Probability Distributions and their Stories

• Story. We perform a series of Bernoulli trials. The number of failures, $y$, before we get $α$ successes is Negative Binomially distributed. An equivalent story is that the sum of $α$ independent and identically Gamma distributed variables is Negative Binomial distributed.
• Example. Bursty gene expression can give mRNA count distributions that are Negative Binomially distributed. Here, "success" is that a burst in gene expression stops. So, the parameter $β$ is the mean length of a burst in expression. The parameter α is related to the frequency of the bursts. If multiple bursts are possible within the lifetime of mRNA, then $α>1$. Then, the number of "failures" is the number of mRNA transcripts that are made in the characteristic lifetime of mRNA.
• Parameters. There are two parameters: $α$, the desired number of successes, and $β$, which is the mean of the α identical Gamma distributions that give the Negative Binomial. The probability of success of each Bernoulli trial is given by $β/(1+β)$.
• Support. The Negative-Binomial distribution is supported on the set of nonnegative integers.
• Probability mass function.

• Usage

• Related distributions.
  • The Geometric distribution is a special case of the Negative Binomial distribution in which $α=1$ and $θ=β/(1+β)$.
  • The continuous analog of the Negative Binomial distribution is the Gamma distribution.
  • In a certain limit, which is easier implemented using the $(μ,ϕ)$ parametrization below, the Negative Binomial distribution becomes a Poisson distribution.

• Notes.
  • The Negative Binomial distribution may be parametrized such that the probability mass function is
    
    
    $\begin{align}\\ \phantom{blah} f(y;\mu,\phi) = \frac{\Gamma(y+\phi)}{\Gamma(\phi) \, y!}\,\left(\frac{\phi}{\mu+\phi}\right)^\phi\left(\frac{\mu}{\mu+\phi}\right)^y. \\ \phantom{blah}\end{align}$
    
    
    These parameters are related to the parametrization above by $ϕ=α$ and $μ=α/β$. In the limit of $ϕ→∞$, which can be taken for the PMF, the Negative Binomial distribution becomes Poisson with parameter $μ$. This also gives meaning to the parameters $μ$ and $ϕ$. $μ$ is the mean of the Negative Binomial, and $ϕ$ controls extra width of the distribution beyond Poisson. The smaller $ϕ$ is, the broader the distribution.
  • In Stan, the Negative Binomial distribution using the $(μ,ϕ)$ parametrization is called neg_binomial_2.
  • SciPy and NumPy use yet another parametrization. The PMF for SciPy is
    
    
    $\begin{align}\\ \phantom{blah} f(y;n, p) = \frac{\Gamma(y+n)}{\Gamma(n) \, y!}\,p^n \left(1-p\right)^y. \\ \phantom{blah}\end{align}$
    
    
    The parameter $p$ is the probability of success of a Bernoulli trial. The parameters are related to the others we have defined by $n=α=ϕ$ and $p=β/(1+β)=ϕ/(μ+ϕ)$.
    

params = [dict(name='α', start=1, end=20, value=5, step=1),
          dict(name='β', start=0, end=5, value=1, step=0.01)]
app = distribution_plot_app(x_min=0,
                            x_max=50,
                            scipy_dist=st.nbinom,
                            params=params,
                            transform=lambda alpha, beta: (alpha, beta/(1+beta)),
                            x_axis_label='y',
                            title='Negative Binomial')
bokeh.io.show(app, notebook_url=notebook_url)