bayesbay.parameterization.ParameterSpace

class bayesbay.parameterization.ParameterSpace(name, n_dimensions=None, n_dimensions_min=1, n_dimensions_max=10, n_dimensions_init_range=0.3, parameters=None)

Utility class to parameterize the Bayesian inference problem

Parameters:

name (str) – name of the parameter space, for display and storing purposes
n_dimensions (Number, optional) – number of dimensions defining the parameter space. None (default) results in a trans-dimensional space.
n_dimensions_min (Number, optional) – minimum and maximum number of dimensions, by default 1 and 10. These parameters are ignored if n_dimensions is not None, i.e. if the parameter space is not trans-dimensional
n_dimensions_max (Number, optional) – minimum and maximum number of dimensions, by default 1 and 10. These parameters are ignored if n_dimensions is not None, i.e. if the parameter space is not trans-dimensional
n_dimensions_init_range (Number, optional) –
percentage of the range n_dimensions_min - n_dimensions_max used to initialize the number of dimensions (0.3. by default). For example, if n_dimensions_min = 1, n_dimensions_max = 10, and n_dimensions_init_range = 0.5, the maximum number of dimensions at the initialization is
```
int((n_dimensions_max - n_dimensions_min) * n_dimensions_init_range + n_dimensions_max)
```
parameters (List[Prior], optional) – a list of free parameters, by default None

Reference Details

is_leaf

name: name of the parameter space

parameters: all the free parameters defined in this parameter space

perturbation_funcs: a list of perturbation functions allowed in the current parameter space. Each function takes in a state (see State) and returns a new state along with the corresponding partial acceptance probability,

\[\begin{split}\underbrace{\alpha_{p}}_{\begin{array}{c} \text{Partial} \\ \text{acceptance} \\ \text{probability} \end{array}} = \underbrace{\frac{p\left({\bf m'}\right)}{p\left({\bf m}\right)}}_{\text{Prior ratio}} \underbrace{\frac{q\left({\bf m} \mid {\bf m'}\right)}{q\left({\bf m'} \mid {\bf m}\right)}}_{\text{Proposal ratio}} \underbrace{\lvert \mathbf{J} \rvert}_{\begin{array}{c} \text{Jacobian} \\ \text{determinant} \end{array}},\end{split}\]

perturbation_weights

a list of perturbation weights, corresponding to each of the perturbation_funcs() that determines the probability of each of them to be chosen during each step

The weights are not normalized and have the following default values:

Birth/Death perturbations: 3
Parameter values perturbation: 6

trans_d: indicates whether the current configuration allows changes in dimensionality

birth(ps_state)

adds a dimension to the current parameter space and returns the thus obtained new state along with the log of the corresponding partial acceptance probability,

\[\begin{split}\underbrace{\alpha_{p}}_{\begin{array}{c} \text{Partial} \\ \text{acceptance} \\ \text{probability} \end{array}} = \underbrace{\frac{p\left({\bf m'}\right)}{p\left({\bf m}\right)}}_{\text{Prior ratio}} \underbrace{\frac{q\left({\bf m} \mid {\bf m'}\right)}{q\left({\bf m'} \mid {\bf m}\right)}}_{\text{Proposal ratio}} \underbrace{\lvert \mathbf{J} \rvert}_{\begin{array}{c} \text{Jacobian} \\ \text{determinant} \end{array}}.\end{split}\]

In this case, we have

\[\frac{p\left({\bf m'}\right)}{p\left({\bf m}\right)} = \prod_i{p\left({m_i^{k+1}}\right)},\]

because all entries of the \(i\)th free parameter, \(\mathbf{m}_i\), are equal to those of \(\mathbf{m'}_i\) except for the newly born \(m_i^{k+1}\), with \(k\) denoting the number of dimensions in the parameter space prior to the birth perturbation.

Furthermore, since \(m_i^{k+1}\) is drawn from the prior and there are \(k+1\) positions available to randomly remove a dimension or add a new one,

\[\frac{q\left({\bf m} \mid {\bf m'}\right)}{q\left({\bf m'} \mid {\bf m}\right)} = \frac{1}{k+1} \frac{\prod_i{p\left({m_i^{k+1}}\right)}}{\frac{1}{(k+1)}} = \frac{1}{\prod_i{p\left({m_i^{k+1}}\right)}}.\]

Finally, it is easy to shown that in this case \(\lvert \mathbf{J} \rvert = 1\). It follows that

\[\alpha_{p} = \frac{p\left({\bf m'}\right)}{p\left({\bf m}\right)} \frac{q\left({\bf m} \mid {\bf m'}\right)}{q\left({\bf m'} \mid {\bf m}\right)} = \prod_i{p\left({\mathbf{m}_i^{k+1}}\right)} \frac{1}{\prod_i{p\left({m_i^{k+1}}\right)}} = 1,\]

and \(\log(\alpha_{p}) = 0\).

Parameters:

ParameterSpaceState – initial parameter space state

Returns:

ParameterSpaceState – new parameter space state
Number – log of the partial acceptance probability, \(\alpha_{p} = \log( \frac{p({\bf m'})}{p({\bf m})} \frac{q\left({\bf m} \mid {\bf m'}\right)}{q\left({\bf m'} \mid {\bf m}\right)} \lvert \mathbf{J} \rvert) = 0\)

death(ps_state)

removes a dimension from the given parameter space and returns the thus obtained new state along with the log of the corresponding partial acceptance probability,

\[\begin{split}\underbrace{\alpha_{p}}_{\begin{array}{c} \text{Partial} \\ \text{acceptance} \\ \text{probability} \end{array}} = \underbrace{\frac{p\left({\bf m'}\right)}{p\left({\bf m}\right)}}_{\text{Prior ratio}} \underbrace{\frac{q\left({\bf m} \mid {\bf m'}\right)}{q\left({\bf m'} \mid {\bf m}\right)}}_{\text{Proposal ratio}} \underbrace{\lvert \mathbf{J} \rvert}_{\begin{array}{c} \text{Jacobian} \\ \text{determinant} \end{array}}.\end{split}\]

Following a reasoning similar to that explained in the documentation of birth(), in this case \(\log(\alpha_{p}) = 0\).

Parameters:

ParameterSpaceState – initial parameter space state

Returns:

ParameterSpaceState – new parameter space state
Number – log of the partial acceptance probability, \(\alpha_{p} = \log( \frac{p({\bf m'})}{p({\bf m})} \frac{q\left({\bf m} \mid {\bf m'}\right)}{q\left({\bf m'} \mid {\bf m}\right)} \lvert \mathbf{J} \rvert) = 0\)

initialize(position=None)

initializes the parameter space including its parameter values

If position is None, a single parameter space state is initialized; otherwise, a list of parameter space states is initialized randomly.

If self._n_dimensions_init_range is not None, the number of dimensions will be initialized randomly within the range defined by the init range.

Paramters

positionnp.ndarray, optional: initial position of the parameter space, by default None. The type is np.ndarray so as to align with other instances to be initialized, such as the priors; here only the length of position is used.

returns:: an initial parameter space state
rtype:: ParameterSpaceState

log_prior(param_space_state)

sample(*args)

Randomly generate a new parameter space state

The new value is sampled from (the full range of) the prior distribution of each parameter, regardless of the value of self._n_dimensions_init_range set during initialization.

Returns:: randomly generated parameter space state
Return type:: ParameterSpaceState

back to top