log(1 - exp(-x)).

This function is numerically more stable than the naive approch.

pymc3.math.log1pexp(x)

Return log(1 + exp(x)), also called softplus.

This function is numerically more stable than the naive approch.

pymc3.math.logdiffexp(exp(a) - exp(b))
pymc3.math.tround(*args, **kwargs)

Temporary function to silence round warning in Theano. Please remove when the warning disappears.