Binomial proportion confidence interval

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https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval 

In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials). In other words, a binomial proportion confidence interval is an interval estimate of a success probability ${\displaystyle p}$ when only the number of experiments ${\displaystyle n}$ and the number of successes ${\displaystyle n_{\mathsf{s}}}$ are known.

There are several formulas for a binomial confidence interval, but all of them rely on the assumption of a binomial distribution. In general, a binomial distribution applies when an experiment is repeated a fixed number of times, each trial of the experiment has two possible outcomes (success and failure), the probability of success is the same for each trial, and the trials are statistically independent. Because the binomial distribution is a discrete probability distribution (i.e., not continuous) and difficult to calculate for large numbers of trials, a variety of approximations are used to calculate this confidence interval, all with their own tradeoffs in accuracy and computational intensity.

A simple example of a binomial distribution is the set of various possible outcomes, and their probabilities, for the number of heads observed when a coin is flipped ten times. The observed binomial proportion is the fraction of the flips that turn out to be heads. Given this observed proportion, the confidence interval for the true probability of the coin landing on heads is a range of possible proportions, which may or may not contain the true proportion. A 95% confidence interval for the proportion, for instance, will contain the true proportion 95% of the times that the procedure for constructing the confidence interval is employed.

Problems with using a normal approximation or "Wald interval"

Plotting the normal approximation interval on an arbitrary logistic curve reveals problems of overshoot and zero-width intervals.

A commonly used formula for a binomial confidence interval relies on approximating the distribution of error about a binomially-distributed observation, ${\displaystyle \hat{p},}$ with a normal distribution. The normal approximation depends on the de Moivre–Laplace theorem (the original, binomial-only version of the central limit theorem) and becomes unreliable when it violates the theorems' premises, as the sample size becomes small or the success probability grows close to either 0 or 1 .

Using the normal approximation, the success probability ${\displaystyle p}$ is estimated by

$${\displaystyle p\approx \hat{p}\pm \frac{z_{\alpha }}{\sqrt{n}}\sqrt{\hat{p}\left(1-\hat{p}\right)},}$$

where ${\displaystyle \hat{p}\equiv \frac{n_{\mathsf{s}}}{n}}$ is the proportion of successes in a Bernoulli trial process and an estimator for ${\displaystyle p}$ in the underlying Bernoulli distribution. The equivalent formula in terms of observation counts is

$${\displaystyle p\approx \frac{n_{\mathsf{s}}}{n}\pm \frac{z_{\alpha }}{\sqrt{n}}\sqrt{\frac{n_{\mathsf{s}}}{n}\frac{n_{\mathsf{f}}}{n}},}$$

where the data are the results of ${\displaystyle n}$ trials that yielded ${\displaystyle n_{\mathsf{s}}}$ successes and ${\displaystyle n_{\mathsf{f}}=n-n_{\mathsf{s}}}$ failures. The distribution function argument ${\displaystyle z_{\alpha }}$ is the ${\displaystyle 1-\frac{\alpha }{2}}$ quantile of a standard normal distribution (i.e., the probit) corresponding to the target error rate ${\displaystyle \alpha .}$ For a 95% confidence level, the error ${\displaystyle \alpha =1-0.95=0.05,}$ so that ${\displaystyle 1-\frac{\alpha }{2}=0.975}$ and ${\displaystyle z_{.05}=1.96.}$

When using the Wald formula to estimate ${\displaystyle p,}$ or just considering the possible outcomes of this calculation, two problems immediately become apparent:

• First, for ${\displaystyle \hat{p}}$ approaching either 1 or 0, the interval narrows to zero width (falsely implying certainty).
• Second, for values of ${\displaystyle \hat{p}<\frac{1}{1+n/z_{\alpha }^2}}$ (probability too low / too close to 0), the interval boundaries exceed ${\displaystyle [0,1]}$ (overshoot).

(Another version of the second, overshoot problem, arises when instead ${\displaystyle 1-\hat{p}}$ falls below the same upper bound: probability too high / too close to 1 .)

An important theoretical derivation of this confidence interval involves the inversion of a hypothesis test. Under this formulation, the confidence interval represents those values of the population parameter that would have large p-values if they were tested as a hypothesized population proportion. The collection of values, ${\displaystyle \theta ,}$ for which the normal approximation is valid can be represented as

$${\displaystyle \left\{\theta |y_{\alpha }\le \frac{\hat{p}-\theta }{\sqrt{\frac{1}{n}\hat{p}\left(1-\hat{p}\right)}}\le z_{\alpha }\right\},}$$

where ${\displaystyle y_{\alpha }}$ is the lower ${\displaystyle \frac{\alpha }{2}}$ quantile of a standard normal distribution, vs. ${\displaystyle z_{\alpha },}$ which is the upper ( i.e., ${\displaystyle 1-\frac{\alpha }{2}}$) quantile.

Since the test in the middle of the inequality is a Wald test, the normal approximation interval is sometimes called the Wald interval or Wald method, after Abraham Wald, but it was first described by Laplace (1812).

Bracketing the confidence interval

Extending the normal approximation and Wald-Laplace interval concepts, Michael Short has shown that inequalities on the approximation error between the binomial distribution and the normal distribution can be used to accurately bracket the estimate of the confidence interval around ${\displaystyle p:}$

$${\displaystyle \frac{k+C_{\mathsf{L}\mathsf{1}}-z_{\alpha }\hat{W}}{n+z_{\alpha }^2}\le p\le \frac{k+C_{\mathsf{U}\mathsf{1}}+z_{\alpha }\hat{W}}{n+z_{\alpha }^2}}$$

with $${\displaystyle \hat{W}\equiv \sqrt{\frac{nk-k^2+C_{\mathsf{L}\mathsf{2}}n-C_{\mathsf{L}\mathsf{3}}k+C_{\mathsf{L}\mathsf{4}}}{n}},}$$

