# Designing and Using Models¶

## Introduction¶

The concept of a **model** is key to the use of QInfer. A model defines the
probability distribution over experimental data given hypotheses about the
system of interest, and given a description of the measurement performed. This
distribution is called the *likelihood function*, and it encapsulates the
definition of the model.

In QInfer, likelihood functions are represented as classes inheriting from either
`Model`

, when the likelihood function can be numerically
evaluated, or `Simulatable`

when only samples from the
function can be efficiently generated.

## Using Models and Simulations¶

### Basic Functionality¶

Both `Model`

and `Simulatable`

offer
basic functionality to describe how they are parameterized, what outcomes are
possible, etc. For this example, we will use a premade model,
`SimplePrecessionModel`

. This model implements the likelihood
function

as can be derived from Born’s Rule for a spin-½ particle prepared and measured in the \(\left|+\right\rangle\propto\left|0\right\rangle+\left|1\right\rangle\) state, and evolved under \(H = \omega \sigma_z / 2\) for some time \(t\).

In this way, we see that by defining the likelihood function in terms of the hypothetical outcome \(d\), the model parameter \(\omega\), and the experimental parameter \(t\), we can reason about the experimental data that we would extract from the system.

In order to use this likelihood function, we must instantiate the model that
implements the likelihood. Since `SimplePrecessionModel`

is
provided with QInfer, we can simply import it and make an instance.

```
>>> from qinfer import SimplePrecessionModel
>>> m = SimplePrecessionModel()
```

Once a model or simulator has been created, you can query how many model parameters it admits and how many outcomes a given experiment can have.

```
>>> print(m.n_modelparams)
1
>>> print(m.modelparam_names)
['\\omega']
>>> print(m.is_n_outcomes_constant)
True
>>> print(m.n_outcomes(expparams=0))
2
```

### Model and Experiment Parameters¶

The division between unknown parameters that we are trying to learn (\(\omega\)
in the `SimplePrecessionModel`

example) and the controls that we can use to
design measurements (\(t\)) is generic, and is key to how QInfer handles
the problem of parameter estimation.
Roughly speaking, model parameters are real numbers that represent properties
of the system that we would like to learn, whereas experiment parameters
represent the choices we get to make in performing measurements.

Model parameters are represented by NumPy arrays of dtype `float`

and that
have two indices, one representing which model is being considered and one
representing which parameter. That is, model parameters are defined by matrices
such that the element \(X_{ij}\) is the \(j^{\text{th}}\) parameter of
the model parameter vector \(\vec{x}_i\).

By contrast, since not all experiment parameters are best represented by
the data type `float`

, we cannot use an array of homogeneous dtype unless there
is only one experimental parameter. The alternative is to use NumPy’s
record array functionality to specify the
*heterogeneous* type of the experiment parameters. To do so, instead of using
a second index to refer to specific experiment parameters, we use *fields*.
Each field then has its own dtype.

For instance, a dtype of `[('t', 'float'), ('basis', 'int')]`

specifies that
an array has two fields, named `t`

and `basis`

, having dtypes of `float`

and `int`

, respectively. Such arrays are initialized by passing lists of
*tuples*, one for each field:

```
>>> eps = np.array([
... (12.3, 2),
... (14.1, 1)
... ], dtype=[('t', 'float'), ('basis', 'int')])
>>> print(eps)
[(12.3, 2) (14.1, 1)]
>>> eps.shape == (2,)
True
```

Once we have made a record array, we can then index by field names to get out each field as an array of that field’s value in each record, or we can index by record to get all fields.

```
>>> print(eps['t'])
[ 12.3 14.1]
>>> print(eps['basis'])
[2 1]
>>> print(eps[0])
(12.3, 2)
```

Model classes specify the dtypes of their experimental parameters with the
property `expparams_dtype`

. Thus, a common
idiom is to pass this property to the dtype keyword of NumPy functions. For
example, the model class `BinomialModel`

adds an `int`

field specifying how many times a two-outcome measurement is repeated, so to
specify that we can use its `expparams_dtype`

:

```
>>> from qinfer import BinomialModel
>>> bm = BinomialModel(m)
>>> print(bm.expparams_dtype)
[('x', 'float'), ('n_meas', 'uint')]
>>> eps = np.array([
... (5.0, 10)
... ], dtype=bm.expparams_dtype)
```

### Model Outcomes¶

Given a specific vector of model parameters \(\vec{x}\) and a specific
experimental configuration \(\vec{c}\), the experiment will
yield some *outcome* \(d\) according to the model distribution
\(\Pr(d|\vec{x},\vec{c})\).

