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Updated Documentation
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Al-Ahmadgaid Asaad committed Apr 13, 2017
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2 changes: 1 addition & 1 deletion doc/index.rst
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.. toctree::
:maxdepth: 2

tutorial.rst
mh.rst
hmc.rst
sghmc.rst


Indices
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199 changes: 199 additions & 0 deletions doc/sghmc.rst
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Stochastic Gradient Hamiltonian Monte Carlo
===================

Implementation of the Hamiltonian Monte Carlo sampler for Bayesian inference.

.. function:: HMC(dU::Function, dK::Function, dKΣ::Array{Float64}, C::Array{Float64}, V::Array{Float64}, init_est::Array{Float64}, d::Int64)

Construct a ``Sampler`` object for Hamiltonian Monte Carlo sampling.

**Arguments**

* ``dU`` : the gradient or first derivative of the potential energy ``U``.
* ``dK`` : the gradient or first derivative of the kinetic energy ``K``.
* ``dKΣ`` : the variance-covariance matrix in the gradient of the kinetic energy ``dK``,\
this is set to identity matrix for the case of standard Gaussian distribution.
* ``C`` : the matrix factor in the frictional force term.
* ``V`` : the matrix factor in the random force term.
* ``init_est`` : the initial/starting value for the markov chain.
* ``d`` : the dimension of the posterior distribution.

**Value**

Returns a ``HMC`` type object.

**Example**

In order to illustrate the modeling, the data is simulated from a simple linear regression expectation function. That is the model is given by

.. code-block:: txt
y_i = w_0 + w_1 x_i + e_i, e_i ~ N(0, 1 / a)
To do so, let :code:`B = [w_0, w_1]' = [.2, -.9]', a = 1 / 5`. Generate 200 hypothetical data:

.. code-block:: julia
using DataFrames
using Distributions
using Gadfly
using StochMCMC
Gadfly.push_theme(:dark)
srand(123);
# Define data parameters
w0 = -.3; w1 = -.5; stdev = 5.; a = 1 / stdev
# Generate Hypothetical Data
n = 200;
x = rand(Uniform(-1, 1), n);
A = [ones(length(x)) x];
B = [w0; w1];
f = A * B;
y = f + rand(Normal(0, a), n);
my_df = DataFrame(Independent = round(x, 4), Dependent = round(y, 4));
Next is to plot this data which can be done as follows:

.. code-block:: julia
plot(my_df, x = :Independent, y = :Dependent)
.. image:: figures/plot1.png
:width: 80%
:align: center
:alt: alternate text

|
In order to proceed with the Bayesian inference, the parameters of the model is considered to be random modeled by a standard Gaussian distribution. That is, :code:`B ~ N(0, I)`, where :code:`0` is the zero vector. The likelihood of the data is given by,
.. code-block:: txt
L(w|[x, y], b) = ∏_{i=1}^n N([x_i, y_i]|w, b)
Thus the posterior is given by,

.. code-block:: txt
P(w|[x, y]) ∝ P(w)L(w|[x, y], b)
To start programming, define the probabilities

.. code-block:: julia
"""
The log prior function is given by the following codes:
"""
function logprior(theta::Array{Float64}; mu::Array{Float64} = zero_vec, s::Array{Float64} = eye_mat)
w0_prior = log(pdf(Normal(mu[1, 1], s[1, 1]), theta[1]))
w1_prior = log(pdf(Normal(mu[2, 1], s[2, 2]), theta[2]))
w_prior = [w0_prior w1_prior]
return w_prior |> sum
end
"""
The log likelihood function is given by the following codes:
"""
function loglike(theta::Array{Float64}; alpha::Float64 = a, x::Array{Float64} = x, y::Array{Float64} = y)
yhat = theta[1] + theta[2] * x
likhood = Float64[]
for i in 1:length(yhat)
push!(likhood, log(pdf(Normal(yhat[i], alpha), y[i])))
end
return likhood |> sum
end
"""
The log posterior function is given by the following codes:
"""
function logpost(theta::Array{Float64})
loglike(theta, alpha = a, x = x, y = y) + logprior(theta, mu = zero_vec, s = eye_mat)
end
To start the estimation, define the necessary parameters

.. code-block:: julia
# Hyperparameters
zero_vec = zeros(2)
eye_mat = eye(2)
Setup the necessary paramters including the gradients.

