Bayesian statistical inference

and

Uncertainty quantification (UQ)

2018 q-Bio summer school Southwest Campus

Yen Ting Lin

6/20/2018

Statistical inference

Definition

: A process to deduce the underlying (model, parameters, etc.) probability distributions using

data

.

Uncertainty quantification

Definition

: (less well-defined) The process of quantifying the uncertainty

1. Frequentists' approach: bootstraping, hypothesis testing (

p

-value)...

2.

Bayesian approach

1. Forward propagation UQ: you have done it last week!

2. Backward propagating UQ: you will learn it today!

Why do we want to go beyond maximum likelihood (best fit) picture and perform statistical inference & UQ?

Data and measurement are not perfect; there are always some noises.

The noise can come from intrinsic behaviour of the system, or measurements (sampling). Is it purely out of luck that we obtain a good experimental (data) support, or is it something "significant"?

A toy example to illustrate the methods

A linear equation with independent Gaussian noise:

Task: Given a set of synthetic data,

find out

model parameters.

MLE

: best-fit parameters which minimize the chi-square.

Statistical inference

: find the "distribution" of the model parameters

Bayesian inference framework

A natural generalization of stochastic models

Deterministic models

deterministic parameters, deterministic observables

Frequentists' stochastic model

deterministic parameters, stochastic observables

Bayesian stochastic model

stochastic parameters, stochastic observables

Concept

:

Joint

probability distribution of the

parameters

and the

observed data

, linked by the model prediction using the Bayes formula.

"Forward"

"Backward"

Bayesian inference framework

A natural generalization of stochastic models

Bayesian stochastic model

stochastic parameters

θ

, stochastic observables

D

Understand Bayesian inference through the toy model

Assume we know everything but the slope

a1

, which will be inferred

We need a "prior" , which is often described as the "prior belief" of the parameter

a1.

We assume it's uniform in between (0, 100).

A naïve way to compute the "posterior" distribution by evaluating its value at every possible (or interested)

a1

value.

Understand Bayesian inference through the toy model

Note that the posterior is updated sequentially (online updating)

for independent experiments.

Effect of the prior I

Effect of the prior II

Usage of the posterior II: test for significance

Uncertainty of a1

Usage of the posterior I: "correlation" between parameters

Let's consider a 2-parameter model: inferring

a0

and

a1:

Usage of the posterior II: test for significance

The

joint

posterior distribution is more informative than the

marginalized

Usage of the posterior II: test for signficance

Usage of the posterior III: evidence, marginalized likelihood

quantification of how much a model explains the data

Probability of a model with parameters

θ

to explain data

D

Prior probability of the parameters

θ

Sum over all possible parameters

θ

Total (all possible

θ)

probability that the model can explain the data

Getting technical: high-dimensional inference

Up to now, we adopted the naïve way to compute the "posterior" distribution by evaluating its value at every possible (or interested)

θ

value. The

grid-based

scanning procedure becomes impossible when the dimensionality of the parameter space is larger than 3...

Solution

: use the sampling techniques in statistical physics:

Monte Carlo methods

.

An intuitive method termed as

Markov chain Monte Carlo

constructs a random walk in the parameter space, such that the

stationary distribution

of the random walk coincides with the

posterior distribution.

Using a 2-state, discrete model as an example:

The art of the game: formulating the likelihood

Extension:

sequential Monte Carlo (sMC)

for stochastic dynamical systems

and/or hidden Markov process

**Questions**

?

?

(illustrated by toy models)