In many cases, such as `SimplePrecessionModel`

discussed above,
there will be a finite number of outcomes, which we can
label by some finite set of integers.
For example, we labeled the outcome \(\left|0\right\rangle\) by
\(d=0\) and the outcome \(\left|1\right\rangle\) by
\(d=1\).
If this is the case for you, the rest of this section will likely
not be very relevant, and you may assume your outcomes are
zero-indexed integers ending at some value.

In other cases, there may be an infinite number of possible outcomes. For example, if the measurement returns the total number of photons measured in a time window, which can in principle be arbitrarily large, or if the measuremnt is of a voltage or current, which can be any real number. Or, we may have outcomes which require fancy data types. For instance, perhaps the output of a single experiment is a tuple of numbers rather than a single number.

To accomodate these possible situations, and to have a
systematic way of testing whether or not all possibe outcomes can be
enumerated, `Simulatable`

(and subclasses like `Model`

)
has a method `domain`

which for every given
experimental parameter, returns a `Domain`

object.
One major benifit of explicitly storing these objects is that
certain quantities (like `bayes_risk`

)
can be computed much more efficiently when all possible outcomes
can be enumerated.
`Domain`

has attributes which specify whether or not it
are finite, how many members it has and what they are,
what data type they are, and so on.

For the `BinomialModel`

defined above, there are
`n_meas+1`

possible outcomes, with possible values
the integers between `0`

and `n_meas`

inclusive.

```
>>> bdomain = bm.domain(eps)[0]
>>> bdomain.n_members
11
>>> bdomain.values
array([ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
>>> bdomain.dtype == np.int
True
```

We need to extract the \(0^\text{th}\) element of
`bm.domain(eps)`

above because `eps`

is a vector
of length \(1\) and `domain`

always
returns one domain for every member of `eps`

.
In the case where the domain is completely independent of `eps`

,
it should be possible to call `m.domain(None)`

to return
the unique domain of the model `m`

.

The `MultinomialModel`

requires a fancy
datatype so that outcomes can be tuples of integers.
In the following a single experiment of the model `mm`

consists of throwing a four sided die `n_meas`

times
and recording how many times each side lands facing down.

```
>>> from qinfer import MultinomialModel, NDieModel
>>> mm = MultinomialModel(NDieModel(n=4))
>>> mm.expparams_dtype
[('exp_num', 'int'), ('n_meas', 'uint')]
>>> mmeps = np.array([(1, 3)], dtype=mm.expparams_dtype)
>>> mmdomain = mm.domain(mmeps)[0]
>>> mmdomain.dtype
dtype([('k', '<i8', (4,))])
>>> mmdomain.n_members
20
>>> print(mmdomain.values)
[([3, 0, 0, 0],) ([2, 1, 0, 0],) ([2, 0, 1, 0],) ([2, 0, 0, 1],)
([1, 2, 0, 0],) ([1, 1, 1, 0],) ([1, 1, 0, 1],) ([1, 0, 2, 0],)
([1, 0, 1, 1],) ([1, 0, 0, 2],) ([0, 3, 0, 0],) ([0, 2, 1, 0],)
([0, 2, 0, 1],) ([0, 1, 2, 0],) ([0, 1, 1, 1],) ([0, 1, 0, 2],)
([0, 0, 3, 0],) ([0, 0, 2, 1],) ([0, 0, 1, 2],) ([0, 0, 0, 3],)]
```

We see here all \(20\) possible ways to roll this die four times.

Note

`Model`

inherits from`Simulatable`

, and`FiniteOutcomeModel`

inherits from`Model`

. The subclass`FiniteOutcomeModel`

is able to concretely define some methods (like`simulate_experiment`

) because of the guarantee that all domains have a finite number of elements. Therefore, it is generally a bit less work to construct a`FiniteOutcomeModel`

than it is to construct a`Model`

.Additionally,

`FiniteOutcomeModel`

automatically defines the domain corresponding to the experimental parameter`ep`

by looking at`n_outcomes`

, namely, if`nep=n_outcomes(ep)`

, then the corresponding domain has members`0,1,...,nep`

by default.Finally, make note of the slightly subtle role of the method

`n_outcomes`

. In principle,`n_outcomes`

is completely independent of`domain`

. For`FiniteOutcomeModel`

, it will almost always hold that`m.n_outcomes(ep)==domain(ep)[0].n_members`

. For models with an infinite number of outcomes,`n_members`

is not defined, but`n_outcomes`

is defined and refers to “enough outcomes” (at the user’s discretion) to make estimates of quantities`bayes_risk`

.

