Bayesian Statistics

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Introduction to Bayesian Statistics

(And some computational methods)

Theo Kypraios

http://www.maths.nott.ac.uk/∼tk

MSc in Applied Bioinformatics @ Cranfield University.

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My Background

Bayesian Statistics;

Computational methods, such as Markov Chain and Sequential Monte Carlo (MCMC & SMC);

Large and complex (real) data analysis, mainly Infectious Disease Modelling, Neuroimaging, Time series . . ..

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Outline of the Talk

1. Why (statistical) modelling is useful?

2. The Frequentic/Classical Approach to Inference.

3. The Bayesian Paradigm to Inference.

Theory Examples

4. More Advanced Concepts (e.g. Model Choice/Comparison)

5. Conclusions

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Use of Statistics

Examples include:

Sample Size Determination

Comparison between two (or more) groups

t-tests, Z-tests;

Analysis of variance (ANOVA); tests for proportions etc;

Classification/Clustering; . . .

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Use of Statistics

Examples include:

t-tests, Z-tests;

Analysis of variance (ANOVA); tests for proportions etc; Classification/Clustering; . . .

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Use of Statistics

Examples include:

t-tests, Z-tests;

Classification/Clustering;

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Use of Statistics

Examples include:

t-tests, Z-tests;

Classification/Clustering; . . .

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Why Statistical Modelling?

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Suppose that we are interested in investigating the association

between x and y .

Isn’t just enough to calculate the correlation(ρ) between x and y ?

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Why Statistical Modelling?

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −3 −2 −1 0 1 2 −2 0 2 4 6 8 explanatory (x) response(y)

Suppose that we are interested in investigating the association

between x and y .

Isn’t just enough to calculate the correlation(ρ) between x and y ?

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Why Statistical Modelling?

Perahps, for this dataset it is enough. . .

. . .ρ = 0.83 indicates some strong correlation between x and_b y .

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Why Statistical Modelling?

Perahps, for this dataset it is enough. . .

. . .ρ = 0.83 indicates some strong correlation between x and_b y .

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Why Statistical Modelling?

What about this dataset? The correlation coefficient turns out to be ≈ 0.5. ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● _● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● −3 −2 −1 0 1 2 2 4 6 8 explanatory (x) response(y) 8 / 1

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Why Statistical Modelling?

What about this dataset? The correlation coefficient turns out to be ≈ -0.78. ● ●● ● ●● ● ●● ●● ●●_●● ●● ● ●_●●●● ● ● ● ●●●● ● ●● ●● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ●●●●●_● ●● ● ● ●●● ● ●● ● ● ● ●●_● ●● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● 0 1 2 3 4 5 −0.1 0.0 0.1 0.2 0.3 explanatory (x) response(y)

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Statistical Modelling

One (of the best) ways(s) todescribe some datais by fitting a (statistical) model.

Examples:

1. y = α + β · x + error

2. y = α + β · x + γ · x2 + error

3. y = 1 −_1+α·x1 − β · x + error

Themodel parameters(α, β, γ) tell us much moreabout the relationship between x and y rather than just thecorrelation coefficient. . .

What about amore generalmodel?

4. y = f (θ, x ) + error

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Aims of Statistical Modelling

In statical modelling we are interested in estimatingthe unknown parameters fromdata→ Statistical Inference.

Parameter estimation needs be done in a formal way. In other words we ask ourselves the question: what are the “best”

values, say, for α and β such that the proposed model“bests”

describes the observed data?

And, what do we mean by “best”?

Should we only look for a single estimatefor (α, β)? No!

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Aims of Statistical Modelling

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Aims of Statistical Modelling

Should we only look for a single estimatefor (α, β)?

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Aims of Statistical Modelling

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Least Squares Estimation

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Find the values of α and β which minimize the squared difference (distance) between what themodel predicts and the data, (a.k.a. Least Squares Estimation (LSE) )

What about other pairs (α, β) (perhaps very different from each other) which describe equally well the observed data →

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Classical (or Frequentist)

Inference

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Statistical Approach: The Likelihood Function

Thelikelihood functionplays a fundamental role in statistical inference.

In non-technical terms, the likelihood function is a function that when evaluated at a particular point, say (α0, β0), is the

probability of observing the (observed) datagiven that the parameters (α, β) take the values α0 and β0.

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The Likelihood Function− A Toy Example

Let us think of a very simple example. Consider a Binomial experiment:

n trials (e.g. toss a coin n times)

model the number x successes (e.g. got heads x times).

Suppose we are interested in estimating theprobability of success (denoted by θ) for one particular experiment. Data: Out of 100 times we repeated the experiment we observed 80 successes.