and where ${\displaystyle p}$ is again the (unknown) proportion of successes in a Bernoulli trial process (as opposed to ${\displaystyle \hat{p}\equiv \frac{n_{\mathsf{s}}}{n}}$ that estimates it) measured with ${\displaystyle n}$ trials yielding ${\displaystyle k}$ successes, ${\displaystyle z_{\alpha }}$ is the ${\displaystyle 1-\frac{\alpha }{2}}$ quantile of a standard normal distribution (i.e., the probit) corresponding to the target error rate ${\displaystyle \alpha ,}$ and the constants ${\displaystyle C_{\mathsf{L}\mathsf{1}},}$ ${\displaystyle C_{\mathsf{L}\mathsf{2}},}$ ${\displaystyle C_{\mathsf{L}\mathsf{3}},}$ ${\displaystyle C_{\mathsf{L}\mathsf{4}},}$ ${\displaystyle C_{\mathsf{U}\mathsf{1}},}$ ${\displaystyle C_{\mathsf{U}\mathsf{2}},}$ ${\displaystyle C_{\mathsf{U}\mathsf{3}},}$ and ${\displaystyle C_{\mathsf{U}\mathsf{4}}}$ are simple algebraic functions of ${\displaystyle z_{\alpha }.}$  For a fixed ${\displaystyle \alpha }$ (hence fixed ${\displaystyle z_{\alpha }}$), the above inequalities give easily computed one- or two-sided intervals which bracket the exact binomial upper and lower confidence limits corresponding to the error rate ${\displaystyle \alpha .}$

Standard error of a proportion estimation when using weighted data

Let there be a simple random sample ${\displaystyle X_1,\dots ,X_n}$ where each ${\displaystyle X_i}$ is i.i.d from a Bernoulli( p ) distribution and weight ${\displaystyle w_i}$ is the weight for each observation, with the(positive) weights ${\displaystyle w_i}$ normalized so they sum to 1 . The weighted sample proportion is: $\hat{p}=\sum _{i=1}^n{w}_i{X}_i.$ Since each of the ${\displaystyle X_i}$ is independent from all the others, and each one has variance ${\displaystyle \mathrm{var}\{X_i\}=p\left(1-p\right)}$ for every ${\displaystyle i=1,\dots ,n;}$ the sampling variance of the proportion therefore is:

$${\displaystyle \mathrm{var}\left\{\hat{p}\right\}=\sum _{i=1}^n\mathrm{var}\left\{w_i{X}_i\right\}=p\left(1-p\right)\sum _{i=1}^n{w}_i^2.}$$

The standard error of ${\displaystyle \hat{p}}$ is the square root of this quantity. Because we do not know ${\displaystyle p\left(1-p\right),}$ we have to estimate it. Although there are many possible estimators, a conventional one is to use ${\displaystyle \hat{p},}$ the sample mean, and plug this into the formula. That gives:

$${\displaystyle \mathrm{SE}\left\{\hat{p}\right\}\approx \sqrt{\hat{p}\left(1-\hat{p}\right)\sum _{i=1}^n{w}_i^2}.}$$

For otherwise unweighted data, the effective weights are uniform $w_i=\frac{1}{n},$ giving $\sum _{i=1}^n{w}_i^2=\frac{1}{n}.$ The ${\displaystyle \mathrm{SE}}$ becomes $\sqrt{\frac{1}{n}\hat{p}\left(1-\hat{p}\right)},$ leading to the familiar formulas, showing that the calculation for weighted data is a direct generalization of them.

Wilson score interval

Wilson score intervals plotted on a logistic curve, revealing asymmetry and good performance for small n and where p is at or near 0 or 1.

The Wilson score interval was developed by E.B. Wilson (1927). It is an improvement over the normal approximation interval in multiple respects: Unlike the symmetric normal approximation interval (above), the Wilson score interval is asymmetric, and it doesn't suffer from problems of overshoot and zero-width intervals that afflict the normal interval. It can be safely employed with small samples and skewed observations. The observed coverage probability is consistently closer to the nominal value, ${\displaystyle 1-\alpha .}$

Like the normal interval, the interval can be computed directly from a formula.

Wilson started with the normal approximation to the binomial: $${\displaystyle z_{\alpha }\approx \frac{p-\hat{p}}{{\sigma }_n}}$$ where ${\displaystyle z_{\alpha }}$ is the standard normal interval half-width corresponding to the desired confidence ${\displaystyle 1-\alpha .}$ The analytic formula for a binomial sample standard deviation is $${\displaystyle {\sigma }_n=\sqrt{\frac{p\left(1-p\right)}{n}}.}$$ Combining the two, and squaring out the radical, gives an equation that is quadratic in ${\displaystyle p:}$ $${\displaystyle {\left(p-\hat{p}\right)}^2=\frac{z_{\alpha }^2}{n}p\left(1-p\right)}$$ or $${\displaystyle p^2-2p\hat{p}+{\hat{p}}^2=p\frac{z_{\alpha }^2}{n}-p^2\frac{z_{\alpha }^2}{n}.}$$ Transforming the relation into a standard-form quadratic equation for ${\displaystyle p,}$ treating ${\displaystyle \hat{p}}$ and ${\displaystyle n}$ as known values from the sample (see prior section), and using the value of ${\displaystyle z_{\alpha }}$ that corresponds to the desired confidence ${\displaystyle 1-\alpha }$ for the estimate of ${\displaystyle p}$ gives this: $${\displaystyle \left(1+\frac{z_{\alpha }^2}{n}\right)p^2-\left(2\hat{p}+\frac{z_{\alpha }^2}{n}\right)p+({\hat{p}}^2)=0,}$$ where all of the values bracketed by parentheses are known quantities. The solution for ${\displaystyle p}$ estimates the upper and lower limits of the confidence interval for ${\displaystyle p.}$ Hence the probability of success ${\displaystyle p}$ is estimated by ${\displaystyle \hat{p}}$ and with ${\displaystyle 1-\alpha }$ confidence bracketed in the interval $${\displaystyle p{\underset{\approx }{\in }}_{\alpha }\left(w^{-},w^{+}\right)=\frac{1}{1+z_{\alpha }^2/n}\left(\hat{p}+\frac{z_{\alpha }^2}{2n}\pm \frac{z_{\alpha }}{2n}\sqrt{4n\hat{p}\left(1-\hat{p}\right)+z_{\alpha }^2}\right)}$$

where ${\displaystyle {\underset{\approx }{\in }}_{\alpha }}$ is an abbreviation for

$${\displaystyle \mathbb{P}\left\{p\in \left(w^{-},w^{+}\right)\right\}=1-\alpha .}$$

