title: Beta distribution

text: In conclusion, the maximum likelihood estimates of the shape parameters of a beta distribution are (in general) a complicated function of the sample geometric mean, and of the sample geometric mean based on "(1−X)", the mirror-image of "X". One may ask, if the variance (in addition to the mean) is necessary to estimate two shape parameters with the method of moments, why is the (logarithmic or geometric) variance not necessary to estimate two shape parameters with the maximum likelihood method, for which only the geometric means suffice? The answer is because the mean does not provide as much information as the geometric mean. For a beta distribution with equal shape parameters "α" = "β", the mean is exactly 1/2, regardless of the value of the shape parameters, and therefore regardless of the value of the statistical dispersion (the variance). On the other hand, the geometric mean of a beta distribution with equal shape parameters "α" = "β", depends on the value of the shape parameters, and therefore it contains more information. Also, the geometric mean of a beta distribution does not satisfy the symmetry conditions satisfied by the mean, therefore, by employing both the geometric mean based on "X" and geometric mean based on (1 − "X"), the maximum likelihood method is able to provide best estimates for both parameters "α" = "β", without need of employing the variance.


title: Beta distribution

text: One can express the joint log likelihood per "N" iid observations in terms of the "sufficient statistics" (the sample geometric means) as follows:


title: Beta distribution

text: We can plot the joint log likelihood per "N" observations for fixed values of the sample geometric means to see the behavior of the likelihood function as a function of the shape parameters α and β. In such a plot, the shape parameter estimators formula_302 correspond to the maxima of the likelihood function. See the accompanying graph that shows that all the likelihood functions intersect at α = β = 1, which corresponds to the values of the shape parameters that give the maximum entropy (the maximum entropy occurs for shape parameters equal to unity: the uniform distribution). It is evident from the plot that the likelihood function gives sharp peaks for values of the shape parameter estimators close to zero, but that for values of the shape parameters estimators greater than one, the likelihood function becomes quite flat, with less defined peaks. Obviously, the maximum likelihood parameter estimation method for the beta distribution becomes less acceptable for larger values of the shape parameter estimators, as the uncertainty in the peak definition increases with the value of the shape parameter estimators. One can arrive at the same conclusion by noticing that the expression for the curvature of the likelihood function is in terms of the geometric variances


title: Beta distribution

text: These variances (and therefore the curvatures) are much larger for small values of the shape parameter α and β. However, for shape parameter values α, β > 1, the variances (and therefore the curvatures) flatten out. Equivalently, this result follows from the Cramér–Rao bound, since the Fisher information matrix components for the beta distribution are these logarithmic variances. The Cramér–Rao bound states that the variance of any "unbiased" estimator formula_322 of α is bounded by the reciprocal of the Fisher information:


title: Beta distribution

text: so the variance of the estimators increases with increasing α and β, as the logarithmic variances decrease.


title: Beta distribution

text: Also one can express the joint log likelihood per "N" iid observations in terms of the digamma function expressions for the logarithms of the sample geometric means as follows:


title: Beta distribution

text: this expression is identical to the negative of the cross-entropy (see section on "Quantities of information (entropy)"). Therefore, finding the maximum of the joint log likelihood of the shape parameters, per "N" iid observations, is identical to finding the minimum of the cross-entropy for the beta distribution, as a function of the shape parameters.


title: Beta distribution

text: The procedure is similar to the one followed in the two unknown parameter case. If "Y", ..., "Y" are independent random variables each having a beta distribution with four parameters, the joint log likelihood function for "N" iid observations is:


title: Beta distribution

text: Finding the maximum with respect to a shape parameter involves taking the partial derivative with respect to the shape parameter and setting the expression equal to zero yielding the maximum likelihood estimator of the shape parameters:


title: Beta distribution

text: these equations can be re-arranged as the following system of four coupled equations (the first two equations are geometric means and the second two equations are the harmonic means) in terms of the maximum likelihood estimates for the four parameters formula_253:


title: Beta distribution

text: The parameters formula_260 are embedded inside the geometric mean expressions in a nonlinear way (to the power 1/"N"). This precludes, in general, a closed form solution, even for an initial value approximation for iteration purposes. One alternative is to use as initial values for iteration the values obtained from the method of moments solution for the four parameter case. Furthermore, the expressions for the harmonic means are well-defined only for formula_355, which precludes a maximum likelihood solution for shape parameters less than unity in the four-parameter case. Fisher's information matrix for the four parameter case is positive-definite only for α, β > 2 (for further discussion, see section on Fisher information matrix, four parameter case), for bell-shaped (symmetric or unsymmetric) beta distributions, with inflection points located to either side of the mode. The following Fisher information components (that represent the expectations of the curvature of the log likelihood function) have singularities at the following values:


title: Beta distribution

text: (for further discussion see section on Fisher information matrix). Thus, it is not possible to strictly carry on the maximum likelihood estimation for some well known distributions belonging to the four-parameter beta distribution family, like the uniform distribution (Beta(1, 1, "a", "c")), and the arcsine distribution (Beta(1/2, 1/2, "a", "c")). N.L.Johnson and S.Kotz ignore the equations for the harmonic means and instead suggest "If a and c are unknown, and maximum likelihood estimators of "a", "c", α and β are required, the above procedure (for the two unknown parameter case, with "X" transformed as "X" = ("Y" − "a")/("c" − "a")) can be repeated using a succession of trial values of "a" and "c", until the pair ("a", "c") for which maximum likelihood (given "a" and "c") is as great as possible, is attained" (where, for the purpose of clarity, their notation for the parameters has been translated into the present notation).


title: Beta distribution

text: Let a random variable X have a probability density "f"("x";"α"). The partial derivative with respect to the (unknown, and to be estimated) parameter α of the log likelihood function is called the score. The second moment of the score is called the Fisher information:


title: Beta distribution

text: The expectation of the score is zero, therefore the Fisher information is also the second moment centered on the mean of the score: the variance of the score.


