In my previous blog post I outlined the basic AR(1) model and the necessary maths needed to infer the unknown parameter \(\phi\). In this post I will outline some basic Julia code to build a MCMC sampler for such a model to infer the unknown parameter \(\phi\).

Firstly, we need to simulate some data. From the previous post we know that the data \(y\) comes simply from the previous value, plus some fixed noise. In Julia this is simply writing a for loop and using the Distributions package to sample some white noise.

function simulate_ar(phi, n)

	 dist = Normal()

	 y = [0.0 for i = 1:n]

	 noise = rand(dist, n)

	 for i in 1:(n-1)

	     y[i+1] = phi*y[i] + noise[i] 

	 return y

For 1000 data points with \(\phi=0.5\) such a process looks like:

AR1 Process Plot

Pretty much looks like a random walk around 0 as expected.

Now to compute the statistics for the posterior distribution we need the sum of squares and the lagged sum of squares ( [see here] ( Then using the Distributions package again we can sample from a truncated normal distribution. We have used a prior distribution of a truncated normal distribution with 0 mean and a standard deviation of 5.

function posterior_ar(n, y)
	 n = length(y)
	 ss = sum(y .* y) + 1/25 
	 ss_lagged = sum(y .* vcat(y[2:n],0))
	 dist = Truncated(Normal(ss_lagged/ss, sqrt(1/ss)), -1, 1)
	 smps = rand(dist, n)

	 return smps

Phi Density Plot

So we can see that the posterior distribution for \(\phi\) is close to the true value of 0.5, so it looks like our algorithm is working.

Although its just a toy model in these posts I have shown how to calculate the posterior for an autoregressive process and how to draw from such a distribution using Julia. Next stop, include more parameters and see how flexible autoregressive models can be.