Introduction to statistical concepts used in the course

Nowcasting and forecasting of infectious disease dynamics

Why statistical concepts?

  • We’ll need to estimate things (delays, reproduction numbers, case numbers now and in the future)

  • We’ll want to correctly specify uncertainty

  • We’ll want to incorporate our domain expertise

  • We’ll do this using Bayesian inference

Bayesian inference in 15 minutes

Interlude: probabilities

Laplace, 1812

Probability theory is nothing but common sense reduced to calculation.

Interlude: probabilities (1/3)

  • If \(A\) is a random variable, we write \[ p(A = a)\] for the probability that \(A\) takes value \(a\).
  • We often write \[ p(A = a) = p(a)\]
  • Example: The probability that it rains tomorrow \[ p(\mathrm{tomorrow} = \mathrm{rain}) = p(\mathrm{rain})\]
  • Normalisation \[ \sum_{a} p(a) = 1 \]

Interlude: probabilities (2/3)

  • If \(A\) and \(B\) are random variables, we write \[ p(A = a, B = b) = p(a, b)\] for the joint probability that \(A\) takes value \(a\) and \(B\) takes value \(b\)
  • Example: The probability that it rains today and tomorrow \[ p(\mathrm{tomorrow} = \mathrm{rain}, \mathrm{today} = \mathrm{rain}) = p(\mathrm{rain}, \mathrm{rain})\]
  • We can obtain a marginal probability from joint probabilities by summing \[ p(a) = \sum_{b} p(a, b)\]

Interlude: probabilities (3/3)

  • The conditional probability of getting outcome \(a\) from random variable \(A\), given that the outcome of random variable \(B\) was \(b\), is written as \[ p(A = a | B = b) = p(a| b) \]
  • Example: the probability that it rains tomorrow given that it rains today \[ p(\mathrm{tomorrow} = \mathrm{rain} | \mathrm{today} = \mathrm{rain}) = p(\mathrm{rain} | \mathrm{rain})\]
  • Conditional probabilities are related to joint probabilities as \[ p(a | b) = \frac{p(a, b)}{p(b)}\]
  • We can combine conditional probabilities in the chain rule \[ p(a, b, c) = p(a | b, c) p(b | c) p (c) \]

Probability distributions (discrete)

  • E.g., how many people die of horse kicks if there are 0.61 kicks per year
  • Described by the Poisson distribution

Two directions

  1. Calculate the probability
  2. Randomly sample

Calculate discrete probability

  • E.g., how many people die of horse kicks if there are 0.61 kicks per year
  • Described by the Poisson distribution

What is the probability of 2 deaths in a year?

  dpois(x = 2, lambda = 0.61)
[1] 0.1010904

Two directions

  1. Calculate the probability
  2. Randomly sample

Generate a random (Poisson) sample

  • E.g., how many people die of horse kicks if there are 0.61 kicks per year
  • Described by the Poisson distribution

Generate one random sample from the probability distribution

  rpois(n = 1, lambda = 0.61)
[1] 0

Two directions

  1. Calculate the probability
  2. Randomly sample

Probability distributions (continuous)

  • Extension of probabilities to continuous variables
  • E.g., the temperature in Stockholm tomorrow

Normalisation: \[ \int p(a) da = 1 \]

Marginal probabilities: \[ p(a) = \int_{} p(a, b) db\]

Two directions

  1. Calculate the probability (density)
  2. Randomly sample

Calculate probability density

  • Extension of probabilities to continuous variables
  • E.g., the temperature in Stockholm tomorrow

What is the probability density of \(30^\circ C\) tomorrow, if the mean temperature on the day is \(23^\circ C\) (standard deviation \(2^\circ C\)) ? A naïve model could be:

  dnorm(x = 30,
        mean = 23,
        sd = 2)
[1] 0.0004363413

Two directions

  1. Calculate the probability
  2. Randomly sample

Generate a random (normal) sample

Generate one random sample from the normal probability distribution with mean 23 and standard deviation 2:

  rnorm(n = 1,
        mean = 23,
        sd = 2)
[1] 24.60111

Two directions

  1. Calculate the probability
  2. Randomly sample

Bayesian inference in 15 minutes

Idea of Bayesian inference: treat \(\theta\) as random variables (with a probability distribution) and condition on data: posterior probability \(p(\theta | \mathrm{data})\) as target of inference.

Bayes’ rule

  • We treat the parameters of the a \(\theta\) as random with prior probabilities given by a distribution \(p(\theta)\). Confronting the model with data we obtain posterior probabilities \(p(\theta | \mathrm{data})\), our target of inference. Applying the rule of conditional probabilities, we can write this as

\[ p(\theta | \textrm{data}) = \frac{p(\textrm{data} | \theta) p(\theta)}{p(\textrm{data})}\]

  • \(p(\textrm{data} | \theta)\) is the /likelihood/

  • \(p(\textrm{data})\) is a /normalisation constant/

  • In words, \[\textrm{(posterior)} \propto \textrm{(normalised likelihood)} \times \textrm{(prior)}\]

Bayesian inference

MCMC

  • Markov-chain Monte Carlo (MCMC) is a method to generate samples from the posterior distribution, the target of inference

  • stan is a probabilistic programming language that helps you to write down probabilistic models and to fit them using MCMC samplers and other methods.

Return to the session