title: Beta distribution

text: If the log likelihood function is twice differentiable with respect to the parameter α, and under certain regularity conditions, then the Fisher information may also be written as follows (which is often a more convenient form for calculation purposes):


title: Beta distribution

text: Thus, the Fisher information is the negative of the expectation of the second derivative with respect to the parameter α of the log likelihood function. Therefore, Fisher information is a measure of the curvature of the log likelihood function of α. A low curvature (and therefore high radius of curvature), flatter log likelihood function curve has low Fisher information; while a log likelihood function curve with large curvature (and therefore low radius of curvature) has high Fisher information. When the Fisher information matrix is computed at the evaluates of the parameters ("the observed Fisher information matrix") it is equivalent to the replacement of the true log likelihood surface by a Taylor's series approximation, taken as far as the quadratic terms. The word information, in the context of Fisher information, refers to information about the parameters. Information such as: estimation, sufficiency and properties of variances of estimators. The Cramér–Rao bound states that the inverse of the Fisher information is a lower bound on the variance of any estimator of a parameter α:


title: Beta distribution

text: The precision to which one can estimate the estimator of a parameter α is limited by the Fisher Information of the log likelihood function. The Fisher information is a measure of the minimum error involved in estimating a parameter of a distribution and it can be viewed as a measure of the resolving power of an experiment needed to discriminate between two alternative hypothesis of a parameter.


title: Beta distribution

text: then the Fisher information takes the form of an "N"×"N" positive semidefinite symmetric matrix, the Fisher Information Matrix, with typical element:


title: Beta distribution

text: Under certain regularity conditions, the Fisher Information Matrix may also be written in the following form, which is often more convenient for computation:


title: Beta distribution

text: With "X", ..., "X" iid random variables, an "N"-dimensional "box" can be constructed with sides "X", ..., "X". Costa and Cover show that the (Shannon) differential entropy "h"("X") is related to the volume of the typical set (having the sample entropy close to the true entropy), while the Fisher information is related to the surface of this typical set.


title: Beta distribution

text: For "X", ..., "X" independent random variables each having a beta distribution parametrized with shape parameters "α" and "β", the joint log likelihood function for "N" iid observations is:


title: Beta distribution

text: For the two parameter case, the Fisher information has 4 components: 2 diagonal and 2 off-diagonal. Since the Fisher information matrix is symmetric, one of these off diagonal components is independent. Therefore, the Fisher information matrix has 3 independent components (2 diagonal and 1 off diagonal).


title: Beta distribution

text: Aryal and Nadarajah calculated Fisher's information matrix for the four-parameter case, from which the two parameter case can be obtained as follows:


title: Beta distribution

text: The Fisher information components are equal to the log geometric variances and log geometric covariance. Therefore, they can be expressed as trigamma functions, denoted ψ(α), the second of the polygamma functions, defined as the derivative of the digamma function:


title: Beta distribution

text: These derivatives are also derived in the and plots of the log likelihood function are also shown in that section. contains plots and further discussion of the Fisher information matrix components: the log geometric variances and log geometric covariance as a function of the shape parameters α and β. contains formulas for moments of logarithmically transformed random variables. Images for the Fisher information components formula_373 and formula_374 are shown in .


title: Beta distribution

text: The determinant of Fisher's information matrix is of interest (for example for the calculation of Jeffreys prior probability). From the expressions for the individual components of the Fisher information matrix, it follows that the determinant of Fisher's (symmetric) information matrix for the beta distribution is:


title: Beta distribution

text: From Sylvester's criterion (checking whether the diagonal elements are all positive), it follows that the Fisher information matrix for the two parameter case is positive-definite (under the standard condition that the shape parameters are positive "α" > 0 and "β" > 0).


title: Beta distribution

text: If "Y", ..., "Y" are independent random variables each having a beta distribution with four parameters: the exponents "α" and "β", and also "a" (the minimum of the distribution range), and "c" (the maximum of the distribution range) (section titled "Alternative parametrizations", "Four parameters"), with probability density function:


title: Beta distribution

text: For the four parameter case, the Fisher information has 4*4=16 components. It has 12 off-diagonal components = (4×4 total − 4 diagonal). Since the Fisher information matrix is symmetric, half of these components (12/2=6) are independent. Therefore, the Fisher information matrix has 6 independent off-diagonal + 4 diagonal = 10 independent components. Aryal and Nadarajah calculated Fisher's information matrix for the four parameter case as follows:


title: Beta distribution

text: In the above expressions, the use of "X" instead of "Y" in the expressions var[ln("X")] = ln(var) is "not an error". The expressions in terms of the log geometric variances and log geometric covariance occur as functions of the two parameter "X" ~ Beta("α", "β") parametrization because when taking the partial derivatives with respect to the exponents ("α", "β") in the four parameter case, one obtains the identical expressions as for the two parameter case: these terms of the four parameter Fisher information matrix are independent of the minimum "a" and maximum "c" of the distribution's range. The only non-zero term upon double differentiation of the log likelihood function with respect to the exponents "α" and "β" is the second derivative of the log of the beta function: ln(B("α", "β")). This term is independent of the minimum "a" and maximum "c" of the distribution's range. Double differentiation of this term results in trigamma functions. The sections titled "Maximum likelihood", "Two unknown parameters" and "Four unknown parameters" also show this fact.


title: Beta distribution

text: The Fisher information for "N" i.i.d. samples is "N" times the individual Fisher information (eq. 11.279, page 394 of Cover and Thomas). (Aryal and Nadarajah take a single observation, "N" = 1, to calculate the following components of the Fisher information, which leads to the same result as considering the derivatives of the log likelihood per "N" observations. Moreover, below the erroneous expression for formula_381 in Aryal and Nadarajah has been corrected.)