An equivalent expression using the observation counts ${\displaystyle n_{\mathsf{s}}}$ and ${\displaystyle n_{\mathsf{f}}}$ is $${\displaystyle p{\underset{\approx }{\in }}_{\alpha }\frac{n_{\mathsf{s}}+\frac{1}{2}{z}_{\alpha }^2}{n+z_{\alpha }^2}\pm \frac{z_{\alpha }}{n+z_{\alpha }^2}\sqrt{\frac{n_{\mathsf{s}}{n}_{\mathsf{f}}}{n}+\frac{z_{\alpha }^2}{4}},}$$

with the counts as above: ${\displaystyle n_{\mathsf{s}}\equiv }$ the count of observed "successes", ${\displaystyle n_{\mathsf{f}}\equiv }$ the count of observed "failures", and their sum is the total number of observations ${\displaystyle n=n_{\mathsf{s}}+n_{\mathsf{f}}.}$

In practical tests of the formula's results, users find that this interval has good properties even for a small number of trials and / or the extremes of the probability estimate, ${\displaystyle \hat{p}\equiv \frac{n_{\mathsf{s}}}{n}.}$

Intuitively, the center value of this interval is the weighted average of ${\displaystyle \hat{p}}$ and $\frac{1}{2},$ with ${\displaystyle \hat{p}}$ receiving greater weight as the sample size increases. Formally, the center value corresponds to using a pseudocount of ${\displaystyle \frac{1}{2}{z}_{\alpha }^2,}$ the number of standard deviations of the confidence interval: Add this number to both the count of successes and of failures to yield the estimate of the ratio. For the common two standard deviations in each direction interval (approximately 95% coverage, which itself is approximately 1.96 standard deviations), this yields the estimate ${\displaystyle \frac{n_{\mathsf{s}}+2}{n+4},}$ which is known as the "plus four rule".

Although the quadratic can be solved explicitly, in most cases Wilson's equations can also be solved numerically using the fixed-point iteration $${\displaystyle p_{k+1}=\hat{p}\pm z_{\alpha }\sqrt{\frac{1}{n}{p}_k\left(1-p_k\right)}}$$ with ${\displaystyle p_0=\hat{p}.}$

The Wilson interval can also be derived from the single sample z-test or Pearson's chi-squared test with two categories. The resulting interval,

$${\displaystyle \left\{\theta |y_{\alpha }\le \frac{\hat{p}-\theta }{\sqrt{\frac{1}{n}\theta \left(1-\theta \right)}}\le z_{\alpha }\right\},}$$

(with ${\displaystyle y_{\alpha }}$ the lower ${\displaystyle \alpha }$ quantile) can then be solved for ${\displaystyle \theta }$ to produce the Wilson score interval. The test in the middle of the inequality is a score test.

The interval equality principle

The probability density function (pdf) for the Wilson score interval, plus pdfs at interval bounds. Tail areas are equal.

Since the interval is derived by solving from the normal approximation to the binomial, the Wilson score interval ${\displaystyle \left(w^{-},w^{+}\right)}$ has the property of being guaranteed to obtain the same result as the equivalent z-test or chi-squared test.

This property can be visualised by plotting the probability density function for the Wilson score interval (see Wallis). After that, then also plotting a normal pdf across each bound. The tail areas of the resulting Wilson and normal distributions represent the chance of a significant result, in that direction, must be equal.

The continuity-corrected Wilson score interval and the Clopper-Pearson interval are also compliant with this property. The practical import is that these intervals may be employed as significance tests, with identical results to the source test, and new tests may be derived by geometry.

Wilson score interval with continuity correction

The Wilson interval may be modified by employing a continuity correction, in order to align the minimum coverage probability, rather than the average coverage probability, with the nominal value, ${\displaystyle 1-\alpha .}$

Just as the Wilson interval mirrors Pearson's chi-squared test, the Wilson interval with continuity correction mirrors the equivalent Yates' chi-squared test.

The following formulae for the lower and upper bounds of the Wilson score interval with continuity correction ${\displaystyle \left(w_{\mathsf{c}\mathsf{c}}^{-},w_{\mathsf{c}\mathsf{c}}^{+}\right)}$ are derived from Newcombe:

$${\displaystyle \begin{array}{rl}{w}_{\mathsf{c}\mathsf{c}}^{-}& =max\left\{0,\frac{2n\hat{p}+z_{\alpha }^2-\left[z_{\alpha }\sqrt{z_{\alpha }^2-\frac{1}{n}+4n\hat{p}\left(1-\hat{p}\right)+\left(4\hat{p}-2\right)}+1\right]}{2\left(n+z_{\alpha }^2\right)}\right\},\\ w_{\mathsf{c}\mathsf{c}}^{+}& =min\left\{1,\frac{2n\hat{p}+z_{\alpha }^2+\left[z_{\alpha }\sqrt{z_{\alpha }^2-\frac{1}{n}+4n\hat{p}\left(1-\hat{p}\right)-\left(4\hat{p}-2\right)}+1\right]}{2\left(n+z_{\alpha }^2\right)}\right\},\end{array}}$$ for ${\displaystyle \hat{p}\ne 0}$ and ${\displaystyle \hat{p}\ne 1.}$

If ${\displaystyle \hat{p}=0,}$ then ${\displaystyle w_{\mathsf{c}\mathsf{c}}^{-}}$ must instead be set to ${\displaystyle 0;}$ if ${\displaystyle \hat{p}=1,}$ then ${\displaystyle w_{\mathsf{c}\mathsf{c}}^{+}}$ must be instead set to ${\displaystyle 1.}$

Wallis (2021) identifies a simpler method for computing continuity-corrected Wilson intervals that employs a special function based on Wilson's lower-bound formula: In Wallis' notation, for the lower bound, let $${\displaystyle \mathsf{W}\mathsf{i}\mathsf{l}\mathsf{s}\mathsf{o}{\mathsf{n}}_{\mathsf{l}\mathsf{o}\mathsf{w}\mathsf{e}\mathsf{r}}\left(\hat{p},n,\frac{\alpha }{2}\right)\equiv w^{-}=\frac{1}{1+z_{\alpha }^2/n}(\hat{p}+\frac{z_{\alpha }^2}{2n}-\frac{z_{\alpha }}{2n}\sqrt{4n\hat{p}(1-\hat{p})+z_{\alpha }^2}),}$$

where ${\displaystyle \alpha }$ is the selected tolerable error level for ${\displaystyle z_{\alpha }.}$ Then $${\displaystyle w_{\mathsf{c}\mathsf{c}}^{-}=\mathsf{W}\mathsf{i}\mathsf{l}\mathsf{s}\mathsf{o}{\mathsf{n}}_{\mathsf{l}\mathsf{o}\mathsf{w}\mathsf{e}\mathsf{r}}\left(max\left\{\hat{p}-\frac{1}{2n},0\right\},n,\frac{\alpha }{2}\right).}$$

This method has the advantage of being further decomposable.