What about L(0.1), L(0.7), L(0.99)?

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The Likelihood Function− A Toy Example

What L(0.1) really means, is what is “how likely is to observe 80 times heads out of tossing a coin 100 times if the true (but unknown) probability of success is 0.7?”

In other words,

L(0.7) = P(X = 80|θ) =100 80

·0.780·(1−0.7)100−80= 0.075 But if we can evaluate L(0.7) then wecan evaluate L(θ) for all possible values for θ.

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The Likelihood Function− A Toy Example

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ● ● ● ●_● ● ● ● ● ● ● ● ● ●●●●●●●●●●● 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.04 0.08 theta L(theta) 17 / 1

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Classical (Frequentist) Inference

Frequentist inferencetell us that:

we should for parameter values that maximise the likelihood function → maximum likelihood estimator (MLE) associate parameter’s uncertainty with the calculation of standard errors(SE) . . .

. . . which in turn enable us to construct confidence intervals for the parameters, e.g. 95% CI

b θ ∓ 1.96 · r varθb or θ ∓ 1.96 · SEb b θ

For this example, this turns out to be

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Interval Estimation

Having obtained the variance-covariance matrix, we can then construct confidence intervals for the parameters based on sampling theory.

Such an approach is based on the notion that:

1. if the experiment was to be repeated many times, 2. and a maximum likelihood method is derived as well as a

confidence interval each time

3. then on average, the interval estimates would contain the true parameter (1 − α)% of the time.

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Interval Estimation

1. if the experiment was to be repeated many times,

2. and a maximum likelihood method is derived as well as a confidence interval each time

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Interval Estimation

1. if the experiment was to be repeated many times,

2. and a maximum likelihood method is derived as well as a confidence interval each time

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Interval Estimation

What’s wrong with that?

Nothing, but . . .

. . . it is approximate, counter-intuitive(data is assumed to be random, parameter assumed to be fixed) and mathematically intractable for complex scenarios.

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Interval Estimation

What’s wrong with that?

Nothing, but . . .

. . . it is approximate, counter-intuitive(data is assumed to be random, parameter assumed to be fixed) and mathematically intractable for complex scenarios.

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Some (Other) Issues with this Approach

For instance, recall the previous experiment: twe cannot ask (or even answer!) questions such as

1. “what is the chance that the unknown parameter (i.e.

probability of success) is greater than 0.6?” i.e. compute the quantity P(θ > 0.6|data) . . .

2. or something like, P(0.3 < θ < 0.9|data);

Sometimes we are interested in (not necessarily linear) functions of parameters, e.g.

θ1+ θ2, θ1/(1 − θ1) θ2/(1 − θ2)

Whilst in some cases, thefrequentist approach offers a solution which isnot exact but approximate, there are others, for which it cannot or it is very hard to do so.

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Bayesian Inference

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Bayesian Inference

When drawing inference within aBayesian framework, the data are treated as a fixed quantity and the parameters are treated as random variables.

That allows us to assign to parameters (and models) probabilities, making the inferential framework

far more intuitive and

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Bayesian Inference (2)

Denote by θ the parameters and by y the observed data. Bayes theorem allows to write:

π(θ|y) = π(y|θ)π(θ) π(y) = π(y|θ)π(θ) R θπ(y|θ)π(θ) dθ where

π(θ|y) denotes the posterior distribution of the parameters given the data;

π(y|θ) = L(θ) is the likelihood function;

π(θ) is the prior distribution of θ which express our beliefs about the parameters, before we see the data;

π(y) is often called the marginal likelihood and plays the role of the normalising constant of the density of the posterior distribution.

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Bayesian Inference (2)

In a nutshell theBayesian paradigm provides us with a distribution

as for what we havelearned about the parameter from the data. In contrast to thefrequentist approach with which we aregetting a point estimate (MLE) and a standard error (SE), in the Bayesian world we getting a whole distribution (i.e. we get much more for our money!)

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The Posterior Distribution

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Bayesian Inference (3)

We can write the posterior distribution as follows:

π(θ|y) = R π(y|θ)π(θ) θπ(y|θ)π(θ) dθ

The density of the posterior distribution isproportional to the likelihood times the prior density;

The posterior distribution tells us everythingwe need to know about the parameter;

Statements such as P(θ > k) or P_1+θθ > k) where k is a constant make sense, since θ is a random variable but, in addition, they are very useful in modelling.

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Why Having the Distribution is Very Useful?

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Why Having the Distribution is Very Useful?

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Why Having the Distribution is Very Useful?

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The Prior

Recall that: π(θ|y) = π(y|θ)π(θ) π(y) = π(y|θ)π(θ) R θπ(y|θ)π(θ) dθ

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Bayesian Inference: The Prior

One of thebiggest criticisms to the Bayesian paradigm is the use of the prior distribution.