title: Beta distribution

text: The lower two diagonal entries of the Fisher information matrix, with respect to the parameter "a" (the minimum of the distribution's range): formula_383, and with respect to the parameter "c" (the maximum of the distribution's range): formula_384 are only defined for exponents α > 2 and β > 2 respectively. The Fisher information matrix component formula_383 for the minimum "a" approaches infinity for exponent α approaching 2 from above, and the Fisher information matrix component formula_384 for the maximum "c" approaches infinity for exponent β approaching 2 from above.


title: Beta distribution

text: The Fisher information matrix for the four parameter case does not depend on the individual values of the minimum "a" and the maximum "c", but only on the total range ("c"−"a"). Moreover, the components of the Fisher information matrix that depend on the range ("c"−"a"), depend only through its inverse (or the square of the inverse), such that the Fisher information decreases for increasing range ("c"−"a").


title: Beta distribution

text: The accompanying images show the Fisher information components formula_383 and formula_388. Images for the Fisher information components formula_389 and formula_390 are shown in . All these Fisher information components look like a basin, with the "walls" of the basin being located at low values of the parameters.


title: Beta distribution

text: The following four-parameter-beta-distribution Fisher information components can be expressed in terms of the two-parameter: "X" ~ Beta(α, β) expectations of the transformed ratio ((1-"X")/"X") and of its mirror image ("X"/(1-"X")), scaled by the range ("c"−"a"), which may be helpful for interpretation:


title: Beta distribution

text: These are also the expected values of the "inverted beta distribution" or beta prime distribution (also known as beta distribution of the second kind or Pearson's Type VI) and its mirror image, scaled by the range ("c" − "a").


title: Beta distribution

text: Also, the following Fisher information components can be expressed in terms of the harmonic (1/X) variances or of variances based on the ratio transformed variables ((1-X)/X) as follows:


title: Beta distribution

text: See section "Moments of linearly transformed, product and inverted random variables" for these expectations.


title: Beta distribution

text: The determinant of Fisher's information matrix is of interest (for example for the calculation of Jeffreys prior probability). From the expressions for the individual components, it follows that the determinant of Fisher's (symmetric) information matrix for the beta distribution with four parameters is:


title: Beta distribution

text: Using Sylvester's criterion (checking whether the diagonal elements are all positive), and since diagonal components formula_381 and formula_396 have singularities at α=2 and β=2 it follows that the Fisher information matrix for the four parameter case is positive-definite for α>2 and β>2. Since for α > 2 and β > 2 the beta distribution is (symmetric or unsymmetric) bell shaped, it follows that the Fisher information matrix is positive-definite only for bell-shaped (symmetric or unsymmetric) beta distributions, with inflection points located to either side of the mode. Thus, important well known distributions belonging to the four-parameter beta distribution family, like the parabolic distribution (Beta(2,2,a,c)) and the uniform distribution (Beta(1,1,a,c)) have Fisher information components (formula_397) that blow up (approach infinity) in the four-parameter case (although their Fisher information components are all defined for the two parameter case). The four-parameter Wigner semicircle distribution (Beta(3/2,3/2,"a","c")) and arcsine distribution (Beta(1/2,1/2,"a","c")) have negative Fisher information determinants for the four-parameter case.


title: Beta distribution

text: The use of Beta distributions in Bayesian inference is due to the fact that they provide a family of conjugate prior probability distributions for binomial (including Bernoulli) and geometric distributions. The domain of the beta distribution can be viewed as a probability, and in fact the beta distribution is often used to describe the distribution of a probability value "p":


title: Beta distribution

text: Examples of beta distributions used as prior probabilities to represent ignorance of prior parameter values in Bayesian inference are Beta(1,1), Beta(0,0) and Beta(1/2,1/2).


title: Beta distribution

text: A classic application of the beta distribution is the rule of succession, introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem. It states that, given "s" successes in "n" conditionally independent Bernoulli trials with probability "p," that the estimate of the expected value in the next trial is formula_399. This estimate is the expected value of the posterior distribution over "p," namely Beta("s"+1, "n"−"s"+1), which is given by Bayes' rule if one assumes a uniform prior probability over "p" (i.e., Beta(1, 1)) and then observes that "p" generated "s" successes in "n" trials. Laplace's rule of succession has been criticized by prominent scientists. R. T. Cox described Laplace's application of the rule of succession to the sunrise problem ( p. 89) as "a travesty of the proper use of the principle." Keynes remarks ( Ch.XXX, p. 382) "indeed this is so foolish a theorem that to entertain it is discreditable." Karl Pearson showed that the probability that the next ("n" + 1) trials will be successes, after n successes in n trials, is only 50%, which has been considered too low by scientists like Jeffreys and unacceptable as a representation of the scientific process of experimentation to test a proposed scientific law. As pointed out by Jeffreys ( p. 128) (crediting C. D. Broad ) Laplace's rule of succession establishes a high probability of success ((n+1)/(n+2)) in the next trial, but only a moderate probability (50%) that a further sample (n+1) comparable in size will be equally successful. As pointed out by Perks, "The rule of succession itself is hard to accept. It assigns a probability to the next trial which implies the assumption that the actual run observed is an average run and that we are always at the end of an average run. It would, one would think, be more reasonable to assume that we were in the middle of an average run. Clearly a higher value for both probabilities is necessary if they are to accord with reasonable belief." These problems with Laplace's rule of succession motivated Haldane, Perks, Jeffreys and others to search for other forms of prior probability (see the next ). According to Jaynes, the main problem with the rule of succession is that it is not valid when s=0 or s=n (see rule of succession, for an analysis of its validity).