.. code-block:: julia
function dU(theta::Array{Float64}; alpha::Float64 = a, b::Float64 = eye_mat[1, 1])
[-alpha * sum(y - (theta[1] + theta[2] * x));
-alpha * sum((y - (theta[1] + theta[2] * x)) .* x)] + b * theta
end
dK(p::AbstractArray{Float64}; Σ::Array{Float64} = eye(length(p))) = inv(Σ) * p;
Define the gradient noise and other parameters of the SGHMC:

.. code-block:: julia
function dU_noise(theta::Array{Float64}; alpha::Float64 = a, b::Float64 = eye_mat[1, 1])
[-alpha * sum(y - (theta[1] + theta[2] * x));
-alpha * sum((y - (theta[1] + theta[2] * x)) .* x)] + b * theta + randn(2,1)
end
Run the MCMC:

.. code-block:: julia
srand(123);
SGHMC_object = SGHMC(dU_noise, dK, eye(2), eye(2), eye(2), [0; 0], 2.);
chain3 = mcmc(SGHMC_object, leapfrog_params = Dict([:ɛ => .09, :τ => 20]), r = 10000);
Extract the estimate:

.. code-block:: julia
est3 = mapslices(mean, chain3[(burn_in + 1):thinning:end, :], [1]);
est3
# 1×2 Array{Float64,2}:
# -0.302745 -0.430272
Plot it

.. code-block:: julia
my_df_sghmc = my_df;
my_df_sghmc[:Yhat] = mapslices(mean, chain3[(burn_in + 1):thinning:end, :], [1])[1] + mapslices(mean, chain3[(burn_in + 1):thinning:end, :], [1])[2] * my_df[:Independent];
for i in (burn_in + 1):thinning:10000
my_df_sghmc[Symbol("Yhat_Sample_" * string(i))] = chain3[i, 1] + chain3[i, 2] * my_df_sghmc[:Independent]
end
my_stack_sghmc = DataFrame(X = repeat(Array(my_df_sghmc[:Independent]), outer = length((burn_in + 1):thinning:10000)),
Y = repeat(Array(my_df_sghmc[:Dependent]), outer = length((burn_in + 1):thinning:10000)),
Var = Array(stack(my_df_sghmc[:, 4:end])[1]),
Val = Array(stack(my_df_sghmc[:, 4:end])[2]));
ch1cor_df = DataFrame(x = collect(0:1:(length(autocor(chain3[(burn_in + 1):thinning:10000, 1])) - 1)),
y1 = autocor(chain3[(burn_in + 1):thinning:10000, 1]),
y2 = autocor(chain3[(burn_in + 1):thinning:10000, 2]));
p0 = plot(my_df, x = :Independent, y = :Dependent, Geom.point, style(default_point_size = .05cm), Guide.xlabel("Explanatory"), Guide.ylabel("Response"));
p1 = plot(DataFrame(chain3), x = :x1, xintercept = [-.3], Geom.vline(color = colorant"white"), Geom.histogram(bincount = 30, density = true), Guide.xlabel("1st Parameter"), Guide.ylabel("Density"));
p2 = plot(DataFrame(chain3), x = :x2, xintercept = [-.5], Geom.vline(color = colorant"white"), Geom.histogram(bincount = 30, density = true), Guide.xlabel("2nd Parameter"), Guide.ylabel("Density"));
p3 = plot(DataFrame(chain3), x = collect(1:nrow(DataFrame(chain3))), y = :x1, yintercept = [-.3], Geom.hline(color = colorant"white"), Geom.line, Guide.xlabel("Iterations"), Guide.ylabel("1st Parameter Chain Values"));
p4 = plot(DataFrame(chain3), x = collect(1:nrow(DataFrame(chain1))), y = :x2, yintercept = [-.5], Geom.hline(color = colorant"white"), Geom.line, Guide.xlabel("Iterations"), Guide.ylabel("2nd Parameter Chain Values"));
p5 = plot(DataFrame(chain3), x = :x1, y = :x2, Geom.path, Geom.point, Guide.xlabel("1st Parameter Chain Values"), Guide.ylabel("2nd Parameter Chain Values"));
p6 = plot(layer(my_df_sghmc, x = :Independent, y = :Yhat, Geom.line, style(default_color = colorant"white")),
layer(my_stack_sghmc, x = :X, y = :Val, group = :Var, Geom.line, style(default_color = colorant"orange")),
layer(my_df_sghmc, x = :Independent, y = :Dependent, Geom.point, style(default_point_size = .05cm)),
Guide.xlabel("Explanatory"), Guide.ylabel("Response and Predicted"));
p7 = plot(ch1cor_df, x = :x, y = :y1, Geom.bar, Guide.xlabel("Lags"), Guide.ylabel("1st Parameter Autocorrelations"), Coord.cartesian(xmin = -1, xmax = 36, ymin = -.05, ymax = 1.05));
p8 = plot(ch1cor_df, x = :x, y = :y2, Geom.bar, Guide.xlabel("Lags"), Guide.ylabel("2nd Parameter Autocorrelations"), Coord.cartesian(xmin = -1, xmax = 36, ymin = -.05, ymax = 1.05));
vstack(hstack(p0, p1, p2), hstack(p3, p4, p5), hstack(p6, p7, p8))
.. image:: figures/plot2.png
:width: 100%
:align: center
:alt: alternate text

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