AVh* dZddlZddlZddlZddlZddlZddlmZddlm Z ddlm Z ddlm Z ddl m Z ddl mZdd l mZdd lmZdd lmZd gZGd dej,dgdZGdd ej0Zy)zBijector base.N)dtypes)ops) tensor_shape) tensor_util) array_ops) check_ops)math_ops)util)object_identityBijectorc`eZdZdZd fd ZedZedZd dZd dZ dZ dZ xZ S) _Mappingz?Helper class to make it easier to manage caching in `Bijector`.c2tt| |||||S)aCustom __new__ so namedtuple items have defaults. Args: x: `Tensor`. Forward. y: `Tensor`. Inverse. ildj_map: `Dictionary`. This is a mapping from event_ndims to a `Tensor` representing the inverse log det jacobian. kwargs: Python dictionary. Extra args supplied to forward/inverse/etc functions. Returns: mapping: New instance of _Mapping. )superr__new__)clsxyildj_mapkwargs __class__s a/home/dcms/DCMS/lib/python3.12/site-packages/tensorflow/python/ops/distributions/bijector_impl.pyrz_Mapping.__new__,s 3 'Q8V DDc tj|jf|jt t |j jzS)z$Returns key used for caching Y=g(X).)r Referencer _deep_tupletuplesortedritemsselfs rx_keyz_Mapping.x_key<J & &tvv . 0   U6$++*;*;*=#>? @ ABrc tj|jf|jt t |j jzS)z)Returns key used for caching X=g^{-1}(Y).)r rrrrrrrr s ry_keyz_Mapping.y_keyBr#rc |t||||}n!td||||fDr tdt|j|j|j|j|j |j |j |j|j|j|j|jS)aiReturns new _Mapping with args merged with self. Args: x: `Tensor`. Forward. y: `Tensor`. Inverse. ildj_map: `Dictionary`. This is a mapping from event_ndims to a `Tensor` representing the inverse log det jacobian. kwargs: Python dictionary. Extra args supplied to forward/inverse/etc functions. mapping: Instance of _Mapping to merge. Can only be specified if no other arg is specified. Returns: mapping: New instance of `_Mapping` which has inputs merged with self. Raises: ValueError: if mapping and any other arg is not `None`. rrrrc3$K|]}|du ywN).0args r z!_Mapping.merge..]s AS_ Asz?Cannot simultaneously specify mapping and individual arguments.) rany ValueError_mergerr _merge_dictsrr)r!rrrrmappings rmergez_Mapping.mergeHs&1HVDg AAx(@ A A $ %%  ++dffgii ( ++dffgii (""4=='2B2BC{{4;;7  99rc|in|}|in|}|jD]A\}}|j|d}|#||urtdj|||||||<C|S)z!Helper to merge two dictionaries.NzHFound different value for existing key (key:{} old_value:{} new_value:{})rgetr/format)r!oldnewkvvals rr1z_Mapping._merge_dictsgs "C "C 1 GGAt c S\==CVA>+, ,c!f  Jrc<||S|||urtd|d||S)z0Helper to merge which handles merging one value.zIncompatible values: z != )r/)r!r7r8s rr0z_Mapping._mergets, { j S^ #sC DD Jrcpt|ttfrtt|j|S|S)z,Converts lists of lists to tuples of tuples.) isinstancelistrmaprr!rs rrz_Mapping._deep_tuple|s6!dE]+ #d&&* +4124r)NNNN)NNNNNNN) __name__ __module__ __qualname____doc__rpropertyr"r%r3r1r0r __classcell__)rs@rrr(sLGE  B B  B B 9> 4rrr'ceZdZdZej d(dZedZedZ edZ edZ edZ ed Z ed Zed Zd Z d)d ZdZdZdZ d*dZdZdZdZdZd+dZdZdZd,dZdZdZ d-dZ dZ!dZ" d.dZ#e$jJd/d Z&d!Z'd"Z(d0d#Z)d$Z*d%Z+d&Z,d'Z-y)1r a2Interface for transformations of a `Distribution` sample. Bijectors can be used to represent any differentiable and injective (one to one) function defined on an open subset of `R^n`. Some non-injective transformations are also supported (see "Non Injective Transforms" below). #### Mathematical Details A `Bijector` implements a [smooth covering map]( https://en.wikipedia.org/wiki/Local_diffeomorphism), i.e., a local diffeomorphism such that every point in the target has a neighborhood evenly covered by a map ([see also]( https://en.wikipedia.org/wiki/Covering_space#Covering_of_a_manifold)). A `Bijector` is used by `TransformedDistribution` but can be generally used for transforming a `Distribution` generated `Tensor`. A `Bijector` is characterized by three operations: 1. Forward Useful for turning one random outcome into another random outcome from a different distribution. 2. Inverse Useful for "reversing" a transformation to compute one probability in terms of another. 