title: Beta distribution

text: The beta distribution achieves maximum differential entropy for Beta(1,1): the uniform probability density, for which all values in the domain of the distribution have equal density. This uniform distribution Beta(1,1) was suggested ("with a great deal of doubt") by Thomas Bayes as the prior probability distribution to express ignorance about the correct prior distribution. This prior distribution was adopted (apparently, from his writings, with little sign of doubt) by Pierre-Simon Laplace, and hence it was also known as the "Bayes-Laplace rule" or the "Laplace rule" of "inverse probability" in publications of the first half of the 20th century. In the later part of the 19th century and early part of the 20th century, scientists realized that the assumption of uniform "equal" probability density depended on the actual functions (for example whether a linear or a logarithmic scale was most appropriate) and parametrizations used. In particular, the behavior near the ends of distributions with finite support (for example near "x" = 0, for a distribution with initial support at "x" = 0) required particular attention. Keynes ( Ch.XXX, p. 381) criticized the use of Bayes's uniform prior probability (Beta(1,1)) that all values between zero and one are equiprobable, as follows: "Thus experience, if it shows anything, shows that there is a very marked clustering of statistical ratios in the neighborhoods of zero and unity, of those for positive theories and for correlations between positive qualities in the neighborhood of zero, and of those for negative theories and for correlations between negative qualities in the neighborhood of unity. "


title: Beta distribution

text: The Beta(0,0) distribution was proposed by J.B.S. Haldane, who suggested that the prior probability representing complete uncertainty should be proportional to "p"(1−"p"). The function "p"(1−"p") can be viewed as the limit of the numerator of the beta distribution as both shape parameters approach zero: α, β → 0. The Beta function (in the denominator of the beta distribution) approaches infinity, for both parameters approaching zero, α, β → 0. Therefore, "p"(1−"p") divided by the Beta function approaches a 2-point Bernoulli distribution with equal probability 1/2 at each end, at 0 and 1, and nothing in between, as α, β → 0. A coin-toss: one face of the coin being at 0 and the other face being at 1. The Haldane prior probability distribution Beta(0,0) is an "improper prior" because its integration (from 0 to 1) fails to strictly converge to 1 due to the singularities at each end. However, this is not an issue for computing posterior probabilities unless the sample size is very small. Furthermore, Zellner points out that on the log-odds scale, (the logit transformation ln("p"/1−"p")), the Haldane prior is the uniformly flat prior. The fact that a uniform prior probability on the logit transformed variable ln("p"/1−"p") (with domain (-∞, ∞)) is equivalent to the Haldane prior on the domain [0, 1] was pointed out by Harold Jeffreys in the first edition (1939) of his book Theory of Probability ( p. 123). Jeffreys writes "Certainly if we take the Bayes-Laplace rule right up to the extremes we are led to results that do not correspond to anybody's way of thinking. The (Haldane) rule d"x"/("x"(1−"x")) goes too far the other way. It would lead to the conclusion that if a sample is of one type with respect to some property there is a probability 1 that the whole population is of that type." The fact that "uniform" depends on the parametrization, led Jeffreys to seek a form of prior that would be invariant under different parametrizations.


title: Beta distribution

text: Harold Jeffreys proposed to use an uninformative prior probability measure that should be invariant under reparameterization: proportional to the square root of the determinant of Fisher's information matrix. For the Bernoulli distribution, this can be shown as follows: for a coin that is "heads" with probability "p" ∈ [0, 1] and is "tails" with probability 1 − "p", for a given (H,T) ∈ {(0,1), (1,0)} the probability is "p"(1 − "p"). Since "T" = 1 − "H", the Bernoulli distribution is "p"(1 − "p"). Considering "p" as the only parameter, it follows that the log likelihood for the Bernoulli distribution is


title: Beta distribution

text: The Fisher information matrix has only one component (it is a scalar, because there is only one parameter: "p"), therefore:


title: Beta distribution

text: Thus, for the Bernoulli, and Binomial distributions, Jeffreys prior is proportional to formula_403, which happens to be proportional to a beta distribution with domain variable "x" = "p", and shape parameters α = β = 1/2, the arcsine distribution:


title: Beta distribution

text: It will be shown in the next section that the normalizing constant for Jeffreys prior is immaterial to the final result because the normalizing constant cancels out in Bayes theorem for the posterior probability. Hence Beta(1/2,1/2) is used as the Jeffreys prior for both Bernoulli and binomial distributions. As shown in the next section, when using this expression as a prior probability times the likelihood in Bayes theorem, the posterior probability turns out to be a beta distribution. It is important to realize, however, that Jeffreys prior is proportional to formula_403 for the Bernoulli and binomial distribution, but not for the beta distribution. Jeffreys prior for the beta distribution is given by the determinant of Fisher's information for the beta distribution, which, as shown in the is a function of the trigamma function ψ of shape parameters α and β as follows:


title: Beta distribution

text: As previously discussed, Jeffreys prior for the Bernoulli and binomial distributions is proportional to the arcsine distribution Beta(1/2,1/2), a one-dimensional "curve" that looks like a basin as a function of the parameter "p" of the Bernoulli and binomial distributions. The walls of the basin are formed by "p" approaching the singularities at the ends "p" → 0 and "p" → 1, where Beta(1/2,1/2) approaches infinity. Jeffreys prior for the beta distribution is a "2-dimensional surface" (embedded in a three-dimensional space) that looks like a basin with only two of its walls meeting at the corner α = β = 0 (and missing the other two walls) as a function of the shape parameters α and β of the beta distribution. The two adjoining walls of this 2-dimensional surface are formed by the shape parameters α and β approaching the singularities (of the trigamma function) at α, β → 0. It has no walls for α, β → ∞ because in this case the determinant of Fisher's information matrix for the beta distribution approaches zero.


title: Beta distribution

text: It will be shown in the next section that Jeffreys prior probability results in posterior probabilities (when multiplied by the binomial likelihood function) that are intermediate between the posterior probability results of the Haldane and Bayes prior probabilities.


