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A user-centered Python package for differentiable probabilistic inference

# AI-DI/Brancher

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# Brancher: A user-centered Python package for differentiable probabilistic inference

Brancher allows to design and train differentiable Bayesian models using stochastic variational inference. Brancher is based on the deep learning framework PyTorch.

## Building probabilistic models

Probabilistic models are defined symbolically. Random variables can be created as follows:

```a = NormalVariable(loc = 0., scale = 1., name = 'a')
b = NormalVariable(loc = 0., scale = 1., name = 'b')```

It is possible to chain together random variables by using arithmetic and mathematical functions:

```c = NormalVariable(loc = a**2 + BF.sin(b),
scale = BF.exp(b),
name = 'a')```

In this way, it is possible to create arbitrarely complex probabilistic models. It is also possible to use all the deep learning tools of PyTorch in order to define probabilistic models with deep neural networks.

## Example: Autoregressive modeling

### Probabilistic model

Probabilistic models are defined symbolically:

```T = 20
driving_noise = 1.
measure_noise = 0.3
x0 = NormalVariable(0., driving_noise, 'x0')
y0 = NormalVariable(x0, measure_noise, 'x0')
b = LogitNormalVariable(0.5, 1., 'b')

x = [x0]
y = [y0]
x_names = ["x0"]
y_names = ["y0"]
for t in range(1,T):
x_names.append("x{}".format(t))
y_names.append("y{}".format(t))
x.append(NormalVariable(b*x[t-1], driving_noise, x_names[t]))
y.append(NormalVariable(x[t], measure_noise, y_names[t]))
AR_model = ProbabilisticModel(x + y)```

### Observe data

Once the probabilistic model is define, we can decide which variable is observed:

`[yt.observe(data[yt][:, 0, :]) for yt in y]`

### Autoregressive variational distribution

The variational distribution can have an arbitrary structure:

```Qb = LogitNormalVariable(0.5, 0.5, "b", learnable=True)
logit_b_post = DeterministicVariable(0., 'logit_b_post', learnable=True)
Qx = [NormalVariable(0., 1., 'x0', learnable=True)]
Qx_mean = [DeterministicVariable(0., 'x0_mean', learnable=True)]
for t in range(1, T):
Qx_mean.append(DeterministicVariable(0., x_names[t] + "_mean", learnable=True))
Qx.append(NormalVariable(BF.sigmoid(logit_b_post)*Qx[t-1] + Qx_mean[t], 1., x_names[t], learnable=True))
variational_posterior = ProbabilisticModel([Qb] + Qx)
model.set_posterior_model(variational_posterior)```

### Inference

Now that the models are specified we can perform approximate inference using stochastic gradient descent:

```inference.perform_inference(AR_model,
number_iterations=500,
number_samples=300,
optimizer="SGD",
lr=0.001)```

A user-centered Python package for differentiable probabilistic inference

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