### Simulation¶

Both models and simulators allow for simulated data to be drawn from the
model distribution using the `simulate_experiment()`

method. This method takes a matrix of model parameters and a vector of experiment
parameter records or scalars (depending on the model or simulator),
then returns an array of sample data, one sample for each combination of model
and experiment parameters.

```
>>> modelparams = np.linspace(0, 1, 100)
>>> expparams = np.arange(1, 10) * np.pi / 2
>>> D = m.simulate_experiment(modelparams, expparams, repeat=3)
>>> print(isinstance(D, np.ndarray))
True
>>> D.shape == (3, 100, 9)
True
```

If exactly one datum is requested, `simulate_experiment()`

will return a scalar:

```
>>> print(m.simulate_experiment(np.array([0.5]), np.array([3.5 * np.pi]), repeat=1).shape)
()
```

Note that in NumPy, a shape tuple of length zero indicates a scalar value, as such an array has no indices.

Note

For models with fancy outcome datatypes, it is demanded
that the outcome data types, `[d.dtype for d in m.domain(expparams)]`

,
be identical for every experimental parameter `expparams`

being
simulated. This can be checked with
`are_expparam_dtypes_consistent`

.

### Likelihooods¶

The core functionality of `Model`

, however, is the
`likelihood()`

method. This takes vectors of outcomes,
model parameters and experiment parameters, then returns for each combination
of the three the corresponding probability \(\Pr(d | \vec{x}; \vec{e})\).

```
>>> modelparams = np.linspace(0, 1, 100)
>>> expparams = np.arange(1, 10) * np.pi / 2
>>> outcomes = np.array([0], dtype=int)
>>> L = m.likelihood(outcomes, modelparams, expparams)
```

The return value of `likelihood()`

is a three-index
array of probabilities whose shape is given by the lengths of `outcomes`

,
`modelparams`

and `expparams`

.
In particular, `likelihood()`

returns a rank-three
tensor \(L_{ijk} := \Pr(d_i | \vec{x}_j; \vec{e}_k)\).

```
>>> print(isinstance(L, np.ndarray))
True
>>> L.shape == (1, 100, 9)
True
```

## Implementing Custom Simulators and Models¶

In order to implement a custom simulator or model, one must specify metadata describing the number of outcomes, model parameters, experimental parameters, etc. in addition to implementing the simulation and/or likelihood methods.

Here, we demonstrate how to do so by walking through a simple subclass of
`FiniteOutcomeModel`

. For more detail, please see the
API Reference.

Suppose we wish to implement the likelihood function

as may arise in looking, for instance, at an experiment inspired by 2D NMR.
This model has two model parameters, \(\omega_1\) and \(\omega_2\), and
so we start by creating a new class and declaring the number of model
parameters as a `property`

:

```
from qinfer import FiniteOutcomeModel
import numpy as np
class MultiCosModel(FiniteOutcomeModel):
@property
def n_modelparams(self):
return 2
```

Next, we proceed to add a property and method indicating that this model always
admits two outcomes, irrespective of what measurement is performed.
This will also automatically define the `domain`

method.

```
@property
def is_n_outcomes_constant(self):
return True
def n_outcomes(self, expparams):
return 2
```

We indicate the valid range for model parameters by returning an array of
dtype `bool`

for each of an input matrix of model parameters, specifying
whether each model vector is valid or not (this is important in resampling,
for instance, to make sure particles don’t move to bad locations). Typically,
this will look like some typical bounds checking, combined using
`logical_and`

and `all`

. Here, we follow that model by insisting
that *all* elements of each model parameter vector must be at least 0, *and*
must not exceed 1.

```
def are_models_valid(self, modelparams):
return np.all(np.logical_and(modelparams > 0, modelparams <= 1), axis=1)
```