title: Beta distribution

text: Jeffreys prior may be difficult to obtain analytically, and for some cases it just doesn't exist (even for simple distribution functions like the asymmetric triangular distribution). Berger, Bernardo and Sun, in a 2009 paper defined a reference prior probability distribution that (unlike Jeffreys prior) exists for the asymmetric triangular distribution. They cannot obtain a closed-form expression for their reference prior, but numerical calculations show it to be nearly perfectly fitted by the (proper) prior


title: Beta distribution

text: where θ is the vertex variable for the asymmetric triangular distribution with support [0, 1] (corresponding to the following parameter values in Wikipedia's article on the triangular distribution: vertex "c" = "θ", left end "a" = 0,and right end "b" = 1). Berger et al. also give a heuristic argument that Beta(1/2,1/2) could indeed be the exact Berger–Bernardo–Sun reference prior for the asymmetric triangular distribution. Therefore, Beta(1/2,1/2) not only is Jeffreys prior for the Bernoulli and binomial distributions, but also seems to be the Berger–Bernardo–Sun reference prior for the asymmetric triangular distribution (for which the Jeffreys prior does not exist), a distribution used in project management and PERT analysis to describe the cost and duration of project tasks.


title: Beta distribution

text: Clarke and Barron prove that, among continuous positive priors, Jeffreys prior (when it exists) asymptotically maximizes Shannon's mutual information between a sample of size n and the parameter, and therefore "Jeffreys prior is the most uninformative prior" (measuring information as Shannon information). The proof rests on an examination of the Kullback–Leibler divergence between probability density functions for iid random variables.


title: Beta distribution

text: If samples are drawn from the population of a random variable "X" that result in "s" successes and "f" failures in "n" Bernoulli trials "n" = "s" + "f", then the likelihood function for parameters "s" and "f" given "x" = "p" (the notation "x" = "p" in the expressions below will emphasize that the domain "x" stands for the value of the parameter "p" in the binomial distribution), is the following binomial distribution:


title: Beta distribution

text: If beliefs about prior probability information are reasonably well approximated by a beta distribution with parameters "α" Prior and "β" Prior, then:


title: Beta distribution

text: According to Bayes' theorem for a continuous event space, the posterior probability is given by the product of the prior probability and the likelihood function (given the evidence "s" and "f" = "n" − "s"), normalized so that the area under the curve equals one, as follows:


title: Beta distribution

text: appears both in the numerator and the denominator of the posterior probability, and it does not depend on the integration variable "x", hence it cancels out, and it is irrelevant to the final result. Similarly the normalizing factor for the prior probability, the beta function B(αPrior,βPrior) cancels out and it is immaterial to the final result. The same posterior probability result can be obtained if one uses an un-normalized prior


title: Beta distribution

text: because the normalizing factors all cancel out. Several authors (including Jeffreys himself) thus use an un-normalized prior formula since the normalization constant cancels out. The numerator of the posterior probability ends up being just the (un-normalized) product of the prior probability and the likelihood function, and the denominator is its integral from zero to one. The beta function in the denominator, B("s" + "α" Prior, "n" − "s" + "β" Prior), appears as a normalization constant to ensure that the total posterior probability integrates to unity.


title: Beta distribution

text: The ratio "s"/"n" of the number of successes to the total number of trials is a sufficient statistic in the binomial case, which is relevant for the following results.


title: Beta distribution

text: From the above expressions it follows that for "s"/"n" = 1/2) all the above three prior probabilities result in the identical location for the posterior probability mean = mode = 1/2. For "s"/"n" < 1/2, the mean of the posterior probabilities, using the following priors, are such that: mean for Bayes prior > mean for Jeffreys prior > mean for Haldane prior. For "s"/"n" > 1/2 the order of these inequalities is reversed such that the Haldane prior probability results in the largest posterior mean. The "Haldane" prior probability Beta(0,0) results in a posterior probability density with "mean" (the expected value for the probability of success in the "next" trial) identical to the ratio "s"/"n" of the number of successes to the total number of trials. Therefore, the Haldane prior results in a posterior probability with expected value in the next trial equal to the maximum likelihood. The "Bayes" prior probability Beta(1,1) results in a posterior probability density with "mode" identical to the ratio "s"/"n" (the maximum likelihood).


title: Beta distribution

text: In the case that 100% of the trials have been successful "s" = "n", the "Bayes" prior probability Beta(1,1) results in a posterior expected value equal to the rule of succession ("n" + 1)/("n" + 2), while the Haldane prior Beta(0,0) results in a posterior expected value of 1 (absolute certainty of success in the next trial). Jeffreys prior probability results in a posterior expected value equal to ("n" + 1/2)/("n" + 1). Perks (p. 303) points out: "This provides a new rule of succession and expresses a 'reasonable' position to take up, namely, that after an unbroken run of n successes we assume a probability for the next trial equivalent to the assumption that we are about half-way through an average run, i.e. that we expect a failure once in (2"n" + 2) trials. The Bayes–Laplace rule implies that we are about at the end of an average run or that we expect a failure once in ("n" + 2) trials. The comparison clearly favours the new result (what is now called Jeffreys prior) from the point of view of 'reasonableness'."


title: Beta distribution

text: Conversely, in the case that 100% of the trials have resulted in failure ("s" = 0), the "Bayes" prior probability Beta(1,1) results in a posterior expected value for success in the next trial equal to 1/("n" + 2), while the Haldane prior Beta(0,0) results in a posterior expected value of success in the next trial of 0 (absolute certainty of failure in the next trial). Jeffreys prior probability results in a posterior expected value for success in the next trial equal to (1/2)/("n" + 1), which Perks (p. 303) points out: "is a much more reasonably remote result than the Bayes-Laplace result 1/("n" + 2)".


