### Ben Mather PRO

Computational Geophysicist at the Sydney Informatics Hub, University of Sydney

Dr. Ben Mather

EarthByte Group

University of Sydney

Essentially, all models are wrong,

but some are useful.

- Box & Draper

Empirical Model-Building and Response Surfaces (1987)

@BenRMather

A Newspoll conducted shortly before the federal election predicted a Labor victory **53%** to the Coalition's **47%** on a two-party preferred preference

@BenRMather

When it comes to the opinion polling, something’s obviously gone really crook with the sampling

both internally and externally.

- ABC political editor Andrew Probyn

- Linear?
- Quadratic?
- Sinusoidal?

@BenRMather

- Everyday life
- Whenever we interpret data
- When we predict something based on data

@BenRMather

- Nature of trends
- Constant, linear, quadratic, etc.
- Correlation length scales

- Distinct populations in data
- Socio-economic classes, smokers
- geochemists, palaeontologists, flat earthers

- Presence of bias in a sample group
- People who respond to Newspoll surveys

- When we predict something based on prior experience

- Make an objective observation.
- Infer something (a hypothesis) from that observation.

@BenRMather

- Formulate a hypothesis
- Find / assume all data that fits their hypothesis

@BenRMather

- Newton's 3 laws of motion
- Greenhouse effect
- The first dice-roll has no effect on the second dice-roll
- The temperature in Newtown is the same as that in Marrickville
- John Farnham will perform at least one more goodbye tour

@BenRMather

There are a lot of words here and most of them mean the same things.

**Machine Learning = Inference**

- Does it pass the common sense test?
- "Bad" models can also tell you something interesting.
- Are there alternatives?
- What are you going to do with your model?

@BenRMather

Generate 50%, 95%, 99% confidence intervals using randomly drawn models

There may be many solutions that fit the same set of observations.

- Formally describes the link between
*observations, model, & prior information.* - Where these intersect is called the
*posterior*

P(\mathbf{m}|\mathbf{d}) \propto P(\mathbf{d}|\mathbf{m}) \cdot P(\mathbf{m})

*posterior*

*likelihood*

*prior*

model

data

example of an ill-posed problem

@BenRMather

example of a well-posed problem

- Use the data to "drive" the model.
- Infer what input parameters you need to satisfy your data and prior information

*Input parameters*

*Model being solved*

*Compare data & priors*

\mathbf{m} : [H_1,H_2,H_3,\ldots, H_n]

\nabla ( k \nabla T) =-H

P(\mathbf{m}|\mathbf{d}) \propto P(\mathbf{d}|\mathbf{m}) \cdot P(\mathbf{m})

**FORWARD MODEL**

Prior

P(\mathbf{m})

Likelihood

P(\mathbf{d}|\mathbf{m})

Posterior

P(\mathbf{m}|\mathbf{d})

Inverse Model

We can estimate the value of pi with monte carlo sampling.

```
from random import random
n = int(input("Enter number of darts"))
c = 0
for i in range(n):
x = 2*random()-1
y = 2*random()-1
if x*x + y*y <= 1:
c += 1
print("Pi is {}".format(4.0*c/n))
```

\pi

Python code to run simulation

Global

minimum

Local

maximum

Local

minimum

Monte Carlo sampling

Global

minimum

Local

maximum

Local

minimum

Markov-Chain Monte Carlo sampling (MCMC)

Global

minimum

Local

maximum

Local

minimum

MCMC with gradient

Global

minimum

Local

maximum

Local

minimum

MCMC with gradient (caveat emptor!)

*trapped!*

- Assimilate heat flow data

- Vary rates of heat production and geometry of each layer to match data

- Plug
**m**and**d**into Bayes' theorem

P(\mathbf{m}|\mathbf{d}) \propto P(\mathbf{d}|\mathbf{m}) \cdot P(\mathbf{m})

\mathbf{d} = q_s

\mathbf{m} = [H_1, H_2, H_3, z_1, z_2, z_3]

- Ascertain the difference between reconstructions
- Does not take into account data uncertainty
- Sensitivity analysis / "bootstrapping"

There are

knownknowns;there are things we know we know.

We also know there are

knownunknowns; that is to say we know there are some things we do not know.But there are also

unknownunknowns- the ones we don't know we don't know.

- Donald Rumsfeld

Former US Secretary of Defense

@BenRMather

- Europeans thought all swans were white... until they came to Australia
- How can you ever model what you can't imagine?
- How can you test assumptions without rare events that prove them wrong?

@BenRMather

**Dr. Ben Mather**

Madsen Building, School of Geosciences,

The University of Sydney, NSW 2006

By Ben Mather

- 327

Computational Geophysicist at the Sydney Informatics Hub, University of Sydney

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