# Appendix A

**Distributions Used in Uncertainty Analysis**

## The Beta distribution

The Beta distribution is one of three distributions associated with a
binomial stochastic process. A binomial process is a random counting
system where there are a discrete number of opportunities (trials) of
some particular event happening (successes) and where each trial has the
same probability of being a success. This means that each trial must be
independent of every other trial.

There are many systems that closely approximate a binomial process.
Random processes like the tossing of a coin are binomial, since one face
of the coin can be defined as being a success and the probability of
each coin being a success remains constant for all tosses. No matter how
many "heads" there have been in a row, the probability of a "tails" for
the next toss remains the same (e.g. 50% for a fair coin).

Random sampling from a population may also closely approximate a
binomial process, where we are concerned with determining what
proportion of that population has some characteristic of interest. If
the population is much larger than the sample size (a rule of thumb is
that the population should be at least 10 times the size of the sample)
then the probability of an individual randomly sampled from the
population having the characteristic of interest remains fairly constant
and equivalent to the proportion of the population with that
characteristic. So, for example, if we are interested in the proportion
of U.S. citizens that eat meat, we can do a random survey of U.S.
citizens. Providing that our sample is much smaller than the population
size, the probability that each consecutive randomly selected person
eats meat remains reasonably constant, though there are actually a
finite number of people who eat meat and there is a finite population
too, which means that the probability that the next randomly sampled
person eats meat does, in fact, depend on the previous samples.

The example above for meat eaters assumes that we are sampling without
replacement: in other words, we would not survey any person more than
once. It also implicitly assumes that there are a fixed number of people
who eat meat in the population and a fixed population size too. This
would be a static system with a constant proportion of meat eaters.

The Beta distribution can be used to model the confidence one has about
the probability of success of a binomial trial *p* where one has
observed *n* independent trials of which *s* were successes (41, 66, 94,
118, 119) so that it is said that *p* is distributed as a
Beta(*s*+1,*n*-*s*+1).

This distribution is the result of applying Bayes' Theorem with a
Uniform(0,1) prior distribution and a binomial likelihood function. In
layman's terms, Bayes' Theorem works as follows:

1.  A prior statement of the knowledge of the variable to be modelled is
    given. In this case, we are saying that the probability *p* lies
    somewhere between zero and one, but we would not like to say that
    any value within that range was anymore likely than any other value
    (hence the Uniform(0,1) prior distribution.

2.  For each allowed value within the prior distribution's range, we
    calculate the probability of observing the *s* successes we observed
    from the *n* trials. This probability is simply the binomial
    probability:

$P(s;n,p) =_{n}C_{s}p^{s}(1 - p)^{n - s}$

3.  These binomial probabilities then become the weightings given to
    each value of *p* in the prior distribution. By normalizing these
    weightings we arrive at a posterior (i.e. final answer)
    distribution.

Thus, the posterior distribution *f(p)* for *p* is given by the product
of the prior distribution density and the likelihood function:

$f(p;s,n) = \frac{1\text{.}_{n}C_{s}p^{s}(1 - p)^{n - s}}{1\text{.}\int_{0}^{1}{{}_{n}C_{s}t^{s}(1 - t)^{n - s}\text{dt}}}$
$0 < p < 1$

where the 1 in the equation is the probability density of the
Uniform(0,1) prior distribution, the ${}_{n}C_{s}p^{s}(1 - p)^{n - s}$
is the weighting given to the *p* value (the likelihood function) and
the integral in the denominator normalizes the distribution so that the
area under the curve equals unity. By omitting the 1, cancelling out the
*~n~C~s~* and using *α~1~*=*s*+1, *α~2~*=*n*-*s*+1, we arrive at the
equation for the Beta(*α~1~*, *α~2~*) distribution's probability density
*f(x;α~1~,α~2~)*:

$$f(x;\alpha_{1},\alpha_{2}) = \frac{p^{\alpha_{1} - 1}(1 - p)^{\alpha_{2} - 1}}{\int_{0}^{1}{t^{\alpha_{1} - 1}(1 - t)^{\alpha_{2} - 1}\text{dt}}}$$