title: Beta distribution

text: Jaynes questions (for the uniform prior Beta(1,1)) the use of these formulas for the cases "s" = 0 or "s" = "n" because the integrals do not converge (Beta(1,1) is an improper prior for "s" = 0 or "s" = "n"). In practice, the conditions 0 (p. 303) shows that, for what is now known as the Jeffreys prior, this probability is (("n" + 1/2)/("n" + 1))(("n" + 3/2)/("n" + 2))...(2"n" + 1/2)/(2"n" + 1), which for "n" = 1, 2, 3 gives 15/24, 315/480, 9009/13440; rapidly approaching a limiting value of formula_416 as n tends to infinity. Perks remarks that what is now known as the Jeffreys prior: "is clearly more 'reasonable' than either the Bayes-Laplace result or the result on the (Haldane) alternative rule rejected by Jeffreys which gives certainty as the probability. It clearly provides a very much better correspondence with the process of induction. Whether it is 'absolutely' reasonable for the purpose, i.e. whether it is yet large enough, without the absurdity of reaching unity, is a matter for others to decide. But it must be realized that the result depends on the assumption of complete indifference and absence of knowledge prior to the sampling experiment."


title: Beta distribution

text: Following are the variances of the posterior distribution obtained with these three prior probability distributions:


title: Beta distribution

text: So, as remarked by Silvey, for large "n", the variance is small and hence the posterior distribution is highly concentrated, whereas the assumed prior distribution was very diffuse. This is in accord with what one would hope for, as vague prior knowledge is transformed (through Bayes theorem) into a more precise posterior knowledge by an informative experiment. For small "n" the Haldane Beta(0,0) prior results in the largest posterior variance while the Bayes Beta(1,1) prior results in the more concentrated posterior. Jeffreys prior Beta(1/2,1/2) results in a posterior variance in between the other two. As "n" increases, the variance rapidly decreases so that the posterior variance for all three priors converges to approximately the same value (approaching zero variance as "n" → ∞). Recalling the previous result that the "Haldane" prior probability Beta(0,0) results in a posterior probability density with "mean" (the expected value for the probability of success in the "next" trial) identical to the ratio s/n of the number of successes to the total number of trials, it follows from the above expression that also the "Haldane" prior Beta(0,0) results in a posterior with "variance" identical to the variance expressed in terms of the max. likelihood estimate s/n and sample size (in ):


title: Beta distribution

text: In Bayesian inference, using a prior distribution Beta("α"Prior,"β"Prior) prior to a binomial distribution is equivalent to adding ("α"Prior − 1) pseudo-observations of "success" and ("β"Prior − 1) pseudo-observations of "failure" to the actual number of successes and failures observed, then estimating the parameter "p" of the binomial distribution by the proportion of successes over both real- and pseudo-observations. A uniform prior Beta(1,1) does not add (or subtract) any pseudo-observations since for Beta(1,1) it follows that ("α"Prior − 1) = 0 and ("β"Prior − 1) = 0. The Haldane prior Beta(0,0) subtracts one pseudo observation from each and Jeffreys prior Beta(1/2,1/2) subtracts 1/2 pseudo-observation of success and an equal number of failure. This subtraction has the effect of smoothing out the posterior distribution. If the proportion of successes is not 50% ("s"/"n" ≠ 1/2) values of "α"Prior and "β"Prior less than 1 (and therefore negative ("α"Prior − 1) and ("β"Prior − 1)) favor sparsity, i.e. distributions where the parameter "p" is closer to either 0 or 1. In effect, values of "α"Prior and "β"Prior between 0 and 1, when operating together, function as a concentration parameter.


title: Beta distribution

text: The accompanying plots show the posterior probability density functions for sample sizes "n" ∈ {3,10,50}, successes "s" ∈ {"n"/2,"n"/4} and Beta("α"Prior,"β"Prior) ∈ {Beta(0,0),Beta(1/2,1/2),Beta(1,1)}. Also shown are the cases for "n" = {4,12,40}, success "s" = {"n"/4} and Beta("α"Prior,"β"Prior) ∈ {Beta(0,0),Beta(1/2,1/2),Beta(1,1)}. The first plot shows the symmetric cases, for successes "s" ∈ {n/2}, with mean = mode = 1/2 and the second plot shows the skewed cases "s" ∈ {"n"/4}. The images show that there is little difference between the priors for the posterior with sample size of 50 (characterized by a more pronounced peak near "p" = 1/2). Significant differences appear for very small sample sizes (in particular for the flatter distribution for the degenerate case of sample size = 3). Therefore, the skewed cases, with successes "s" = {"n"/4}, show a larger effect from the choice of prior, at small sample size, than the symmetric cases. For symmetric distributions, the Bayes prior Beta(1,1) results in the most "peaky" and highest posterior distributions and the Haldane prior Beta(0,0) results in the flattest and lowest peak distribution. The Jeffreys prior Beta(1/2,1/2) lies in between them. For nearly symmetric, not too skewed distributions the effect of the priors is similar. For very small sample size (in this case for a sample size of 3) and skewed distribution (in this example for "s" ∈ {"n"/4}) the Haldane prior can result in a reverse-J-shaped distribution with a singularity at the left end. However, this happens only in degenerate cases (in this example "n" = 3 and hence "s" = 3/4 < 1, a degenerate value because s should be greater than unity in order for the posterior of the Haldane prior to have a mode located between the ends, and because "s" = 3/4 is not an integer number, hence it violates the initial assumption of a binomial distribution for the likelihood) and it is not an issue in generic cases of reasonable sample size (such that the condition 1 < "s" < "n" − 1, necessary for a mode to exist between both ends, is fulfilled).


title: Beta distribution

text: In Chapter 12 (p. 385) of his book, Jaynes asserts that the "Haldane prior" Beta(0,0) describes a "prior state of knowledge of complete ignorance", where we are not even sure whether it is physically possible for an experiment to yield either a success or a failure, while the "Bayes (uniform) prior Beta(1,1) applies if" one knows that "both binary outcomes are possible". Jaynes states: ""interpret the Bayes-Laplace (Beta(1,1)) prior as describing not a state of complete ignorance", but the state of knowledge in which we have observed one success and one failure...once we have seen at least one success and one failure, then we know that the experiment is a true binary one, in the sense of physical possibility." Jaynes does not specifically discuss Jeffreys prior Beta(1/2,1/2) (Jaynes discussion of "Jeffreys prior" on pp. 181, 423 and on chapter 12 of Jaynes book refers instead to the improper, un-normalized, prior "1/"p" "dp"" introduced by Jeffreys in the 1939 edition of his book, seven years before he introduced what is now known as Jeffreys' invariant prior: the square root of the determinant of Fisher's information matrix. ""1/p" is Jeffreys' (1946) invariant prior for the exponential distribution, not for the Bernoulli or binomial distributions"). However, it follows from the above discussion that Jeffreys Beta(1/2,1/2) prior represents a state of knowledge in between the Haldane Beta(0,0) and Bayes Beta (1,1) prior.


