Probability distributions

Generates a nonparametric distribution.

## Usage

```
LogNormal(meanlog, sdlog, mean, sd, max = Inf)
Gamma(shape, rate, scale, mean, sd, max = Inf)
Normal(mean, sd, max = Inf)
Fixed(value, max = Inf)
NonParametric(pmf)
```

## Arguments

- meanlog, sdlog
mean and standard deviation of the distribution on the log scale with default values of

`0`

and`1`

respectively.- mean, sd
mean and standard deviation of the distribution

- max
Numeric, maximum value of the distribution. The distribution will be truncated at this value. Default:

`Inf`

, i.e. no maximum.- shape, scale
shape and scale parameters. Must be positive,

`scale`

strictly.- rate
an alternative way to specify the scale.

- value
Value of the fixed (delta) distribution

- pmf
Probability mass of the given distribution; this is passed as a zero-indexed numeric vector (i.e. the fist entry represents the probability mass of zero). If not summing to one it will be normalised to sum to one internally.

## Details

Probability distributions are ubiquitous in EpiNow2, usually representing epidemiological delays (e.g., the generation time for delays between becoming infecting and infecting others; or reporting delays)

They are generated using functions that have a name corresponding to the
probability distribution that is being used. They generated `dist_spec`

objects that are then passed to the models underlying EpiNow2.
All parameters can be given either as fixed values (a numeric value) or as
uncertain values (a `dist_sepc`

). If given as uncertain values, currently
only normally distributed parameters (generated using `Normal()`

) are
supported.

Each distribution has a representation in terms of "natural" parameters (the ones used in stan) but can sometimes also be specified using other parameters such as the mean or standard deviation of the distribution. If not given as natural parameters then these will be calculated from the given parameters. If they have uncertainty, this will be done by random sampling from the given uncertainty and converting resulting parameters to their natural representation.

Currently available distributions are lognormal, gamma, normal, fixed (delta) and nonparametric. The nonparametric is a special case where the probability mass function is given directly as a numeric vector.

## Examples

```
LogNormal(mean = 4, sd = 1)
#> - lognormal distribution:
#> meanlog:
#> 1.4
#> sdlog:
#> 0.25
LogNormal(mean = 4, sd = 1, max = 10)
#> - lognormal distribution (max: 10):
#> meanlog:
#> 1.4
#> sdlog:
#> 0.25
LogNormal(mean = Normal(4, 1), sd = 1, max = 10)
#> Warning: Uncertain lognormal distribution specified in terms of parameters that are not the "natural" parameters of the distribution (meanlog, sdlog). Converting using a crude and very approximate method that is likely to produce biased results. If possible, it is preferable to specify the distribution directly in terms of the natural parameters.
#> - lognormal distribution (max: 10):
#> meanlog:
#> - normal distribution:
#> mean:
#> 1.4
#> sd:
#> 0.41
#> sdlog:
#> - normal distribution:
#> mean:
#> 0.25
#> sd:
#> 0.18
Gamma(mean = 4, sd = 1)
#> - gamma distribution:
#> shape:
#> 16
#> rate:
#> 4
Gamma(shape = 16, rate = 4)
#> - gamma distribution:
#> shape:
#> 16
#> rate:
#> 4
Gamma(shape = Normal(16, 2), rate = Normal(4, 1))
#> - gamma distribution:
#> shape:
#> - normal distribution:
#> mean:
#> 16
#> sd:
#> 2
#> rate:
#> - normal distribution:
#> mean:
#> 4
#> sd:
#> 1
Gamma(shape = Normal(16, 2), scale = Normal(1/4, 1))
#> Warning: Uncertain gamma distribution specified in terms of parameters that are not the "natural" parameters of the distribution (shape, rate). Converting using a crude and very approximate method that is likely to produce biased results. If possible, it is preferable to specify the distribution directly in terms of the natural parameters.
#> - gamma distribution:
#> shape:
#> - normal distribution:
#> mean:
#> 16
#> sd:
#> 5.8
#> rate:
#> - normal distribution:
#> mean:
#> 4
#> sd:
#> 2.9
Normal(mean = 4, sd = 1)
#> - normal distribution:
#> mean:
#> 4
#> sd:
#> 1
Normal(mean = 4, sd = 1, max = 10)
#> - normal distribution (max: 10):
#> mean:
#> 4
#> sd:
#> 1
Fixed(value = 3)
#> - fixed value:
#> 3
Fixed(value = 3.5)
#> - fixed value:
#> 3.5
NonParametric(c(0.1, 0.3, 0.2, 0.4))
#> - nonparametric distribution
#> PMF: [0.1 0.3 0.2 0.4]
NonParametric(c(0.1, 0.3, 0.2, 0.1, 0.1))
#> - nonparametric distribution
#> PMF: [0.12 0.37 0.25 0.12 0.12]
```