Consider an experiment that is executed $N$ times. The outcome $A$ ( this could be a single event or a set of events ) occurs in $M$ of these cases. As $N\to\infin$ the ratio $M/N$ tends to a limit which is defined as the probability $p(A)$ of $A$.

The experiment may be repeated $N$ times sequentially or $N$ identical experiments may be carried out in parallel. The set of all $N$ outcomes is called collective or ensemble

Bayesian statistics

$$ P(A|B) = \frac{P(B|A)P(A)}{P(B|A)P(A) + P(B|\overline{A})[1-P(A)]} $$

Subjective probability

$$ P({\rm theo}\,|\,{\rm res})=\frac{P({\rm res}\,|\,{\rm theo})}{P({\rm res}\,|\,{\rm theo})P({\rm theo})+P({\rm res}\,|\,{\rm not~theo})[1-P({\rm theo})]}P({\rm theo}) $$

where “theo” is theory and “res” is result

• If $P({\rm result}\,|\,{\rm theory})=P({\rm result}\,|\,{\rm not~theory})$ there is no information gaines as this results in $P({\rm theory}\,|\,{\rm result})=P({\rm theory})$
• If $P({\rm result}\,|\,{\rm theory})\gg P({\rm result}\,|\,{\rm not~theory})$ which leads to the observation of the result being highly predictive $P({\rm theory}\,|\,{\rm result})\approx 1$

Confidence intervals in estimation

Confidence belt construction

Interpretation:

We can now make the statement that the true value of $\mu$ lies between $\mu_-$ and $\mu_+$ with 90% probability.

🗒️ Note: this is only a statement about the boundaries $\mu_-$ and $\mu_+$ and not about $\mu$ itself

Gaussian confidence intervals

In the case of Gaussian distribution functions with constant mean and standard deviation, the construction becomes very simple. The $x_−$ and $x_+$ curves become straight lines and the limits are obtained simply by $μ_±=x±nσ$, where $n=1$ for 68% confidence level, $n=1.64$ for 90% confidence level, and so on.

Example of confidence intervals

• Binomial confidence intervals

  The confidence interval covered by the range $k_-$ to $k_+$ is at least $C$. This is given by the following

  $$ \begin{aligned}\sum_{k=0}^{k_+}P(k;p,n)&\geq 1-(1-C)/2 \\ \sum_{k=k_-}^{n}P(k;p,n)&\geq 1-(1-C)/2 \end{aligned} $$

  🚴‍♀️ Example: if we are to construct bands with $C=0.9$ these two equations mean that we have to construct one-sided intervals that each cover at lest $0.95$. Their intersection, i.e. the range $k_-$ to $k_+$ will then cover at least $0.9$

  If $m$ successes are observed the limits on the true probability intervals can be assigned with $p_-$ and $p_+$ give by

  $$ \begin{aligned} \sum_{k=m+1}^{n}P(k;p_+,n)&= 1-(1-C)/2 \\ \sum_{k=0}^{m-1}P(k;p_-,n)&= 1-(1-C)/2 \end{aligned} $$

  In practical these are the outward-facing corners of the confidence belt at position $k=m$. There are also known as the Clopper-Pearson confidence limits

• Poisson confidence intervals

  To construct intervals of confidence level $C$, we need the greatest value of $k_-$ that satisfies for a given $\lambda$

  $$ \sum_{k=k_-}^\infty P(k;\lambda)\geq 1-(1-C)/2 $$

  which is equivalent to

  $$ \sum_{k=0}^{k_--1} P(k;\lambda) \leq (1-C)/2 $$

  Accordingly we require the smallest $k_+$ that satisfies for a given $\lambda$

  $$ \sum_{k=0}^{k_+} P(k;\lambda) \geq 1-(1-C)/2 $$