Jeffreys interval

Jeffreys intervals plotted on a logistic curve, revealing asymmetry and good performance for small n and where p is at or near 0 or 1.

The Jeffreys interval has a Bayesian derivation, but good frequentist properties (outperforming most frequentist constructions). In particular, it has coverage properties that are similar to those of the Wilson interval, but it is one of the few intervals with the advantage of being equal-tailed (e.g., for a 95% confidence interval, the probabilities of the interval lying above or below the true value are both close to 2.5%). In contrast, the Wilson interval has a systematic bias such that it is centred too close to ${\displaystyle p=0.5}$.

The Jeffreys interval is the Bayesian credible interval obtained when using the non-informative Jeffreys prior for the binomial proportion ${\displaystyle p.}$ The Jeffreys prior for this problem is a Beta distribution with parameters ${\displaystyle \left(\frac{1}{2},\frac{1}{2}\right),}$ a conjugate prior. After observing ${\displaystyle x}$ successes in ${\displaystyle n}$ trials, the posterior distribution for ${\displaystyle p}$ is a Beta distribution with parameters ${\displaystyle \left(x+\frac{1}{2},n-x+\frac{1}{2}\right).}$

When ${\displaystyle x\ne 0}$ and ${\displaystyle x\ne n,}$ the Jeffreys interval is taken to be the ${\displaystyle 100\left(1-\alpha \right)\mathrm{\%}}$ equal-tailed posterior probability interval, i.e., the ${\displaystyle \frac{1}{2}\alpha }$ and ${\displaystyle 1-\frac{1}{2}\alpha }$ quantiles of a Beta distribution with parameters ${\displaystyle \left(x+\frac{1}{2},n-x+\frac{1}{2}\right).}$

In order to avoid the coverage probability tending to zero when ${\displaystyle p\to 0}$ or 1 , when ${\displaystyle x=0}$ the upper limit is calculated as before but the lower limit is set to 0 , and when ${\displaystyle x=n}$ the lower limit is calculated as before but the upper limit is set to 1 .

Jeffreys' interval can also be thought of as a frequentist interval based on inverting the p-value from the G-test after applying the Yates correction to avoid a potentially-infinite value for the test statistic.

Clopper–Pearson interval

The Clopper–Pearson interval is an early and very common method for calculating binomial confidence intervals. This is often called an 'exact' method, as it attains the nominal coverage level in an exact sense, meaning that the coverage level is never less than the nominal ${\displaystyle 1-\alpha }$.

The Clopper–Pearson interval can be written as

$${\displaystyle S_{\le }\cap S_{\ge }}$$

or equivalently,

$${\displaystyle \left(inf{S}_{\ge },sup{S}_{\le }\right)}$$

with

$${\displaystyle S_{\le }\equiv \left\{p|\mathbb{P}\left\{\mathrm{Bin}\left(n;p\right)\le x\right\}>\frac{\alpha }{2}\right\}}$$ and $${\displaystyle S_{\ge }\equiv \left\{p|\mathbb{P}\left\{\mathrm{Bin}\left(n;p\right)\ge x\right\}>\frac{\alpha }{2}\right\},}$$

where ${\displaystyle 0\le x\le n}$ is the number of successes observed in the sample and ${\displaystyle \mathsf{B}\mathsf{i}\mathsf{n}\left(n;p\right)}$ is a binomial random variable with ${\displaystyle n}$ trials and probability of success ${\displaystyle p.}$

Equivalently we can say that the Clopper–Pearson interval is $\left(\frac{x}{n}-{\epsilon }_1,\frac{x}{n}+{\epsilon }_2\right)$ with confidence level ${\displaystyle 1-\alpha }$ if ${\displaystyle {\epsilon }_i}$ is the infimum of those such that the following tests of hypothesis succeed with significance $\frac{\alpha }{2}:$

1. H₀: ${\displaystyle p=\frac{x}{n}-{\epsilon }_1}$ with H_A: ${\displaystyle p>\frac{x}{n}-{\epsilon }_1}$
2. H₀: ${\displaystyle p=\frac{x}{n}+{\epsilon }_2}$ with H_A: ${\displaystyle p<\frac{x}{n}+{\epsilon }_2.}$

Because of a relationship between the binomial distribution and the beta distribution, the Clopper–Pearson interval is sometimes presented in an alternate format that uses quantiles from the beta distribution.

$${\displaystyle B\left(\frac{\alpha }{2};x,n-x+1\right)<p<B\left(1-\frac{\alpha }{2};x+1,n-x\right)}$$

where ${\displaystyle x}$ is the number of successes, ${\displaystyle n}$ is the number of trials, and ${\displaystyle B\left(p;v,w\right)}$ is the pth quantile from a beta distribution with shape parameters ${\displaystyle v}$ and ${\displaystyle w.}$

Thus, ${\displaystyle p_{min}<p<p_{max},}$ where: $${\displaystyle \frac{\mathrm{\Gamma }(n+1)}{\mathrm{\Gamma }(x)\mathrm{\Gamma }(n-x+1)}{\int }_0^{p_{min}}{t}^{x-1}(1-t{)}^{n-x}\mathrm{d}t=\frac{\alpha }{2},}$$ $${\displaystyle \frac{\mathrm{\Gamma }(n+1)}{\mathrm{\Gamma }(x+1)\mathrm{\Gamma }(n-x)}{\int }_0^{p_{max}}{t}^x(1-t{)}^{n-x-1}\mathrm{d}t=1-\frac{\alpha }{2}.}$$ The binomial proportion confidence interval is then ${\displaystyle \left(p_{min},p_{max}\right),}$ as follows from the relation between the Binomial distribution cumulative distribution function and the regularized incomplete beta function.

