Month: September 2021

A nice Limit

From @infseriesbot, prove the identity: \(\gamma_n = \displaystyle\int_1^\infty \frac{\{x\}}{x^2} (n-\log(x)) \log^{n-1} \, dx \).

We have \(\{x\}=x-i \, \text{for} \, i \leq x \leq i+1\),

so

\(\text{rhs}=\displaystyle\sum _{i=1}^{\infty } \displaystyle\int_i^{i+1} \frac{(x-i) (n-\log (x)) \log ^{n-1}(x)}{x^2} \, dx,\)

and since \(\displaystyle\int \frac{(x-i) (n-x \log ) \left(x \log ^{n-1}\right)}{x^2} \, dx=\log ^n(x) \left(-\frac{i}{x}-\frac{\log (x)}{n+1}+1\right)\)

\(\displaystyle \int_i^{i+1} \frac{(x-i) (n-\log (x)) \log ^{n-1}(x)}{x^2} \, dx=\frac{(i+1) \log ^{n+1}(i)+(-(i+1) \log (i+1)+n+1) \log ^n(i+1)}{(i+1) (n+1)}\),

allora:

\(\text{rhs}=\underset{T\to \infty }{\text{lim}}\left(\frac{(n-(T+1) \log (T+1)+1) \log ^n(T+1)}{(n+1) (T+1)}+\sum _{i=2}^T \frac{\log ^n(i)}{i}\right)\),

and since \(\frac{\log ^n(T+1) (n-(T+1) \log (T+1)+1)}{(n+1) (T+1)}\overset{T}{\to}\frac{\log ^{n+1}(T)}{n+1}\),

From the series representation of the Stieltjes Gamma function, \(\gamma_n\):

\(\underset{T\to \infty }{\text{lim}}\left(\sum _{i=1}^T \frac{\log ^n i}{i}-\frac{\log ^{n+1}(T)}{n+1}\right)=\gamma _n\)

Maximum Ignorance Probability, with application to surgery’s error rates

Introduction and Result

A maximum entropy alternative to Bayesian methods for the estimation of independent Bernouilli sums.

Let \(X=\{x_1,x_2,\ldots, x_n\}\), where \(x_i \in \{0,1\}\) be a vector representing an n sample of independent Bernouilli distributed random variables \(\sim \mathcal{B}(p)\). We are interested in the estimation of the probability p.

We propose that the probablity that provides the best statistical overview, \(p_m\) (by reflecting the maximum ignorance point) is

\(p_m= 1-I_{\frac{1}{2}}^{-1}(n-m, m+1)\), (1)

where \(m=\sum_i^n x_i \) and \(I_.(.,.)\) is the beta regularized function.

Comparison to Alternative Methods

EMPIRICAL: The sample frequency corresponding to the “empirical” distribution \(p_s=\mathbb{E}(\frac{1}{n} \sum_i^n x_i)\), which clearly does not provide information for small samples.

BAYESIAN: The standard Bayesian approach is to start with, for prior, the parametrized Beta Distribution \(p \sim Beta(\alpha,\beta)\), which is not trivial: one is contrained by the fact that matching the mean and variance of the Beta distribution constrains the shape of the prior. Then it becomes convenient that the Beta, being a conjugate prior, updates into the same distribution with new parameters. Allora, with n samples and m realizations:

\(p_b \sim Beta(\alpha+m, \beta+n-m)\) (2)

with mean \(p_b = \frac{\alpha +m}{\alpha +\beta +n}\). We will see below how a low variance beta has too much impact on the result.

Derivations

Let \(F_p(x)\) be the CDF of the binomial \( \mathcal{B} in(n,p)\). We are interested in \(\{ p: F_p(x)=q \}\) the maximum entropy probability. First let us figure out the target value q.

To get the maximum entropy probability, we need to maximize \(H_q=-\left(\;q \; log(q) +(1-q)\; log (1-q)\right)\). This is a very standard result: taking the first derivative w.r. to q, \(\log (q)+\log (1-q)=0, 0\leq q\leq 1\) and since \(H_q\) is concave to q, we get \(q =\frac{1}{2}\).

Now we must find p by inverting the CDF. Allora for the general case,

\(p= 1-I_{\frac{1}{2}}^{-1}(n-x,x+1)\).

And note that as in the graph below (thanks to comments below by