Note that the prior distribution for *p* is a Uniform(0,1) distribution,
shown in Figure A1. The Uniform(0,1) prior distribution is an uninformed
prior, meaning that no subjective opinion or any other information has
been involved in determining the prior. This is logically the most
conservative approach one could take where conservatism here means
expressing the maximum degree of uncertainty possible. The selection of
an appropriate prior is sometimes slightly contentious. For example,
Beta(0,0) is sometimes suggested as an uninformed prior, though it does
not in theory exist. One criticism for using a Beta(1,1) prior is that
the mean of the estimated probability is biased towards 50%, away from
the observed proportion. In fact, for all applications of the Beta
distribution in this analysis, the information contained in the prior
distribution is generally overwhelmed by the information contained in
the sample data and the results are essentially equivalent to a more
traditional frequentist statistics approach which would give the
confidence distribution for the probability *p* as: $\hat{p}$=
Binomial(*n*, *s*/*n*) / *n*. The frequentist also uses the central
limit theorem in situations where a large number *n* of samples were
taken, say *n* \> 30. By the central limit theorem $\hat{p}$ has a
Normal distribution Normal(*p*, √\[*p*(1-*p*)/*n*\]), where *p* is the
true value of the probability of success. One then assumes (i.e.
estimates) the confidence distribution for *p* to be :

$$\hat{p} = \text{Normal}\left( \frac{s}{n},\sqrt{\frac{s(n - s)}{n^{3}}} \right)$$

Thus, from a practical viewpoint, there is little difference in using a
Bayesian or frequentist approach to uncertainty estimating in this
model, except that the Bayesian approach allows one to combine
information from dissimilar data (66).

Perhaps the easiest way to understand the Beta distribution used in this
manner is to look at a few plots of its shape for varying values of
*α~1~* and *α~2~*. Figure A1, the Uniform(0,1) distribution, is also the
Beta(1,1), i.e. the Beta distribution where we have observed *s* = 0
successes and (*n-s*) = 0 failures: this is the distribution when we
have not yet done any trials and hence remains the prior distribution.

![](media/image1.wmf){width="4.997222222222222in"
height="3.077777777777778in"}

**Figure A1**: The Uniform(0,1) distribution, which is also the
Beta(1,1) distribution

Figure A2 show the Beta(1,11)and the Beta(11,1) distributions: the
former where we have observed zero successes in ten trials and the
latter where we have observed ten successes in ten trials. Note that
they are simply the reflection of each other since they essentially
represent the same thing: one needs only to reverse the definition of a
success to its opposite. Also note that, since all trials have been a
success or a failure, the distributions peak at zero and one
respectively. If this pattern continues with more trials, the
distributions will become progressively more concentrated at zero and
one.

![](media/image2.wmf){width="4.997222222222222in"
height="3.077777777777778in"}

**Figure A2**: Examples of the Beta distribution where all trials are
successes (peak at *p*=1)

or all are failures (peak at *p*=0)

Figure A3 shows the Beta(3,3), Beta(11,11) and Beta(21,21) distributions
representing four, 20 and 40 trials where 50% have been successes. Note
that as the number of trials increases, the distribution becomes
progressively narrower: in other words, one is becoming progressively
more confident about what the true value of *p* must be.

![](media/image3.wmf){width="4.997222222222222in"
height="3.077777777777778in"}

**Figure A3**: Examples of the Beta distribution where there are equal
successes and failures (i.e. *α~1~*=*α~2~*)

The figures below show that except for the case with very small *n*, the
Binomial distributions or the Normal approximations to them that a
frequentist would use to model the confidence distributions modeled as
Beta posterior distributions above are practically indistinguishable
from the Beta distributions. The cumulative probability distributions
for the Binomial(4,0.5)/4, the Normal(0.5, 0.25), and the Beta(3,3)
which are the ones for the case *n*=4, are shown in Figure A4a. One can
see that for the proportion 0.5 the deviations between the Binomial and
the Normal or Beta are fairly evenly arranged around the center of the
graph. For a proportion of success less than 0.5, the deviations between
the Binomial and the Normal or Beta are larger to the left half of the
graph and very small toward the right side of the graph. Conversely, for
a proportion of success greater than 0.5, the deviations between the
Binomial and the Normal or Beta are larger to the right half of the
graph and very small toward the left side of the graph. The major source
of the deviation is the relative size of the steps in the Binomial when
*n* is small. The cumulative distribution for the Binomial is a step
function. As sample size increases the deviations between the discrete
Binomial cumulative probability distributions and those of the
corresponding continuous Normal or Beta distributions diminish. Figure
A4b displays the cumulative probability distributions for the case
*n*=20 and Figure A4c, for the case *n*=40. Because the Beta
distribution is able to model success probability using non-integer
numbers of successes it is extremely useful for risk assessment modeling
where inputs are weighted survey results. Bayesian modeling is more
conducive to combining various types of probability distributions than
frequentist modeling is.