When ${\displaystyle x}$ is either 0 or ${\displaystyle n,}$ closed-form expressions for the interval bounds are available: when ${\displaystyle x=0}$ the interval is $\left(0,1-{\left(\frac{\alpha }{2}\right)}^{1/n}\right)$ and when ${\displaystyle x=n}$ it is $\left({\left(\frac{\alpha }{2}\right)}^{1/n},1\right).$

The beta distribution is, in turn, related to the F-distribution so a third formulation of the Clopper–Pearson interval can be written using F quantiles:

$${\displaystyle {\left(1+\frac{n-x+1}{xF\left[\frac{\alpha }{2};2x,2(n-x+1)\right]}\right)}^{-1}<p<{\left(1+\frac{n-x}{(x+1)F\left[1-\frac{\alpha }{2};2(x+1),2(n-x)\right]}\right)}^{-1}}$$

where ${\displaystyle x}$ is the number of successes, ${\displaystyle n}$ is the number of trials, and ${\displaystyle F\left(c;d_1,d_2\right)}$ is the ${\displaystyle c}$ quantile from an F-distribution with ${\displaystyle d_1}$ and ${\displaystyle d_2}$ degrees of freedom.

The Clopper–Pearson interval is an 'exact' interval, since it is based directly on the binomial distribution rather than any approximation to the binomial distribution. This interval never has less than the nominal coverage for any population proportion, but that means that it is usually conservative. For example, the true coverage rate of a 95% Clopper–Pearson interval may be well above 95%, depending on ${\displaystyle n}$ and ${\displaystyle p.}$ Thus the interval may be wider than it needs to be to achieve 95% confidence, and wider than other intervals. In contrast, it is worth noting that other confidence interval may have coverage levels that are lower than the nominal ${\displaystyle 1-\alpha ,}$ i.e., the normal approximation (or "standard") interval, Wilson interval, Agresti–Coull interval, etc., with a nominal coverage of 95% may in fact cover less than 95%, even for large sample sizes.

The definition of the Clopper–Pearson interval can also be modified to obtain exact confidence intervals for different distributions. For instance, it can also be applied to the case where the samples are drawn without replacement from a population of a known size, instead of repeated draws of a binomial distribution. In this case, the underlying distribution would be the hypergeometric distribution.

The interval boundaries can be computed with numerical functions qbeta in R and scipy.stats.beta.ppf in Python.

from scipy.stats import beta
import numpy as np

k = 20
n = 400
alpha = 0.05
p_u, p_o = beta.ppf([alpha / 2, 1 - alpha / 2], [k, k + 1], [n - k + 1, n - k])
if np.isnan(p_o):
    p_o = 1
if np.isnan(p_u):
    p_u = 0

Agresti–Coull interval

The Agresti–Coull interval is also another approximate binomial confidence interval.

Given ${\displaystyle n_{\mathsf{s}}}$ successes in ${\displaystyle n}$ trials, define $${\displaystyle \tilde{n}\equiv n+z_{\alpha }^2}$$

and $${\displaystyle \tilde{p}=\frac{1}{\tilde{n}}\left(n_{\mathsf{s}}+\frac{z_{\alpha }^2}{2}\right)}$$

Then, a confidence interval for ${\displaystyle p}$ is given by

$${\displaystyle p\approx \tilde{p}\pm z_{\alpha }\sqrt{\frac{\tilde{p}}{\tilde{n}}\left(1-\tilde{p}\right)}}$$

where ${\displaystyle z_{\alpha }={\mathrm{\Phi }}^{\text{-}1}\left(1-\frac{\alpha }{2}\right)}$ is the quantile of a standard normal distribution, as before (for example, a 95% confidence interval requires ${\displaystyle \alpha =0.05,}$ thereby producing ${\displaystyle z_{.05}=1.96}$). According to Brown, Cai, & DasGupta (2001), taking ${\displaystyle z=2}$ instead of 1.96 produces the "add 2 successes and 2 failures" interval previously described by Agresti & Coull.

This interval can be summarised as employing the centre-point adjustment, ${\displaystyle \tilde{p},}$ of the Wilson score interval, and then applying the Normal approximation to this point.

$${\displaystyle \tilde{p}=\frac{\hat{p}+\frac{z_{\alpha }^2}{2n}}{1+\frac{z_{\alpha }^2}{n}}}$$

Arcsine transformation

Main article: Arcsine transformation

Further information: Cohen's h

The arcsine transformation has the effect of pulling out the ends of the distribution. While it can stabilize the variance (and thus confidence intervals) of proportion data, its use has been criticized in several contexts.

Let ${\displaystyle X}$ be the number of successes in ${\displaystyle n}$ trials and let ${\displaystyle p=\frac{1}{n}X.}$ The variance of ${\displaystyle p}$ is

$${\displaystyle \mathrm{var}\{p\}=\frac{1}{n}p(1-p).}$$

Using the arc sine transform, the variance of the arcsine of ${\displaystyle \sqrt{p}}$ is

$${\displaystyle \mathrm{var}\left\{\arcsin \sqrt{p}\right\}\approx \frac{\mathrm{var}\{p\}}{4p(1-p)}=\frac{p(1-p)}{4np(1-p)}=\frac{1}{4n}.}$$

So, the confidence interval itself has the form

$${\displaystyle {\sin }^2\left(-\frac{z_{\alpha }}{2\sqrt{n}}+\arcsin \sqrt{p}\right)<\theta <{\sin }^2\left(+\frac{z_{\alpha }}{2\sqrt{n}}+\arcsin \sqrt{p}\right),}$$

where ${\displaystyle z_{\alpha }}$ is the ${\displaystyle 1-\frac{\alpha }{2}}$ quantile of a standard normal distribution.

This method may be used to estimate the variance of ${\displaystyle p}$ but its use is problematic when ${\displaystyle p}$ is close to 0 or 1 .

t_a transform

Let ${\displaystyle p}$ be the proportion of successes. For ${\displaystyle 0\le a\le 2,}$

$${\displaystyle t_a=\log \left(\frac{p^a}{(1-p{)}^{2-a}}\right)=a\log p-(2-a)\log (1-p)}$$

This family is a generalisation of the logit transform which is a special case with a = 1 and can be used to transform a proportional data distribution to an approximately normal distribution. The parameter a has to be estimated for the data set.

Rule of three — for when no successes are observed

The rule of three is used to provide a simple way of stating an approximate 95% confidence interval for ${\displaystyle p}$, in the special case that no successes (${\displaystyle \hat{p}=0}$) have been observed. The interval is ${\displaystyle \left(0,\frac{3}{n}\right)}$.

By symmetry, in the case of only successes (${\displaystyle \hat{p}=1}$), the interval is ${\displaystyle \left(1-\frac{3}{n},1\right)}$.

