1. Probabilistic models Jouni Tuomisto THL

2. Outline • Deterministic models with probabilistic parameters • Hierarchical Bayesian models • Bayesian belief nets

3. Deterministic models with probabilistic parameters • Inputs are uncertain, but causal relations are assumed certain. • Works well with established situations, especially if physical foundations. • Exposure = ∑ (ci ti) / ∑ ti • i = microenvironment • c = concentration • t = time

4. Functional vs. probabilistic dependency Va1=2.54*Ch1^2 Va2=normal(2.54*Ch1^2,2)

5. Hierarchical Bayesian models • Relations are probabilistic • Gibbs Sampler • Another MCMC (Markov chain Monte Carlo) Method • Update a single parameter at a time • Sample from conditional distribution • When other parameters are fixed

6. Gibbs sampling • To introduce the Gibbs sampler, consider a bivariate random variable (x; y), and suppose we wish to compute one or both marginals, p(x) and p(y). • The idea behind the sampler is that it is far easier to consider a sequence of conditional distributions, p(x |y) and p(y |x), than it is to obtain the marginal by integration of the joint density p(x; y), e.g., • p(x) = ∫ p(x; y)dy.

7. Gibbs sampling in practice • The sampler starts with some initial value y0 for y and obtains x0 by generating a random variable from the conditional distribution p(x |y = y0). • The sampler then uses x0 to generate a new value of y1, drawing from the conditional distribution based on the value x0, p(y j x = x0). The sampler proceeds as follows • xi ≈p(x |y = yi-1) (proportionality) • yi ≈p(y |x = xi) • Repeating this process k times, generates a Gibbs sequence of length k, where a subset of points (xj; yj) for 1 ≤j ≤m < k are taken as our simulated draws from the full joint distribution.

8. Hierarchical model with parameters and hyperparameters • A useful graphical tool for representing hierarchical Bayes models is the directed acyclic graph, or DAG. In this diagram, the likelihood function is represented as the root of the graph; each prior is represented as a separate node pointing to the node that depends on it.

9. Bayesian belief nets • Relations are described either with conditional probabilities P(x|y), P(y) or marginal probabilities P(x), P(y) and a rank correlation between them. • You need to get the conditional probabilities from somewhere. • Unlike hierarchical Bayes model, belief nets are not developed for updating when new data comes out. • The model is used to make inference.

10. Bayesian belief nets P(rain) P(sprinkler | rain) P(grass wet | sprinkler, rain)

11. Uninet: diagram view • V1: rain (mm/day) • V2: sprinkler on (h/day) • V3 ”wetness” of grass (range 0-1)

12. Uninet: variable definition view

13. Bayes belief network: unconditional situation

14. Conditioning on input variables

15. Conditioning on outcome