“Couldn’t I choose a very informative prior and come up with favorable result”?

Yes, but this is bad science!

“I know nothing about the parameter; what prior should I choose”?

Choose an uninformative (or vague) prior. → more details shortly.

If there is a lot of data available then the posterior distribution would not be influenced so much by the prior and vice versa;

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Bayesian Inference: The Prior

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Bayesian Inference: The Prior

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Bayesian Inference: The Prior

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Some Examples on the Effect of the Prior

83/100 successes: interested in probability of success θ

0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 theta poster ior posterior lik prior 33 / 1

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Some Examples on the Effect of the Prior

0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 poster ior posterior lik prior

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Some Examples on the Effect of the Prior

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Some Examples on the Effect of the Prior

0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 poster ior posterior lik prior

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Some Examples on the Effect of the Prior

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The Prior Distribution

Take−home message:

Be rather careful with the choice of prior, which of

course is (or can be) subjective!

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(Back to) The Posterior

Distribution

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Taking a Closer a Look at the Formulas

We can write the posterior distribution as follows:

π(θ|y) = R π(y|θ)π(θ) θπ(y|θ)π(θ) dθ

= π(y|θ)π(θ) π(y)

∝ π(y|θ)π(θ) (1)

where π(y) is often called the marginal likelihood andplays the role of the normalising constantof the density of the posterior distribution, i.e. makes the area under the curve π(y|θ)π(θ) to integrate to one, i.e. aproper probability density function

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How to Deal with the Normalising Constant?

If we were only interested in finding bθMAP for which π(θ|y) is maximised, then there is no need to compute the normalising constant, π(y).

Nevertheless, suppose that we want to get as a summary statistic from our posterior and compute, for instance, a posterior expectation, e.g.

E [θ|y] = Z

θ

θ · π(θ|y) dθ

or the posterior variance etc. That, of course, requires

knowledge of the full expression of π(θ|y), i.e. not just up to a normalising constant.

How to compute this integral then? Numerical integration techniques? Can we “guess”? Or . . .

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How to Deal with the Normalising Constant?

E [θ|y] = Z

θ

θ · π(θ|y) dθ

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How to Deal with the Normalising Constant?

E [θ|y] = Z

θ

θ · π(θ|y) dθ

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Do we Really Need to Compute the

Normalising Constant?

Instead of deriving the full expression of the probability density function of θ|y explicitly, we could draw samples from π(θ|y).

If we have samples from π(θ|y) then we can approximate the posterior expectation as follows:

E [π(θ|y )] ≈ 1 M M X i =1 θ0_j, θj 0 ∼ π(θ|y)

Therefore, the only thing we need to come up with, is a method which will allow us to draw samples from π(θ|y) without the need of evaluating the normalising constant.

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Do we Really Need to Compute the

Normalising Constant?

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Do we Really Need to Compute the

Normalising Constant?

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Derive the Posterior Distribution via

Sampling−Based Inference

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Sampling−Based Inference

If we can draw samples from the posterior distribution π(θ|y) then we can do everything we want/need:

Estimate the density (kernel density estimation, histogram);

Estimate moments (eg means, variances), probabilities etc; Derive the distribution of (not necessarily linear) functions of the parameters g (θ) in a very straightforward manner; Visualise the relationship of two or more model parameters.

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Sampling−Based Inference

Estimate the density (kernel density estimation, histogram); Estimate moments (eg means, variances), probabilities etc;

Derive the distribution of (not necessarily linear) functions of the parameters g (θ) in a very straightforward manner; Visualise the relationship of two or more model parameters.

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Sampling−Based Inference

Estimate the density (kernel density estimation, histogram); Estimate moments (eg means, variances), probabilities etc; Derive the distribution of (not necessarily linear) functions of the parameters g (θ) in a very straightforward manner;

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Sampling−Based Inference

Estimate the density (kernel density estimation, histogram); Estimate moments (eg means, variances), probabilities etc; Derive the distribution of (not necessarily linear) functions of the parameters g (θ) in a very straightforward manner; Visualise the relationship of two or more model parameters.

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A Toy Example on Sampling−Based Inference

Suppose that the random variable X comes from a Gamma distribution with the following probability density function (pdf)

fX(x |α, β) = βα Γ(α)x

α−1_{· exp{−βx},} _{α, β > 0} For any given α and β the expectation, i.e the mean of X is derived by doing this integral

E [X ] = Z

X

x · fX(x ) dx and we also know thatthe probability

P(X < 0.5) = Z 0.5

0

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A Toy Example on Sampling−Based Inference

1. Suppose that, somehow, we have a way of simulating

realizations from the Gamma distribution . . .