Comparison and discussion

There are several research papers that compare these and other confidence intervals for the binomial proportion.

Both Ross (2003) and Agresti & Coull (1998) point out that exact methods such as the Clopper–Pearson interval may not work as well as some approximations. The normal approximation interval and its presentation in textbooks has been heavily criticised, with many statisticians advocating that it not be used. The principal problems are overshoot (bounds exceed [0, 1]), zero-width intervals at ${\displaystyle \hat{p}=0}$ or 1 (falsely implying certainty), and overall inconsistency with significance testing.

Of the approximations listed above, Wilson score interval methods (with or without continuity correction) have been shown to be the most accurate and the most robust, though some prefer Agresti & Coulls' approach for larger sample sizes. Wilson and Clopper–Pearson methods obtain consistent results with source significance tests, and this property is decisive for many researchers.

Many of these intervals can be calculated in R using packages like binom.

Gödel, Escher, Bach

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

Cover of the first edition

Gödel, Escher, Bach: an Eternal Golden Braid (1979) by Douglas Hofstadter, is a book about the intellectual themes common to the lives and the works of the logician Kurt Gödel, the artist M. C. Escher, and the composer Johann Sebastian Bach, and shows the thematic connections among mathematics, symmetry, and intelligence. Through short stories, illustrations, and analyses, Gödel, Escher, Bach: an Eternal Golden Braid explains how systems acquire meaningful context from the "meaningless" elements that compose a system; self-reference and formal rules; isomorphism; the meaning of communication; how knowledge can be represented and stored; the methods and limitations of symbolic representation; and the notion of "meaning".

As a cognitive scientist, Hofstadter said that Gödel, Escher, Bach is not about the relationships of mathematics, art, and music, but about how cognition emerges from hidden neurological mechanisms, e.g. how individual neurons in the brain coordinate to create a coherent mind.

Gödel, Escher, Bach won the Pulitzer Prize for General Nonfiction and the National Book Award for Science Hardcover.

Structure

Gödel, Escher, Bach is in interweaving narratives and the main chapters alternate dialogues among fictional characters, usually Achilles and the tortoise, first used by Zeno of Elea and later by Lewis Carroll in "What the Tortoise Said to Achilles". These origins are related in the first two dialogues, and later dialogues introduce new characters such as the Crab. These narratives often are self-referential and metafictional.

Word play, such as puns are used to connect ideas, such as the "Magnificrab, Indeed" with Bach's Magnificat in D; "SHRDLU, Toy of Man's Designing" with Bach's "Jesu, Joy of Man's Desiring"; and "Typographical Number Theory", or "TNT", which inevitably reacts explosively when it attempts to make statements about itself. One dialogue contains a story about a genie (from the Arabic "Djinn") and various "tonics" (of both the liquid and musical varieties), which is titled "Djinn and Tonic". Sometimes word play has no significant connection, such as the dialogue "A Mu Offering", which has no close affinity to Bach's The Musical Offering.

One dialogue in the book is written in the form of a crab canon, in which every line before the midpoint corresponds to an identical line past the midpoint. The conversation still makes sense due to uses of common phrases that can be used as either greetings or farewells ("Good day") and the positioning of lines that double as an answer to a question in the next line. Another is a sloth canon, where one character repeats the lines of another, but slower and negated.

Themes

The book contains many instances of recursion and self-reference, where objects and ideas speak about or refer back to themselves. One is Quining, a term Hofstadter invented in homage to Willard Van Orman Quine, referring to programs that produce their own source code. Another is the presence of a fictional author in the index, Egbert B. Gebstadter, a man with initials E, G, and B and a surname that partially matches Hofstadter. A phonograph dubbed "Record Player X" destroys itself by playing a record titled I Cannot Be Played on Record Player X (an analogy to Gödel's incompleteness theorems), an examination of canon form in music, and a discussion of Escher's lithograph of two hands drawing each other.

To describe such self-referencing objects, Hofstadter coins the term "strange loop", a concept he examines in more depth in his follow-up book I Am a Strange Loop. To escape many of the logical contradictions brought about by these self-referencing objects, Hofstadter discusses Zen koans. He attempts to show readers how to perceive reality outside their own experience and embrace such paradoxical questions by rejecting the premise, a strategy also called "unasking".

Elements of computer science such as call stacks are also discussed in Gödel, Escher, Bach, as one dialogue describes the adventures of Achilles and the Tortoise as they make use of "pushing potion" and "popping tonic" involving entering and leaving different layers of reality. The same dialogue has a genie with a lamp containing another genie with another lamp and so on. Subsequent sections discuss the basic tenets of logic, self-referring statements, ("typeless") systems, and even programming. Hofstadter further creates BlooP and FlooP, two simple programming languages, to illustrate his point.

Puzzles

The book is filled with puzzles, including Hofstadter's MU puzzle, which contrasts reasoning within a defined logical system with reasoning about that system. Another example can be found in the chapter titled Contracrostipunctus, which combines the words acrostic and contrapunctus (counterpoint). In this dialogue between Achilles and the Tortoise, the author hints that there is a contrapunctal acrostic in the chapter that refers both to the author (Hofstadter) and Bach. This can be spelled out by taking the first word of each paragraph, to reveal "Hofstadter's Contracrostipunctus Acrostically Backwards Spells J. S. Bach". The second acrostic is found by taking the first letters of the words of the first, and reading them backwards to get "J S Bach", as the acrostic sentence self-referentially states.

Reception and impact

Gödel, Escher, Bach won the Pulitzer Prize for General Nonfiction and the National Book Award for Science Hardcover.

Martin Gardner's July 1979 column in Scientific American stated, "Every few decades, an unknown author brings out a book of such depth, clarity, range, wit, beauty and originality that it is recognized at once as a major literary event."

For Summer 2007, the Massachusetts Institute of Technology created an online course for high school students built around the book.

In its February 19, 2010, investigative summary on the 2001 anthrax attacks, the Federal Bureau of Investigation suggested that Bruce Edwards Ivins was inspired by the book to hide secret codes based upon nucleotide sequences in the anthrax-laced letters he allegedly sent in September and October 2001, using bold letters, as suggested on page 404 of the book. It was also suggested that he attempted to hide the book from investigators by throwing it in the trash.

In 2019, British mathematician Marcus du Sautoy curated a series of events at London's Barbican Centre to celebrate the book's fortieth anniversary.

