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Bayesian Statistics Probabilities Are Subjective

This document discusses Bayesian statistics and the Bayesian approach to statistical inference. It introduces the key concepts of the prior distribution, which represents knowledge about a parameter before observing data, and the posterior distribution, which represents updated knowledge about the parameter after observing data. The posterior is calculated by combining the prior with the likelihood of the data using Bayes' rule. Examples are given where the prior and posterior distributions for the probability of success in binomial experiments are beta distributions. The mathematical details of how the beta prior leads to a beta posterior are also outlined.

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Dorian Grey
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14 views6 pages

Bayesian Statistics Probabilities Are Subjective

This document discusses Bayesian statistics and the Bayesian approach to statistical inference. It introduces the key concepts of the prior distribution, which represents knowledge about a parameter before observing data, and the posterior distribution, which represents updated knowledge about the parameter after observing data. The posterior is calculated by combining the prior with the likelihood of the data using Bayes' rule. Examples are given where the prior and posterior distributions for the probability of success in binomial experiments are beta distributions. The mathematical details of how the beta prior leads to a beta posterior are also outlined.

Uploaded by

Dorian Grey
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Bayesian statistics

So far we have thought of probabilities as the long term


“success frequency”: #successes / #trails → P(success).

In Bayesian statistics probabilities are subjective!


Examples
* Probability that two companies merge
* Probability that a stock goes up
* Probability that it rains tomorrow

We typically want to make inference for a parameter θ, for


example µ, σ2 or π. How is this done using subjective
probabilities?

1 lecture 8
Bayesian statistics
Bayesian idea: We describe our “knowledge” about the
parameter of interest, θ, in terms of a distribution π(θ). This
is known as the prior distriubtion (or just prior) – as it
describes the situation before we see any data.

Example: Assume θ is the probability of success.


Prior distributions describing what value we think θ has:

2 lecture 8
Bayesian statistics
Posterior
Let x denote our data. The conditional distribution of θ
given data x is denoted the posterior distribution:
f (x | θ )π (θ )
π (θ | x ) =
g (x )

Here f(x|θ) how data is specified conditional on θ.

Example:
Let x denote the number of successes in n trail.
Conditional θ, x follows a binomial distribution:
n x
f ( x | θ ) =  θ (1 − θ ) n − x
 x
3 lecture 8
Bayesian statistics
Posterior – some data
We now observe n=10 experiment with x=3 successes, i.e.
x/n=0.3
Posterior distributions – our “knowledge” after having seen
data.

Shaded area: Prior distribution


Solid line: Posterior distribution
Notice that the posteriors are moving towards 0.3.
4 lecture 8
Bayesian statistics
Posterior – some data
We now observe n = 100 experiment with x = 30 successes,
i.e. x/n = 0.3
Posterior distributions – our “knowledge” after having seen
data.

Shaded area: Prior distribution


Solid line: Posterior distribution
Notice that the posteriors are almost identical.
5 lecture 8
Bayesian statistics
Mathematical details
The prior is given by a so-called Beta distribution with
parameters α > 0 and β > 0:
Γ(α + β ) α −1
π (θ ) = θ (1 − θ ) β −1 for 0 ≤ θ ≤ 1
Γ(α )Γ( β )

The posterior then becomes

Γ(α + β + n)
π (θ | x) = θ α + x −1 (1 − θ ) β + n − x −1
Γ(α + x)Γ( β + n − x)

a Beta distribution with parameters α+x and β+n−x.

6 lecture 8

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