3. `log_det_jacobian(x)` "The log of the absolute value of the determinant of the matrix of all first-order partial derivatives of the inverse function." Useful for inverting a transformation to compute one probability in terms of another. Geometrically, the Jacobian determinant is the volume of the transformation and is used to scale the probability. We take the absolute value of the determinant before log to avoid NaN values. Geometrically, a negative determinant corresponds to an orientation-reversing transformation. It is ok for us to discard the sign of the determinant because we only integrate everywhere-nonnegative functions (probability densities) and the correct orientation is always the one that produces a nonnegative integrand. By convention, transformations of random variables are named in terms of the forward transformation. The forward transformation creates samples, the inverse is useful for computing probabilities. #### Example Uses - Basic properties: ```python x = ... # A tensor. # Evaluate forward transformation. fwd_x = my_bijector.forward(x) x == my_bijector.inverse(fwd_x) x != my_bijector.forward(fwd_x) # Not equal because x != g(g(x)). ``` - Computing a log-likelihood: ```python def transformed_log_prob(bijector, log_prob, x): return (bijector.inverse_log_det_jacobian(x, event_ndims=0) + log_prob(bijector.inverse(x))) ``` - Transforming a random outcome: ```python def transformed_sample(bijector, x): return bijector.forward(x) ``` #### Example Bijectors - "Exponential" ```none Y = g(X) = exp(X) X ~ Normal(0, 1) # Univariate. ``` Implies: ```none g^{-1}(Y) = log(Y) |Jacobian(g^{-1})(y)| = 1 / y Y ~ LogNormal(0, 1), i.e., prob(Y=y) = |Jacobian(g^{-1})(y)| * prob(X=g^{-1}(y)) = (1 / y) Normal(log(y); 0, 1) ``` Here is an example of how one might implement the `Exp` bijector: ```python class Exp(Bijector): def __init__(self, validate_args=False, name="exp"): super(Exp, self).__init__( validate_args=validate_args, forward_min_event_ndims=0, name=name) def _forward(self, x): return math_ops.exp(x) def _inverse(self, y): return math_ops.log(y) def _inverse_log_det_jacobian(self, y): return -self._forward_log_det_jacobian(self._inverse(y)) def _forward_log_det_jacobian(self, x): # Notice that we needn't do any reducing, even when`event_ndims > 0`. # The base Bijector class will handle reducing for us; it knows how # to do so because we called `super` `__init__` with # `forward_min_event_ndims = 0`. return x ``` - "Affine" ```none Y = g(X) = sqrtSigma * X + mu X ~ MultivariateNormal(0, I_d) ``` Implies: ```none g^{-1}(Y) = inv(sqrtSigma) * (Y - mu) |Jacobian(g^{-1})(y)| = det(inv(sqrtSigma)) Y ~ MultivariateNormal(mu, sqrtSigma) , i.e., prob(Y=y) = |Jacobian(g^{-1})(y)| * prob(X=g^{-1}(y)) = det(sqrtSigma)^(-d) * MultivariateNormal(inv(sqrtSigma) * (y - mu); 0, I_d) ``` #### Min_event_ndims and Naming Bijectors are named for the dimensionality of data they act on (i.e. without broadcasting). We can think of bijectors having an intrinsic `min_event_ndims` , which is the minimum number of dimensions for the bijector act on. For instance, a Cholesky decomposition requires a matrix, and hence `min_event_ndims=2`. Some examples: `AffineScalar: min_event_ndims=0` `Affine: min_event_ndims=1` `Cholesky: min_event_ndims=2` `Exp: min_event_ndims=0` `Sigmoid: min_event_ndims=0` `SoftmaxCentered: min_event_ndims=1` Note the difference between `Affine` and `AffineScalar`. `AffineScalar` operates on scalar events, whereas `Affine` operates on vector-valued events. More generally, there is a `forward_min_event_ndims` and an `inverse_min_event_ndims`. In most cases, these will be the same. However, for some shape changing bijectors, these will be different (e.g. a bijector which pads an extra dimension at the end, might have `forward_min_event_ndims=0` and `inverse_min_event_ndims=1`. #### Jacobian Determinant The Jacobian determinant is a reduction over `event_ndims - min_event_ndims` (`forward_min_event_ndims` for `forward_log_det_jacobian` and `inverse_min_event_ndims` for `inverse_log_det_jacobian`). To see this, consider the `Exp` `Bijector` applied to a `Tensor` which has sample, batch, and event (S, B, E) shape semantics. Suppose the `Tensor`'s partitioned-shape