I Am a Strange Loop

Main article: I Am a Strange Loop

Hofstadter has expressed some frustration with how Gödel, Escher, Bach was received. He felt that readers did not fully grasp that strange loops were supposed to be the central theme of the book, and attributed this confusion to the length of the book and the breadth of the topics covered.

To remedy this issue, Hofstadter published I Am a Strange Loop in 2007, which had a more focused discussion of the idea.

Translation

Hofstadter claims the idea of translating his book "never crossed [his] mind" when he was writing it—but when his publisher brought it up, he was "very excited about seeing [the] book in other languages, especially… French." He knew, however, that "there were a million issues to consider" when translating, since the book relies not only on word-play, but on "structural puns" as well—writing where the form and content of the work mirror each other (such as the "Crab canon" dialogue, which reads almost exactly the same forwards as backwards).

Hofstadter gives an example of translation trouble in the paragraph "Mr. Tortoise, Meet Madame Tortue", saying translators "instantly ran headlong into the conflict between the feminine gender of the French noun tortue and the masculinity of my character, the Tortoise." Hofstadter agreed to the translators' suggestions of naming the French character Madame Tortue, and the Italian version Signorina Tartaruga. Because of other troubles translators might have retaining meaning, Hofstadter "painstakingly went through every sentence of Gödel, Escher, Bach, annotating a copy for translators into any language that might be targeted."

Translation also gave Hofstadter a way to add new meaning and puns. For instance, in Chinese, the subtitle is not a translation of an Eternal Golden Braid, but a seemingly unrelated phrase Jí Yì Bì (集异璧, literally "collection of exotic jades"), which is homophonic to GEB in Chinese. Some material regarding this interplay is in Hofstadter's later book, Le Ton beau de Marot, which is mainly about translation.

Subjectivity and objectivity (philosophy)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Subjectivity_and_objectivity_(philosophy)

The distinction between subjectivity and objectivity is a basic idea of philosophy, particularly epistemology and metaphysics. Various understandings of this distinction have evolved through the work of philosophers over centuries. One basic distinction is:

• Something is subjective if it is dependent on minds (such as biases, perception, emotions, opinions, imaginary objects, or conscious experiences). If a claim is true exclusively when considering the claim from the viewpoint of a sentient being, it is subjectively true. For example, one person may consider the weather to be pleasantly warm, and another person may consider the same weather to be too hot; both views are subjective.
• Something is objective if it can be confirmed or assumed independently of any minds. If a claim is true even when considering it outside the viewpoint of a sentient being, then it may be labelled objectively true. For example, many people would regard "2 + 2 = 4" as an objective statement of mathematics.

Both ideas have been given various and ambiguous definitions by differing sources as the distinction is often a given but not the specific focal point of philosophical discourse. The two words are usually regarded as opposites, though complications regarding the two have been explored in philosophy: for example, the view of particular thinkers that objectivity is an illusion and does not exist at all, or that a spectrum joins subjectivity and objectivity with a gray area in-between, or that the problem of other minds is best viewed through the concept of intersubjectivity, developing since the 20th century.

The distinction between subjectivity and objectivity is often related to discussions of consciousness, agency, personhood, philosophy of mind, philosophy of language, reality, truth, and communication (for example in narrative communication and journalism).

Etymology

The root of the words subjectivity and objectivity are subject and object, philosophical terms that mean, respectively, an observer and a thing being observed. The word subjectivity comes from subject in a philosophical sense, meaning an individual who possesses unique conscious experiences, such as perspectives, feelings, beliefs, and desires, or who (consciously) acts upon or wields power over some other entity (an object).

In different fields

In Ancient philosophy

Aristotle's teacher Plato considered geometry to be a condition of his idealist philosophy concerned with universal truth.  In Plato's Republic, Socrates opposes the sophist Thrasymachus's relativistic account of justice, and argues that justice is mathematical in its conceptual structure, and that ethics was therefore a precise and objective enterprise with impartial standards for truth and correctness, like geometry. The rigorous mathematical treatment Plato gave to moral concepts set the tone for the western tradition of moral objectivism that came after him. His contrasting between objectivity and opinion became the basis for philosophies intent on resolving the questions of reality, truth, and existence. He saw opinions as belonging to the shifting sphere of sensibilities, as opposed to a fixed, eternal and knowable incorporeality. Where Plato distinguished between how we know things and their ontological status, subjectivism such as George Berkeley's depends on perception. In Platonic terms, a criticism of subjectivism is that it is difficult to distinguish between knowledge, opinions, and subjective knowledge.

Platonic idealism is a form of metaphysical objectivism, holding that the ideas exist independently from the individual. Berkeley's empirical idealism, on the other hand, holds that things only exist as they are perceived. Both approaches boast an attempt at objectivity. Plato's definition of objectivity can be found in his epistemology, which is based on mathematics, and his metaphysics, where knowledge of the ontological status of objects and ideas is resistant to change.

In Western philosophy

In Western philosophy, the idea of subjectivity is thought to have its roots in the works of the European Enlightenment thinkers Descartes and Kant though it could also stem as far back as the Ancient Greek philosopher Aristotle's work relating to the soul. The idea of subjectivity is often seen as a peripheral to other philosophical concepts, namely skepticism, individuals and individuality, and existentialism.The questions surrounding subjectivity have to do with whether or not people can escape the subjectivity of their own human existence and whether or not there is an obligation to try to do so.

Important thinkers who focused on this area of study include Descartes, Locke, Kant, Hegel, Kierkegaard, Husserl, Foucault, Derrida, Nagel, and Sartre.

Subjectivity was rejected by Foucault and Derrida in favor of constructionism, but Sartre embraced and continued Descartes' work in the subject by emphasizing subjectivity in phenomenology. Sartre believed that, even within the material force of human society, the ego was an essentially transcendent being—posited, for instance, in his opus Being and Nothingness through his arguments about the 'being-for-others' and the 'for-itself' (i.e., an objective and subjective human being).

The innermost core of subjectivity resides in a unique act of what Fichte called "self-positing", where each subject is a point of absolute autonomy, which means that it cannot be reduced to a moment in the network of causes and effects.

