flowchart TD S__and__X_0 -- 1.0 --> S__and__X_0__and__Y1_false["Y1=false"] S__and__X_2__and__Y1_true -- 1.0 --> S__and__X_2__and__Y1_true__and__Y2_true["Y2=true"] S -- 0.33 --> S__and__X_0["X=0"] S__and__X_1__and__Y1_true__and__Y2_true__and__Y3_true -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_true__and__Y3_true__and__Y4_false["Y4=false"] S__and__X_1__and__Y1_true__and__Y2_false__and__Y3_true -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_false__and__Y3_true__and__Y4_false["Y4=false"] S__and__X_0__and__Y1_false__and__Y2_false__and__Y3_false -- 1.0 --> S__and__X_0__and__Y1_false__and__Y2_false__and__Y3_false__and__Y4_false["Y4=false"] S__and__X_1__and__Y1_false__and__Y2_false__and__Y3_false -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_false__and__Y3_false__and__Y4_true["Y4=true"] S -- 0.33 --> S__and__X_1["X=1"] S__and__X_2__and__Y1_true__and__Y2_true__and__Y3_true -- 1.0 --> S__and__X_2__and__Y1_true__and__Y2_true__and__Y3_true__and__Y4_true["Y4=true"] S__and__X_1__and__Y1_true__and__Y2_false__and__Y3_false -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_false__and__Y3_false__and__Y4_true["Y4=true"] S__and__X_1__and__Y1_false__and__Y2_true -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_true__and__Y3_false["Y3=false"] S__and__X_1__and__Y1_true__and__Y2_true -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_true__and__Y3_false["Y3=false"] S__and__X_1__and__Y1_false__and__Y2_false__and__Y3_false -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_false__and__Y3_false__and__Y4_false["Y4=false"] S__and__X_1__and__Y1_true__and__Y2_true__and__Y3_false -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_true__and__Y3_false__and__Y4_false["Y4=false"] S__and__X_1__and__Y1_true__and__Y2_false__and__Y3_true -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_false__and__Y3_true__and__Y4_true["Y4=true"] S__and__X_1 -- 0.5 --> S__and__X_1__and__Y1_false["Y1=false"] S__and__X_1__and__Y1_false -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_false["Y2=false"] S__and__X_1__and__Y1_false__and__Y2_true -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_true__and__Y3_true["Y3=true"] S__and__X_1__and__Y1_true -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_true["Y2=true"] S__and__X_0__and__Y1_false -- 1.0 --> S__and__X_0__and__Y1_false__and__Y2_false["Y2=false"] S__and__X_1__and__Y1_true -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_false["Y2=false"] S__and__X_1__and__Y1_false -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_true["Y2=true"] S__and__X_2 -- 1.0 --> S__and__X_2__and__Y1_true["Y1=true"] S__and__X_1__and__Y1_false__and__Y2_true__and__Y3_true -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_true__and__Y3_true__and__Y4_false["Y4=false"] S__and__X_1 -- 0.5 --> S__and__X_1__and__Y1_true["Y1=true"] S -- 0.33 --> S__and__X_2["X=2"] S__and__X_1__and__Y1_true__and__Y2_false -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_false__and__Y3_false["Y3=false"] S__and__X_2__and__Y1_true__and__Y2_true -- 1.0 --> S__and__X_2__and__Y1_true__and__Y2_true__and__Y3_true["Y3=true"] S__and__X_1__and__Y1_false__and__Y2_false -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_false__and__Y3_true["Y3=true"] S__and__X_1__and__Y1_false__and__Y2_true__and__Y3_false -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_true__and__Y3_false__and__Y4_false["Y4=false"] S__and__X_1__and__Y1_true__and__Y2_true -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_true__and__Y3_true["Y3=true"] S__and__X_1__and__Y1_true__and__Y2_true__and__Y3_true -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_true__and__Y3_true__and__Y4_true["Y4=true"] S__and__X_1__and__Y1_false__and__Y2_true__and__Y3_false -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_true__and__Y3_false__and__Y4_true["Y4=true"] S__and__X_1__and__Y1_true__and__Y2_true__and__Y3_false -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_true__and__Y3_false__and__Y4_true["Y4=true"] S__and__X_1__and__Y1_false__and__Y2_false__and__Y3_true -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_false__and__Y3_true__and__Y4_false["Y4=false"] S__and__X_1__and__Y1_true__and__Y2_false -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_false__and__Y3_true["Y3=true"] S__and__X_1__and__Y1_true__and__Y2_false__and__Y3_false -- 0.5 --> S__and__X_1__and__Y1_true__and__Y2_false__and__Y3_false__and__Y4_false["Y4=false"] S__and__X_1__and__Y1_false__and__Y2_false__and__Y3_true -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_false__and__Y3_true__and__Y4_true["Y4=true"] S__and__X_1__and__Y1_false__and__Y2_true__and__Y3_true -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_true__and__Y3_true__and__Y4_true["Y4=true"] S__and__X_1__and__Y1_false__and__Y2_false -- 0.5 --> S__and__X_1__and__Y1_false__and__Y2_false__and__Y3_false["Y3=false"] S__and__X_0__and__Y1_false__and__Y2_false -- 1.0 --> S__and__X_0__and__Y1_false__and__Y2_false__and__Y3_false["Y3=false"]
Forward sampling
Outline
Topics
- Notion of forward sampling (also known as forward simulation)
- How to do it in practice
- Useful functions
- Graphical models
Rationale
- Sampling is the main way Bayesian inference is performed nowadays.
- We introduce here the simplest flavour of sampling, forward sampling.
- Bayesian inference mostly uses a more complicated type of sampling called posterior sampling which we will cover later.
- But forward sampling is still helpful to help debug Bayesian inference software as we will see soon.
Forward sampling as depth-first traversal
Recall our recurring bag sampling example, with its corresponding decision tree:
- Forward simulation is a type of tree traversal. I.e. moving from node to node in the tree.
- Forward simulation is a recursive process initialized at the root of the decision tree (labelled \(S\)).
- When we are a node \(v\) in the tree, we pick one of \(v\)’s children at random.
- More precisely, we use methods discussed in the previous section on simulation from PMFs
- We recurse until we reach a leaf.
- From this leaf we obtain an outcome and hence a realization for all random variables, both “observed” and “unobserved.”
- When we are a node \(v\) in the tree, we pick one of \(v\)’s children at random.
Forward sampling as specifying a model
We have encountered that notation earlier:
\[ \begin{align*} X &\sim {\mathrm{Unif}}\{0, 1, 2\} \\ Y_i | X &\sim {\mathrm{Bern}}(X/2). \end{align*} \]
- This notation is a recipe providing all the information required to perform forward sampling.
- Specifically, the PMF to use at each recursion step.
- In continuous models, it will be the same idea except that we will have a probability density instead of a PMF.
Example 1
You will practice forward sampling in Ex1, Q1.1.2
Example 2
Consider the following model:
\[ \begin{align*} X &\sim {\mathrm{Unif}}\{1, 2, 3, 4\} \\ Y | X &\sim {\mathrm{Unif}}\{1, \dots, X\}. \end{align*} \]
Interpretation: you roll a D&D d4 dice (blue one on the image), then pick an integer uniformly between 1 and the number on the dice.
Example of forward simulation code for the above “censored dice”:
require(extraDistr)Loading required package: extraDistrset.seed(4)
forward_simulate_roll_and_pick <- function() {
x <- rdunif(1, min=1, max=4) # 1 uniform between 1-6, i.e. a dice
y <- rdunif(1, min=1, max=x)
c(x, y) # return a vector with these two realizations
}
forward_simulate_roll_and_pick()[1] 3 1forward_simulate_roll_and_pick()[1] 2 1forward_simulate_roll_and_pick()[1] 4 2