Next, we specify what a measurement looks like by defining `expparams_dtype`

.
In this case, we want one field that is an array of two `float`

elements:

```
@property
def expparams_dtype(self):
return [('ts', 'float', 2)]
```

Finally, we write the likelihood itself. Since this is a two-outcome model,
we can calculate the rank-two tensor
\(p_{jk} = \Pr(0 | \vec{x}_j; \vec{e}_k)\) and let
`pr0_to_likelihood_array()`

add an index over
outcomes for us so \(L_{0jk}=p_{jk}\) and \(L_{1jk}=1-p_{jk}\).
To compute \(p_{jk}\) efficiently, it is helpful to do a bit of index
gymnastics using NumPy’s powerful broadcasting rules. In this example, we
set up the calculation to produce terms of the form
\(\cos^2(x_{j,l} e_{k,l} / 2)\) for \(l \in \{0, 1\}\) indicating
whether we’re referring to \(\omega_1\) or \(\omega_2\), respectively.
Multiplying along this axis then gives us the product of the two cosine
functions, and in a way that very nicely generalizes to likelihood functions of
the form

Running through the index gymnastics, we can implement the likelihood function as:

```
def likelihood(self, outcomes, modelparams, expparams):
# We first call the superclass method, which basically
# just makes sure that call count diagnostics are properly
# logged.
super(MultiCosModel, self).likelihood(outcomes, modelparams, expparams)
# Next, since we have a two-outcome model, everything is defined by
# Pr(0 | modelparams; expparams), so we find the probability of 0
# for each model and each experiment.
#
# We do so by taking a product along the modelparam index (len 2,
# indicating omega_1 or omega_2), then squaring the result.
pr0 = np.prod(
np.cos(
# shape (n_models, 1, 2)
modelparams[:, np.newaxis, :] *
# shape (n_experiments, 2)
expparams['ts']
), # <- broadcasts to shape (n_models, n_experiments, 2).
axis=2 # <- product over the final index (len 2)
) ** 2 # square each element
# Now we use pr0_to_likelihood_array to turn this two index array
# above into the form expected by SMCUpdater and other consumers
# of likelihood().
return FiniteOutcomeModel.pr0_to_likelihood_array(outcomes, pr0)
```

Our new custom model is now ready to use! To simulate data from this model, we
set up `modelparams`

and `expparams`

as before, taking care to conform to
the `expparams_dtype`

of our model:

```
>>> mcm = MultiCosModel()
>>> modelparams = np.dstack(np.mgrid[0:1:100j,0:1:100j]).reshape(-1, 2)
>>> expparams = np.empty((81,), dtype=mcm.expparams_dtype)
>>> expparams['ts'] = np.dstack(np.mgrid[1:10,1:10] * np.pi / 2).reshape(-1, 2)
>>> D = mcm.simulate_experiment(modelparams, expparams, repeat=2)
>>> print(isinstance(D, np.ndarray))
True
>>> D.shape == (2, 10000, 81)
True
```

Note

Creating `expparams`

as an empty array and filling it by field name is a
straightforward way to make sure it matches `expparams_dtype`

, but it
comes with the risk of forgetting to initialize a field, so take care when
using this method.

## Adding Functionality to Models with Other Models¶

QInfer also provides model classes which add functionality or otherwise modify
other models. For instance, the `BinomialModel`

class accepts instances
of two-outcome models and then represents the likelihood for many repeated
measurements of that model. This is especially useful in cases where
experimental concerns make switching experiments costly, such that repeated
measurements make sense.

To use `BinomialModel`

, simply provide an instance of another model
class:

```
>>> from qinfer import SimplePrecessionModel
>>> from qinfer import BinomialModel
>>> bin_model = BinomialModel(SimplePrecessionModel())
```

Experiments for `BinomialModel`

have an additional field from the
underlying models, called `n_meas`

. If the original model used scalar
experiment parameters (e.g.: `expparams_dtype`

is `float`

), then the original
scalar will be referred to by a field `x`

.

```
>>> eps = np.array([(12.1, 10)], dtype=bin_model.expparams_dtype)
>>> print(eps['x'], eps['n_meas'])
[ 12.1] [10]
```

Another model which *decorates* other models in this way is `PoisonedModel`

,
which is discussed in more detail in Performance and Robustness Testing. Roughly,
this model causes the likeihood functions calculated by its underlying model
to be subject to random noise, so that the robustness of an inference algorithm
against such noise can be tested.