2. . . . and draw N samples where N is a large number, e.g. 100, 000.

3. If weplot the histogram of these N values by doing something like this in R

hist(rgamma(10^5, 5, 3), prob=TRUE, main="Samples from Gamma(5,3)", xlab=expression(x),col=2)

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A Toy Example on Sampling−Based Inference

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A Toy Example on Sampling−Based Inference

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A Toy Example on Sampling−Based Inference

We get something like this:

Samples from Gamma(5,3)

Density 0 1 2 3 4 5 6 7 0.0 0.1 0.2 0.3 0.4 0.5

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A Toy Example on Sampling−Based Inference

We get something like this and drawfX(x ) on top:

Samples from Gamma(5,3)

x Density 0 1 2 3 4 5 6 7 0.0 0.1 0.2 0.3 0.4 0.5 48 / 1

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A Toy Example on Sampling−Based Inference

That means that we can:

“approximate“ (or “estimate”) the mean E[X] by the sample mean, i.e. [ E [X ] = 1 N N X i =1 xi

and the probability P(X < 0.5) by the proportion of the values in the sample which are less than 0.5, i.e.

\ P(X < 0.5) = 1 N N X i =1 1(xi < 0.5)

Of course, these “approximations” are getting better and better as N →> ∞.

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An Example of a Bivariate Distribution

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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 0 2 4 −2 0 2 4 6 x y 50 / 1

(74)

Derive the Posterior Distribution via

Analytic Integration

(75)

Obtaining the Posterior Distribution

Analytically

Thebasic idearelies on the following simple observation: Consider for example the random variable X which follows a Gamma distribution (as in the example before) with pdf:

fX(x |α, β) = βα Γ(α)x

α−1_{· exp{−βx}.} Since this is aproper pdf then it holds that

Z X

βα Γ(α)x

α−1_{· exp{−βx} dx = 1}

andre-arrangement give us that

Z X

xα−1· exp{−βx} dx = Γ(α) βα

Of course, the same idea applies for otherwell-known distributions and their densities

(76)

Obtaining the Posterior Distribution

Analytically

By making use of that idea, it is often the case that with asuitable choice of prior distributionwe can avoid calculatingthe

normalising constant.

Recall the Binomial example(n, θ):

Our parameter of interest is a probability, and therefore can take values only between 0 and 1.

A suitable prior distribution which takes the above into account is the Beta distribution with some parameters, say λα and λβ:

π(θ) = 1

B(λα, λβ)

θλα−1_(1−θ)λβ−1_, _λ

(77)

Obtaining the Posterior Distribution

Analytically

The prior density is

π(θ) = 1

B(λα, λβ)θ

λα−1_{(1 − θ)}λβ−1_, _λ

α The likelihood function is:

π(x |θ) = L(θ) =n x

· θx· (1 − θ)n−x

To derive the posterior density we just multiply the two terms: π(θ|x ) ∝ θx_{· (1 − θ)}n−x _{· θ}λα−1_{· (1 − θ)}λβ−1

π(θ|x ) ∝ θx +λα−1_{· (1 − θ)}n+λβ−x−1

Note that we have derive π(θ|x ) up to proportionality!

(78)

Obtaining the Posterior Distribution

Analytically

In this case Z θ θx +λα−1_{· (1 − θ)}n+λβ−x−1_{θ dθ} is equal to B(x + λα− 1, n + λ_β− x − 1)

because we aremaking use of the basic idea we described earlier,

i.e. that _Z x 1 B(A, C )x A−1_{· (1 − x)}C −1_{= 1} and hence Z x xA−1· (1 − x)C −1_{= B(A, C )} which holds for any x ,A and C !

(79)

Obtaining the Posterior Distribution

Analytically

That allows us to say that

θ|x ∼ Beta(x + λα, n + λβ− x)

which is veryconvenientbecause we know a lot of things about the Beta distribution, such as the mean, the variance etc

Summary: Using a Beta distribution as a prior for θ led to a Beta posterior distribution for θ.

The above is a special case of what is called conjugate priors.

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Bayesian vs Frequentist Inference

Everything is assigned distributions (prior, posterior);

we are allowed to incorporate prior information about the parameter . . .

which is then updated by using the likelihood function . . . leading to the posterior distribution which tell us everything we need about the parameter.

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Bayesian vs Frequentist Inference

Everything is assigned distributions (prior, posterior); we are allowed to incorporate prior information about the parameter . . .

which is then updated by using the likelihood function . . . leading to the posterior distribution which tell us everything we need about the parameter.