Implement different signal priors

[sofia-calgaro]

Some ideas

• [x] flat
• [x] flat on square(S)
• [x] flat on log(S) - can be done with Distributions.jl

[tdixon97]

To do this we need to create custom distributions for the cases we care about (I guess they are not defined in Distributions.jl) This can be done like this:

# Define the custom distribution type
struct CustomDist <: ContinuousUnivariateDistribution end

# Define the support as (0, ∞)
Distributions.support(::CustomDist) = (0.0, Inf)

# Define the PDF (Exponential-like PDF)
Distributions.pdf(::CustomDist, x::Float64) = x > 0 ? exp(-x) : 0

# Define how to draw random samples (inverse CDF method for exponential)
Distributions.rand(::CustomDist) = -log(rand())

# Define the log-PDF
Distributions.logpdf(::CustomDist, x::Float64) = x > 0 ? -x : -Inf

# Define the CDF
Distributions.cdf(::CustomDist, x::Float64) = x > 0 ? 1 - exp(-x) : 0

# Create an instance of the distribution
dist = CustomDist()

[tdixon97]

I tried quickly and it seems to work, Now for the distributions a uniform distribution of log(S) means:

$$ u= \log{(S)}$$

$$g(u) = \text{Uniform}[a,b]$$

So:

$$f(S) dS = g(u) du \implies f(S) = g(u) \frac{du}{dS} = g(u)/s $$

i.e. its a function of $1/S$ but cut from $e^a$ to $e^b$, evidently this distributions cannot be defined to ($S=0$) which is a known thing and we need some cutoff. Similar maths works for $\sqrt{(S)}$ but this time the transformation is to $1/\sqrt{(S)}$. Both of these require a lower cutoff I guess we can see what GERDA used

[tdixon97]

I tried the following code to get a prior flat on log(S): Several of the methods are annoying to define (maths:

• CDF : comes from integrating the PDF
• sampling: comes from inverting the CDF as in $y=CDF(x) \implies x= CDF^{-}(y)$ and replacing y with rand() in the code.

  
  # Define the custom distribution type
  struct MyCustomDist <: ContinuousUnivariateDistribution 
  a::Float64
  b::Float64
  end

Define the support as (0, ∞)

Distributions.support(::MyCustomDist) = (0.0, Inf)

Define the PDF (1/S like)

Distributions.pdf(d::MyCustomDist, x) = (x > d.a) && (x<d.b) ? 1/(x*(log(d.b)-log(d.a))) : 0

Define how to draw random samples (inverse CDF method for exponential)

Distributions.rand(d::MyCustomDist) = exp(rand()*(log(d.b)-log(d.a))+log(d.a))

Define the log-PDF

Distributions.logpdf(d::MyCustomDist, x::Float64) = x > d.a && x<d.b ? -log(x)-log(log(d.b)-log(d.a)) : 0

Define the CDF

Distributions.cdf(d::MyCustomDist, x::Float64) = x < d.a ? 0 : x<d.b ? (log(x)-log(d.a))/(log(d.b)-log(d.a)) : 1

Create an instance of the distribution

dist = MyCustomDist(1,10)

The CDF and PDF (along with some sampled data) look ok, but we have to try to see what happens in BAT.

![image](https://github.com/user-attachments/assets/531af282-76f6-4f62-934a-0df8f6705d61)
![image](https://github.com/user-attachments/assets/397e3bff-9aeb-4133-83bf-75f87b3f0a49)

[tdixon97]

Ok I tried it in BAT didnt seems to work Also found that in Distributions.jl they already have a log uniform: https://juliastats.org/Distributions.jl/stable/univariate/#Continuous-Distributions but the uniform sqrt(x) isnt there. Need to think of other ideas?

[tdixon97]

We can try to follow it replacing the 1/x with 1/sqrt(x)

https://github.com/JuliaStats/Distributions.jl/blob/00b7fad421c606c1f7d58c38ab0114472e76a747/src/univariate/continuous/loguniform.jl#L1-L12