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Introduction to Likelihood

Adapted from slides by:

Cari van Schalkwyk, PhD

South African Centre for Epidemiological Modelling and Analysis

Stellenbosch University, Stellenbosch, South Africa

MMED 2024

Delivered by:

Bobby Reiner, PhD

University of Washington

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Goals

Define probability distribution function
Define Likelihood
Define MLE
Discuss some uses

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Public Health, Epidemiology, and Models (Day 1)

Simple Models (Day 1)

Foundations of Dynamic Modeling (Day 1)

(Hidden) Assumptions of Simple ODE’s (Day 2)

Breaking Assumptions!

Consequences of Heterogeneity (Day 6)

Introduction stochastic simulation models (Day 3)

Heterogeneity tutorial (Day 6)

Introduction to Infectious Disease Data (Day 1)

Thinking about Data (Day 2)

Data management and cleaning (Day 9)

Creating a Model World (Day 4)

Study design and analysis in epidemiology (Day 3)

Introduction to Statistical Philosophy (Day 4)

Variability, Sampling Distributions, & Simulation (Day 10)

HIV in Harare tutorial (Day 3)

Integration!

Introduction to Likelihood (Day 4)

Fitting Dynamic Models I – III (Day 5, 8, & 9)

Modeling for Policy (Day 11)

Model Assessment (Day 10)

MCMC Lab (Day 9)

MLE Fitting SIR model to prevalence data (Day 5)

Likelihood Lab (Day 4)

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Binomial Distribution

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Number of Heads	Probability
0	6.25%
1	25%
2	37.5%
3	25.5%
4	6.25%

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Binomial Distribution

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If we tossed our coin 4 times and observed 3 heads, would we think our coin is fair?

If someone told us the coin was unfair, what would our best guess be for the altered chance of observing a head?

If we tossed our coin 100 times and observed 75 heads, would we think our coin is fair?

What probability of heads would best explain the observed 75 heads out of 100?

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pdf versus likelihood

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Probability Distribution Function (pdf)

Defines the probability of observing a range of outcomes assuming a specific ‘truth’ about the distribution of outcomes

Likelihood function

Defines the probability of observing a specific outcome for a range of possible ‘truths’ about the distribution of outcomes

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pdf versus likelihood

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Instead of coin flips, suppose we thought that seroprevalence of SARS-CoV-2 antibodies in a population was 50%.

Just as with our coin example, we could sample 4 individuals from the population.

If we found 3 of these 4 individuals had antibodies associated with a past SARS-CoV-2 infection (i.e., seroprevalence was 75%), is it plausible that the true value of seroprevalence in the population is 50%?

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pdf versus likelihood

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4.6875%

42.1875%

0%

25%

0.75

0.421875

Maximum Likelihood Estimate (mle)

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pdf versus likelihood

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Now imagine we sampled 100 individuals and found 75 of them had antibodies associated with a past SARS-CoV-2 infection (i.e., again, seroprevalence is 75%).

It seems unlikely that the true seroprevalence is 50%. What values are plausible?

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pdf versus likelihood

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0.75

Much less than 0.421875

mle

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Likelihood versus log-likelihood

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Instead of trying to maximize this

it is frequently easier to maximize this

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Deriving the maximum likelihood estimator

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