title: Beta distribution

text: Similarly, Karl Pearson in his 1892 book The Grammar of Science (p. 144 of 1900 edition) maintained that the Bayes (Beta(1,1) uniform prior was not a complete ignorance prior, and that it should be used when prior information justified to "distribute our ignorance equally"". K. Pearson wrote: "Yet the only supposition that we appear to have made is this: that, knowing nothing of nature, routine and anomy (from the Greek ανομία, namely: a- "without", and nomos "law") are to be considered as equally likely to occur. Now we were not really justified in making even this assumption, for it involves a knowledge that we do not possess regarding nature. We use our "experience" of the constitution and action of coins in general to assert that heads and tails are equally probable, but we have no right to assert before experience that, as we know nothing of nature, routine and breach are equally probable. In our ignorance we ought to consider before experience that nature may consist of all routines, all anomies (normlessness), or a mixture of the two in any proportion whatever, and that all such are equally probable. Which of these constitutions after experience is the most probable must clearly depend on what that experience has been like."


title: Beta distribution

text: If there is sufficient sampling data, "and the posterior probability mode is not located at one of the extremes of the domain" (x=0 or x=1), the three priors of Bayes (Beta(1,1)), Jeffreys (Beta(1/2,1/2)) and Haldane (Beta(0,0)) should yield similar "posterior" probability densities. Otherwise, as Gelman et al. (p. 65) point out, "if so few data are available that the choice of noninformative prior distribution makes a difference, one should put relevant information into the prior distribution", or as Berger (p. 125) points out "when different reasonable priors yield substantially different answers, can it be right to state that there "is" a single answer? Would it not be better to admit that there is scientific uncertainty, with the conclusion depending on prior beliefs?."


title: Beta distribution

text: The beta distribution has an important application in the theory of order statistics. A basic result is that the distribution of the "k"th smallest of a sample of size "n" from a continuous uniform distribution has a beta distribution. This result is summarized as:


title: Beta distribution

text: From this, and application of the theory related to the probability integral transform, the distribution of any individual order statistic from any continuous distribution can be derived.


title: Beta distribution

text: In standard logic, propositions are considered to be either true or false. In contradistinction, subjective logic assumes that humans cannot determine with absolute certainty whether a proposition about the real world is absolutely true or false. In subjective logic the posteriori probability estimates of binary events can be represented by beta distributions.


title: Beta distribution

text: A wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" that promptly decays. Wavelets can be used to extract information from many different kinds of data, including – but certainly not limited to – audio signals and images. Thus, wavelets are purposefully crafted to have specific properties that make them useful for signal processing. Wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. Therefore, standard Fourier Transforms are only applicable to stationary processes, while wavelets are applicable to non-stationary processes. Continuous wavelets can be constructed based on the beta distribution. Beta wavelets can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two shape parameters α and β.


title: Beta distribution

text: The Balding–Nichols model is a two-parameter parametrization of the beta distribution used in population genetics. It is a statistical description of the allele frequencies in the components of a sub-divided population:


title: Beta distribution

text: The beta distribution can be used to model events which are constrained to take place within an interval defined by a minimum and maximum value. For this reason, the beta distribution — along with the triangular distribution — is used extensively in PERT, critical path method (CPM), Joint Cost Schedule Modeling (JCSM) and other project management/control systems to describe the time to completion and the cost of a task. In project management, shorthand computations are widely used to estimate the mean and standard deviation of the beta distribution:


title: Beta distribution

text: where "a" is the minimum, "c" is the maximum, and "b" is the most likely value (the mode for "α" > 1 and "β" > 1).


title: Beta distribution

text: The above estimate for the mean formula_426 is known as the PERT three-point estimation and it is exact for either of the following values of "β" (for arbitrary α within these ranges):


title: Beta distribution

text: The above estimate for the standard deviation "σ"("X") = ("c" − "a")/6 is exact for either of the following values of "α" and "β":


title: Beta distribution

text: Otherwise, these can be poor approximations for beta distributions with other values of α and β, exhibiting average errors of 40% in the mean and 549% in the variance.


title: Beta distribution

text: So one algorithm for generating beta variates is to generate formula_439, where "X" is a gamma variate with parameters (α, 1) and "Y" is an independent gamma variate with parameters (β, 1). In fact, here formula_440 and formula_441 are independent, and formula_442. If formula_443 and formula_444 is independent of formula_445 and formula_446, then formula_447 and formula_448 is independent of formula_440. This shows that the product of independent formula_450 and formula_451 random variables is a formula_452 random variable.


title: Beta distribution

text: Also, the "k"th order statistic of "n" uniformly distributed variates is formula_453, so an alternative if α and β are small integers is to generate α + β − 1 uniform variates and choose the α-th smallest.


title: Beta distribution

text: Another way to generate the Beta distribution is by Pólya urn model. According to this method, one start with an "urn" with α "black" balls and β "white" balls and draw uniformly with replacement. Every trial an additional ball is added according to the color of the last ball which was drawn. Asymptotically, the proportion of black and white balls will be distributed according to the Beta distribution, where each repetition of the experiment will produce a different value.


title: Beta distribution

text: Thomas Bayes, in a posthumous paper published in 1763 by Richard Price, obtained a beta distribution as the density of the probability of success in Bernoulli trials (see ), but the paper does not analyze any of the moments of the beta distribution or discuss any of its properties.


