# What inputs do Monte Carlo algorithms need?

Monte Carlo sampling algorithms (either MCMC or not) have a goal to attain samples from a distribution.  They can be organized by what inputs or prior knowledge about the distribution they require.  This ranges from a low amount of knowledge, as in slice sampling (just give it an unnormalized density function), to a high amount, as in Gibbs sampling (you have to decompose your distribution into individual conditionals).

Typical inputs include $$f(x)$$, an unnormalized density or probability function for the target distribution, which returns a real number for a variable value.  $$g()$$ and $$g(x)$$ represent sample generation procedures (that output a variable value); some generators require an input, some do not.

Here are the required inputs for a few algorithms.  (For an overview, see e.g. Ch 29 of MacKay.)  There are many more out there of course.  I’m leaving off tuning parameters.

Black-box samplers: Slice samplingAffine-invariant ensemble
- unnorm density $$f(x)$$

Metropolis (symmetric proposal)
- unnorm density/pmf $$f(x)$$
- proposal generator $$g(x)$$

Hastings (asymmetric proposal)
- unnorm density/pmf $$f(x)$$
- proposal generator $$g(x)$$
- proposal unnorm density/pmf $$q(x’; x)$$  .
… [For the proposal generator at state $$x$$, probability it generates $$x'$$]

Importance sampling, rejection sampling
- unnorm density/pmf $$f(x)$$
- proposal generator $$g()$$
- proposal unnorm density/pmf $$q(x)$$

Independent Metropolis-Hastings: the proposal is always the same, but still have to worry about asymmetric corrections
- unnorm density/pmf $$f(x)$$
- proposal generator $$g()$$
- proposal unnorm density/pmf $$q(x’; x)$$

Hamiltonian Monte Carlo
- unnorm density $$f(x)$$
- unnorm density gradient $$gf(x)$$

Gibbs Sampling
- local conditional generators $$g_i(x_{-i})$$
… [which have to give samples from $$p(x_i | x_{-i})$$]

Note importance/rejection sampling are stateless, but the MCMC algorithms are stateful.

I’m distinguishing a sampling procedure $$g$$ from a density evaluation function $$f$$ because having the latter doesn’t necessarily give you the former.  (For the one-dimension case, having an inverse CDF indeed gives you a a sampler, but multidimensional gets harder — part of why all these techniques were invented in the first place!)  Shay points out their relationship is analogous to 3-SAT: it’s easy to evaluate a full variable setting, but hard to generate them.  (Or specifically, think about a 3-SAT PMF $$p(x) = 1\{\text{\(x$$ is boolean satisfiable}\}\) where only one $$x$$ has non-zero probability; PMF evaluation is easy but the best known sampler is exponential time.)

And of course there’s a related organization of optimization algorithms.  Here’s a rough look at a few unconstrained optimizers:

Black-box optimizers: Grid search, Nelder-Mead, evolutionary, …
- objective $$f(x)$$

BFGS, CG
- objective $$f(x)$$
- gradient $$gf(x)$$

Newton-Raphson
- objective $$f(x)$$
- gradient $$gf(x)$$
- hessian $$hf(x)$$

Simulated annealing
- objective $$f(x)$$
- proposal generator $$g(x)$$

SGD
- one-example gradient $$gf_i(x)$$

I think it’s neat that Gibbs Sampling and SGD don’t always require you to implement a likelihood/objective function.  It’s nice to do to ensure you’re actually optimizing or exploring the posterior, but strictly speaking the algorithms don’t require it.

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### 2 Responses to What inputs do Monte Carlo algorithms need?

1. Very nice summary.

One question: does SGD actually work in cases in which the objective function doesn’t decompose in an obvious way into the sum of one-example objective functions? My vague recollection of the proofs of convergence is that they make heavy use of the decomposition of the objective functions.

2. brendano says:

Yeah SGD requires decomposition into minibatch gradients you can sum together for the big gradient (and thus the gradient from a minibatch is in expectation same as the full gradient). I’m not aware of SGD for non-sum situations…