Religion

One way that subjectivity has been conceptualized by philosophers such as Kierkegaard is in the context of religion. Religious beliefs can vary quite extremely from person to person, but people often think that whatever they believe is the truth. Subjectivity as seen by Descartes and Sartre was a matter of what was dependent on consciousness, so, because religious beliefs require the presence of a consciousness that can believe, they must be subjective. This is in contrast to what has been proven by pure logic or hard sciences, which does not depend on the perception of people, and is therefore considered objective. Subjectivity is what relies on personal perception regardless of what is proven or objective.

Many philosophical arguments within this area of study have to do with moving from subjective thoughts to objective thoughts with many different methods employed to get from one to the other along with a variety of conclusions reached. This is exemplified by Descartes deductions that move from reliance on subjectivity to somewhat of a reliance on God for objectivity. Foucault and Derrida denied the idea of subjectivity in favor of their ideas of constructs in order to account for differences in human thought. Instead of focusing on the idea of consciousness and self-consciousness shaping the way humans perceive the world, these thinkers would argue that it is instead the world that shapes humans, so they would see religion less as a belief and more as a cultural construction.

Phenomenology

Others like Husserl and Sartre followed the phenomenological approach. This approach focused on the distinct separation of the human mind and the physical world, where the mind is subjective because it can take liberties like imagination and self-awareness where religion might be examined regardless of any kind of subjectivity. The philosophical conversation around subjectivity remains one that struggles with the epistemological question of what is real, what is made up, and what it would mean to be separated completely from subjectivity.

In epistemology

In opposition to philosopher René Descartes' method of personal deduction, natural philosopher Isaac Newton applied the relatively objective scientific method to look for evidence before forming a hypothesis. Partially in response to Kant's rationalism, logician Gottlob Frege applied objectivity to his epistemological and metaphysical philosophies. If reality exists independently of consciousness, then it would logically include a plurality of indescribable forms. Objectivity requires a definition of truth formed by propositions with truth value. An attempt of forming an objective construct incorporates ontological commitments to the reality of objects.

The importance of perception in evaluating and understanding objective reality is debated in the observer effect of quantum mechanics. Direct or naïve realists rely on perception as key in observing objective reality, while instrumentalists hold that observations are useful in predicting objective reality. The concepts that encompass these ideas are important in the philosophy of science. Philosophies of mind explore whether objectivity relies on perceptual constancy.

In historiography

History as a discipline has wrestled with notions of objectivity from its very beginning. While its object of study is commonly thought to be the past, the only thing historians have to work with are different versions of stories based on individual perceptions of reality and memory.

Several history streams developed to devise ways to solve this dilemma: Historians like Leopold von Ranke (19th century) have advocated for the use of extensive evidence –especially archived physical paper documents– to recover the bygone past, claiming that, as opposed to people's memories, objects remain stable in what they say about the era they witnessed, and therefore represent a better insight into objective reality. In the 20th century, the Annales School emphasized the importance of shifting focus away from the perspectives of influential men –usually politicians around whose actions narratives of the past were shaped–, and putting it on the voices of ordinary people. Postcolonial streams of history challenge the colonial-postcolonial dichotomy and critique Eurocentric academia practices, such as the demand for historians from colonized regions to anchor their local narratives to events happening in the territories of their colonizers to earn credibility. All the streams explained above try to uncover whose voice is more or less truth-bearing and how historians can stitch together versions of it to best explain what "actually happened."

Trouillot

The anthropologist Michel-Rolph Trouillot developed the concepts of historicity 1 and 2 to explain the difference between the materiality of socio-historical processes (H1) and the narratives that are told about the materiality of socio-historical processes (H2). This distinction hints that H1 would be understood as the factual reality that elapses and is captured with the concept of "objective truth", and that H2 is the collection of subjectivities that humanity has stitched together to grasp the past. Debates about positivism, relativism, and postmodernism are relevant to evaluating these concepts' importance and the distinction between them.

In his book "Silencing the past", Trouillot wrote about the power dynamics at play in history-making, outlining four possible moments in which historical silences can be created: (1) making of sources (who gets to know how to write, or to have possessions that are later examined as historical evidence), (2) making of archives (what documents are deemed important to save and which are not, how to classify materials, and how to order them within physical or digital archives), (3) making of narratives (which accounts of history are consulted, which voices are given credibility), and (4) the making of history (the retrospective construction of what The Past is).

Because history (official, public, familial, personal) informs current perceptions and how we make sense of the present, whose voice gets to be included in it –and how– has direct consequences in material socio-historical processes. Thinking of current historical narratives as impartial depictions of the totality of events unfolded in the past by labeling them as "objective" risks sealing historical understanding. Acknowledging that history is never objective and always incomplete has a meaningful opportunity to support social justice efforts. Under said notion, voices that have been silenced are placed on an equal footing to the grand and popular narratives of the world, appreciated for their unique insight of reality through their subjective lens.

In social sciences

Being "Objective" has several meaning, it can refer to value neutrality (impartiality) or to truth (as opposed to subjective opinions).

Subjectivity is an inherently social mode that comes about through innumerable interactions within society. As much as subjectivity is a process of individuation, it is equally a process of socialization, the individual never being isolated in a self-contained environment, but endlessly engaging in interaction with the surrounding world. Culture is a living totality of the subjectivity of any given society constantly undergoing transformation. Subjectivity is both shaped by it and shapes it in turn, but also by other things like the economy, political institutions, communities, as well as the natural world.

Though the boundaries of societies and their cultures are indefinable and arbitrary, the subjectivity inherent in each one is palatable and can be recognized as distinct from others. Subjectivity is in part a particular experience or organization of reality, which includes how one views and interacts with humanity, objects, consciousness, and nature, so the difference between different cultures brings about an alternate experience of existence that forms life in a different manner. A common effect on an individual of this disjunction between subjectivities is culture shock, where the subjectivity of the other culture is considered alien and possibly incomprehensible or even hostile.

Political subjectivity is an emerging concept in social sciences and humanities. Political subjectivity is a reference to the deep embeddedness of subjectivity in the socially intertwined systems of power and meaning. "Politicality", writes Sadeq Rahimi in Meaning, Madness and Political Subjectivity, "is not an added aspect of the subject, but indeed the mode of being of the subject, that is, precisely what the subject is."

Scientific objectivity is practicing science while intentionally reducing partiality, biases, or external influences. Moral objectivity is the concept of moral or ethical codes being compared to one another through a set of universal facts or a universal perspective and not through differing conflicting perspectives.

Journalistic objectivity is the reporting of facts and news with minimal personal bias or in an impartial or politically neutral manner.