![](media/image4.wmf){width="5.275in"
height="3.033333333333333in"}**Figure A4a**. Cumulative probability
distributions for *p* based on the Beta, Binomial and Normal for the
case n=4 and the number of observed successes is 2.

![](media/image5.wmf){width="5.366666666666666in"
height="2.966666666666667in"}

**Figure A4b**. Cumulative probability distributions for *p* based on
the Beta, Binomial and Normal for the case n=20 and the number of
observed successes is 10.

![](media/image6.wmf){width="5.258333333333334in"
height="3.0256944444444445in"}

**Figure A4c**. Cumulative probability distributions for *p* based on
the Beta, Binomial and Normal for the case n=40 and the number of
observed successes is 20.

## Use of the Gamma distribution to describe the uncertainty about a Poisson mean

Like the binomial probability *p*, the mean events per period *λ* is a
fundamental property of the stochastic system in question. It can never
be observed and it can never be exactly known. However, we can become
progressively more certain about its value as more data are collected.
Bayesian inference again provides us with a means of quantifying the
state of our knowledge as we accumulate data.

Let us assume an uninformed prior *π(λ)* = 1/* λ .* The Poisson
likelihood function for observing *X* events in period *t* is given by:

$l(X|\lambda,t) = \frac{e^{- \lambda t}(\lambda t)^{X}}{X!} \propto e^{- \lambda t}(\lambda)^{X}$
$$

The posterior distribution density is the product of the prior density
and the likelihood function. We can disregard terms that not involving
*λ*, recognizing that the distribution will be normalized eventually,
and we then get the posterior distribution:

$p(\lambda|X) \propto e^{- \lambda t}\lambda^{X - 1}$ $\lambda > 0$

which has the functional form of, and therefore is, a Gamma(*X*,1/*t*)
distribution where *λ* is the variable.

The shape of the posterior Gamma distribution becomes progressively less
sensitive to the prior distribution as data is collected. In essence,
the sensitivity of the Gamma distribution to the prior amounts to
whether (*X*-1) is approximately the same as *X*. So, if *X* was 100,
the difference would be roughly 1% influenced by the prior and 99%
influenced by the data. In this model, the information contained in the
quantity of data available *always* overpowers the prior.

Gamma distributions used in the risk assessment to model the rate of
invasive infection, λ~i~ which has a Gamma(43,1) distribution and the
rate of enteric infection, λ~e~ which has a Gamma(3985,1) distribution.
Those distributions are shown in the following graphs.

![](media/image7.wmf){width="5.24375in" height="2.8090277777777777in"}

**Figure A5a**. The Gamma(43,1) distribution used to model the rate of
invasive disease, λ~i~.

![](media/image8.wmf){width="5.345833333333333in" height="2.95in"}

**Figure A5b**. The Gamma(3985,1) distribution used to model the rate of
enteric disease, λ~e~.

As was the case for the Beta distribution, there is a distributional
counterpart to the gamma that the frequentist would apply to this
estimation problem. When X, the number of events in one unit of time
(t=1), is distributed Poisson(λ),the maximum likelihood estimator for λ,
$\hat{\lambda}$ is the observed value of X. Then the Poisson(X)
distribution is used as the uncertainty distribution for λ. The Poisson
distribution is the limiting distribution for a Binomial random variable
when *n* is large and *p* is very small. Because of this relationship,
the central limit theorem applies for the Poisson under the same
limiting conditions. By the central limit theorem $\hat{\lambda}$is
normal with mean λ and variance λ, where λ is the true value of the
parameter. This means that the confidence distribution for λ is
estimated as:

$$\text{Normal}(X,\sqrt{X})$$

The three possible choices for the confidence distribution for λ when 43
cases were observed during the year are shown to demonstrate how similar
they are.

![](media/image9.wmf){width="4.889583333333333in"
height="3.0256944444444445in"}

**Figure A6**. The cumulative distribution functions for Gamma(43,1),
Poisson(43), and Normal(43,6.56)