is `(S=[4], B=[2], E=[3, 3])`. The shape of the `Tensor` returned by `forward` and `inverse` is unchanged, i.e., `[4, 2, 3, 3]`. However the shape returned by `inverse_log_det_jacobian` is `[4, 2]` because the Jacobian determinant is a reduction over the event dimensions. Another example is the `Affine` `Bijector`. Because `min_event_ndims = 1`, the Jacobian determinant reduction is over `event_ndims - 1`. It is sometimes useful to implement the inverse Jacobian determinant as the negative forward Jacobian determinant. For example, ```python def _inverse_log_det_jacobian(self, y): return -self._forward_log_det_jac(self._inverse(y)) # Note negation. ``` The correctness of this approach can be seen from the following claim. - Claim: Assume `Y = g(X)` is a bijection whose derivative exists and is nonzero for its domain, i.e., `dY/dX = d/dX g(X) != 0`. Then: ```none (log o det o jacobian o g^{-1})(Y) = -(log o det o jacobian o g)(X) ``` - Proof: From the bijective, nonzero differentiability of `g`, the [inverse function theorem]( https://en.wikipedia.org/wiki/Inverse_function_theorem) implies `g^{-1}` is differentiable in the image of `g`. Applying the chain rule to `y = g(x) = g(g^{-1}(y))` yields `I = g'(g^{-1}(y))*g^{-1}'(y)`. The same theorem also implies `g^{-1}'` is non-singular therefore: `inv[ g'(g^{-1}(y)) ] = g^{-1}'(y)`. The claim follows from [properties of determinant]( https://en.wikipedia.org/wiki/Determinant#Multiplicativity_and_matrix_groups). Generally its preferable to directly implement the inverse Jacobian determinant. This should have superior numerical stability and will often share subgraphs with the `_inverse` implementation. #### Is_constant_jacobian Certain bijectors will have constant jacobian matrices. For instance, the `Affine` bijector encodes multiplication by a matrix plus a shift, with jacobian matrix, the same aforementioned matrix. `is_constant_jacobian` encodes the fact that the jacobian matrix is constant. The semantics of this argument are the following: * Repeated calls to "log_det_jacobian" functions with the same `event_ndims` (but not necessarily same input), will return the first computed jacobian (because the matrix is constant, and hence is input independent). * `log_det_jacobian` implementations are merely broadcastable to the true `log_det_jacobian` (because, again, the jacobian matrix is input independent). Specifically, `log_det_jacobian` is implemented as the log jacobian determinant for a single input. ```python class Identity(Bijector): def __init__(self, validate_args=False, name="identity"): super(Identity, self).__init__( is_constant_jacobian=True, validate_args=validate_args, forward_min_event_ndims=0, name=name) def _forward(self, x): return x def _inverse(self, y): return y def _inverse_log_det_jacobian(self, y): return -self._forward_log_det_jacobian(self._inverse(y)) def _forward_log_det_jacobian(self, x): # The full log jacobian determinant would be array_ops.zero_like(x). # However, we circumvent materializing that, since the jacobian # calculation is input independent, and we specify it for one input. return constant_op.constant(0., x.dtype.base_dtype) ``` #### Subclass Requirements - Subclasses typically implement: - `_forward`, - `_inverse`, - `_inverse_log_det_jacobian`, - `_forward_log_det_jacobian` (optional). The `_forward_log_det_jacobian` is called when the bijector is inverted via the `Invert` bijector. If undefined, a slightly less efficiently calculation, `-1 * _inverse_log_det_jacobian`, is used. If the bijector changes the shape of the input, you must also implement: - _forward_event_shape_tensor, - _forward_event_shape (optional), - _inverse_event_shape_tensor, - _inverse_event_shape (optional). By default the event-shape is assumed unchanged from input. - If the `Bijector`'s use is limited to `TransformedDistribution` (or friends like `QuantizedDistribution`) then depending on your use, you may not need to implement all of `_forward` and `_inverse` functions. Examples: 1. Sampling (e.g., `sample`) only requires `_forward`. 