MC Julia — affine-invariant MCMC sampling in Julia
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MC Julia

MC Julia is an affine-invariant ensemble MCMC sampler, written in the Julia programming language. It implements a highly efficient MCMC algorithm by Goodman & Weare (2010), and is heavily inspired by the emcee Python package.

The algorithm is very efficient at sampling probability distributions which are highly skewed but more or less linearly correlated.


The algorithm is implemented through the Sampler type. It keeps track of the sampled function and some optional parameters, and stores the chain of samples and other sampling output.

See the examples directory for some example programs.


Sampler(k, dim, f)

This is the minimal constructor for a new Sampler object. The parameter k gives the number of walkers in the ensemble, dim gives the dimension of the sampling space and f is the log-probability function. The number of walkers must be even and greater than twice the dimension.

The log-probability function must be defined to take the sampled position as an Array{Float64}, even when the sampling space is one-dimensional.

log_probability(X::Array{Float64}) = -(X[1] - X[2])^2 / 0.5 - (X[1] + X[2])^2 / 1.5

S = Sampler(50, 2, log_probability)

Sampler(k, dim, f, a, args)

Sampler(k, dim, f, args)

Sampler(k, dim, f, a)

It is also possible to give some additional parameters to the sampler. The scale parameter a::Float64 defines the stretch move scale (see Goodman & Weare, 2010). Its default value is 2.0, which is good for most cases, but tuning the parameter may give a performance increase in some situations. The parameter args is an arbitrary tuple which is passed to the log-probability function along with the position vector. The args tuple is expanded in the function call, so a tuple (a,b,c) will lead to a function call f(X, a, b, c).

The following is a simple example of constant parameters with a one-dimensional Gaussian:

log_probability(X::Array{Float64}, mu::Float64, sigma::Float64) = -(X[1] - mu)^2 / (2 * sigma^2)

args = (1.0, 0.5)
S = Sampler(50, 2, log_probability, args)


p = sample(S, p0, N, thin, storechain)

p = sample(S, p0, N, storechain)

p = sample(S, p0, N)

This function does the actual sampling. It takes as parameters a Sampler object S, an initial position p0, the number of samples N, the thinning factor thin and a boolean storechain.

The initial position p0 must be an Array{Float64, 2} with the shape (k, dim), and give the initial position in the dim-dimensional sampling space for each of the k walkers in the ensemble.

The Int64 parameter thin gives thinning factor, the interval by which the positions of the walkers are saved to S.chain. That is, with thin=10, the chain is updated once every ten samping steps.

The boolean parameter storechain tells whether to save the steps of the walkers to the chain. Setting it to false will produce a burn-in, where the walkers move around, settling towards their steady state around the high-probability region of the sampled density, without any of the steps being saved.

The return value p is an Array{Float64, 2} giving the positions of the walkers at the last step of the sampling. It can directly be used as the initial position of another call to sample. It is useful to first do a burn-in with storechains set to false, then perform the actual sampling starting from the positions returned by the first call.


Reset a Sampler object to the state it was in right after construction. This will set S.iterations and S.accepted to zero and also empty the S.chain and S.ln_posterior.


Return the S.chain array flatted along the walker dimension.

save_chain(S, filename)

Save the flattened chain into a plaintext file in CSV format.


These are the fields defined for the Sampler type. The values of some of them will be of interest to the user. The user should not need to change the values of any fields manually.


The number of walkers in the ensemble. Set by the user through the constructor.


The stretch move scale parameter. Set by the user through the constructor.


The dimension of the sampling space. Set by the user through the constructor.


The log-probability function to be sampled. Set by the user through the constructor.


The constant arguments for the log-probability function. Set by the user through the constructor.


An Array{Float64, 3} with shape (k, dim, N), where k is the number of walkers in the ensemble, dim is the dimension of the sampling space and N is the number of samples taken (see the sample function, below). This is what you are mostly interested in acquiring.


An Array{Flaot64, 2} with shape (k, N), giving the log-probability value of each walker at each point in the chain.


An integer giving the number of moves proposed since the last reset (or creation) of the Sampler object. Includes also sampling steps done with storechain == false. One sampling step with k walkers will add k to this variable.


An integer giving the number of accepted moves since the last reset (or creation) of the Sampler object. Includes also sampling steps done with storechain == false. Every accepted move of a walker will add one to this counter. With Sampler.iterations can be used to estimate the acceptance fraction of the full ensemble.

Future work


This is definitely planned as the next major improvement.


The Python package emcee allows the log-probability function to return arbitrary metadata "Blobs" which are also saved for each step in the chain. This may be a useful feature and will be implemented after parallelization.