2. Probability functions (e.g., `prob`, `cdf`, `survival`) only require `_inverse` (and related). 3. Only calling probability functions on the output of `sample` means `_inverse` can be implemented as a cache lookup. See "Example Uses" [above] which shows how these functions are used to transform a distribution. (Note: `_forward` could theoretically be implemented as a cache lookup but this would require controlling the underlying sample generation mechanism.) #### Non Injective Transforms **WARNING** Handing of non-injective transforms is subject to change. Non injective maps `g` are supported, provided their domain `D` can be partitioned into `k` disjoint subsets, `Union{D1, ..., Dk}`, such that, ignoring sets of measure zero, the restriction of `g` to each subset is a differentiable bijection onto `g(D)`. In particular, this implies that for `y in g(D)`, the set inverse, i.e. `g^{-1}(y) = {x in D : g(x) = y}`, always contains exactly `k` distinct points. The property, `_is_injective` is set to `False` to indicate that the bijector is not injective, yet satisfies the above condition. The usual bijector API is modified in the case `_is_injective is False` (see method docstrings for specifics). Here we show by example the `AbsoluteValue` bijector. In this case, the domain `D = (-inf, inf)`, can be partitioned into `D1 = (-inf, 0)`, `D2 = {0}`, and `D3 = (0, inf)`. Let `gi` be the restriction of `g` to `Di`, then both `g1` and `g3` are bijections onto `(0, inf)`, with `g1^{-1}(y) = -y`, and `g3^{-1}(y) = y`. We will use `g1` and `g3` to define bijector methods over `D1` and `D3`. `D2 = {0}` is an oddball in that `g2` is one to one, and the derivative is not well defined. Fortunately, when considering transformations of probability densities (e.g. in `TransformedDistribution`), sets of measure zero have no effect in theory, and only a small effect in 32 or 64 bit precision. For that reason, we define `inverse(0)` and `inverse_log_det_jacobian(0)` both as `[0, 0]`, which is convenient and results in a left-semicontinuous pdf. ```python abs = tfp.distributions.bijectors.AbsoluteValue() abs.forward(-1.) ==> 1. abs.forward(1.) ==> 1. abs.inverse(1.) ==> (-1., 1.) # The |dX/dY| is constant, == 1. So Log|dX/dY| == 0. abs.inverse_log_det_jacobian(1., event_ndims=0) ==> (0., 0.) # Special case handling of 0. abs.inverse(0.) ==> (0., 0.) abs.inverse_log_det_jacobian(0., event_ndims=0) ==> (0., 0.) ``` Nc|xsg|_| | td||}n||}t|ts-t dj t |jt|ts-t dj t |j|dkr td|dkr td||_||_ ||_ i|_ ||_ ||_ i|_i|_|r||_n2d}|t |jj#d |_t%|jD],\} } | t'j(| rtd | | fzy) aConstructs Bijector. A `Bijector` transforms random variables into new random variables. Examples: ```python # Create the Y = g(X) = X transform. identity = Identity() # Create the Y = g(X) = exp(X) transform. exp = Exp() ``` See `Bijector` subclass docstring for more details and specific examples. Args: graph_parents: Python list of graph prerequisites of this `Bijector`. is_constant_jacobian: Python `bool` indicating that the Jacobian matrix is not a function of the input. validate_args: Python `bool`, default `False`. Whether to validate input with asserts. If `validate_args` is `False`, and the inputs are invalid, correct behavior is not guaranteed. dtype: `tf.dtype` supported by this `Bijector`. `None` means dtype is not enforced. forward_min_event_ndims: Python `integer` indicating the minimum number of dimensions `forward` operates on. inverse_min_event_ndims: Python `integer` indicating the minimum number of dimensions `inverse` operates on. Will be set to `forward_min_event_ndims` by default, if no value is provided. name: The name to give Ops created by the initializer. Raises: ValueError: If neither `forward_min_event_ndims` and `inverse_min_event_ndims` are specified, or if either of them is negative. ValueError: If a member of `graph_parents` is not a `Tensor`. 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