title: Beta distribution

text: The first systematic modern discussion of the beta distribution is probably due to Karl Pearson. In Pearson's papers the beta distribution is couched as a solution of a differential equation: Pearson's Type I distribution which it is essentially identical to except for arbitrary shifting and re-scaling (the beta and Pearson Type I distributions can always be equalized by proper choice of parameters). In fact, in several English books and journal articles in the few decades prior to World War II, it was common to refer to the beta distribution as Pearson's Type I distribution. William P. Elderton in his 1906 monograph "Frequency curves and correlation" further analyzes the beta distribution as Pearson's Type I distribution, including a full discussion of the method of moments for the four parameter case, and diagrams of (what Elderton describes as) U-shaped, J-shaped, twisted J-shaped, "cocked-hat" shapes, horizontal and angled straight-line cases. Elderton wrote "I am chiefly indebted to Professor Pearson, but the indebtedness is of a kind for which it is impossible to offer formal thanks." Elderton in his 1906 monograph provides an impressive amount of information on the beta distribution, including equations for the origin of the distribution chosen to be the mode, as well as for other Pearson distributions: types I through VII. Elderton also included a number of appendixes, including one appendix ("II") on the beta and gamma functions. In later editions, Elderton added equations for the origin of the distribution chosen to be the mean, and analysis of Pearson distributions VIII through XII.


title: Beta distribution

text: As remarked by Bowman and Shenton "Fisher and Pearson had a difference of opinion in the approach to (parameter) estimation, in particular relating to (Pearson's method of) moments and (Fisher's method of) maximum likelihood in the case of the Beta distribution." Also according to Bowman and Shenton, "the case of a Type I (beta distribution) model being the center of the controversy was pure serendipity. A more difficult model of 4 parameters would have been hard to find." The long running public conflict of Fisher with Karl Pearson can be followed in a number of articles in prestigious journals. For example, concerning the estimation of the four parameters for the beta distribution, and Fisher's criticism of Pearson's method of moments as being arbitrary, see Pearson's article "Method of moments and method of maximum likelihood" (published three years after his retirement from University College, London, where his position had been divided between Fisher and Pearson's son Egon) in which Pearson writes "I read (Koshai's paper in the Journal of the Royal Statistical Society, 1933) which as far as I am aware is the only case at present published of the application of Professor Fisher's method. To my astonishment that method depends on first working out the constants of the frequency curve by the (Pearson) Method of Moments and then superposing on it, by what Fisher terms "the Method of Maximum Likelihood" a further approximation to obtain, what he holds, he will thus get, "more efficient values" of the curve constants."


title: Beta distribution

text: David and Edwards's treatise on the history of statistics cites the first modern treatment of the beta distribution, in 1911, using the beta designation that has become standard, due to Corrado Gini, an Italian statistician, demographer, and sociologist, who developed the Gini coefficient. N.L.Johnson and S.Kotz, in their comprehensive and very informative monograph on leading historical personalities in statistical sciences credit Corrado Gini as "an early Bayesian...who dealt with the problem of eliciting the parameters of an initial Beta distribution, by singling out techniques which anticipated the advent of the so-called empirical Bayes approach."


title: Into the Woods (film)

text: Into the Woods is a 2014 American musical fantasy film directed by Rob Marshall, and adapted to the screen by James Lapine from his and Stephen Sondheim's 1986 Broadway musical of the same name. Produced by Walt Disney Pictures, it features an ensemble cast that includes Meryl Streep, Emily Blunt, James Corden, Anna Kendrick, Chris Pine, Tracey Ullman, Christine Baranski, Lilla Crawford, Daniel Huttlestone, MacKenzie Mauzy, Billy Magnussen, and Johnny Depp. Inspired by the Grimm Brothers' fairy tales of "Little Red Riding Hood", "Cinderella", "Jack and the Beanstalk", and "Rapunzel", the film is centered on a childless couple who set out to end a curse placed on them by a vengeful witch. Ultimately, the characters are forced to experience the unintended consequences of their actions.


title: Into the Woods (film)

text: After several unsuccessful attempts by other studios and producers to adapt the musical to film, Disney announced in 2012 that it was producing an adaptation, with Marshall directing and John DeLuca serving as producer. Principal photography commenced in September 2013, and took place entirely in the United Kingdom, including at Shepperton Studios in London.


title: Into the Woods (film)

text: "Into the Woods" had its world premiere at the Ziegfeld Theatre in New York City on December 8, 2014, and was released theatrically in the United States on December 25, 2014. It was commercially successful and received generally positive reviews, receiving praise for its acting performances (particularly Streep's), visual style, production merits, and musical numbers, and criticism for its lighter tone compared to the source material, and the changes made for the film adaptation. It grossed $213 million worldwide. "Into the Woods" was named one of the top 11 best films of 2014 by the American Film Institute; the film received three Academy Award nominations, including a Best Supporting Actress nomination for Streep, and three Golden Globe Award nominations, including Best Motion Picture – Musical or Comedy.


title: Into the Woods (film)

text: A Baker and his wife wish for a child but suffer under a curse laid upon their family by a Witch. The Witch had found the baker's father robbing her garden when the baker's mother was pregnant, and demanded their baby in return. Because the Baker's father also stole some magic beans, the Witch's own mother cursed her with ugliness. The Witch is able to lift the curse and allow the Baker and his wife to have a child only if they obtain four items for a potion to break her curse: a cow as white as milk, a cape as red as blood, hair as yellow as corn, and a slipper as pure as gold—none of which she is allowed to touch.


title: Into the Woods (film)

text: The Witch's demands bring the Baker and his wife into contact with Jack, who is selling his beloved cow, Milky-White, and to whom the Baker offers the magic beans his father left him, which Jack's mother accidentally grows into a large beanstalk; with Red Riding Hood, whose red cape the couple noticed when she stopped by the bakery earlier to buy and steal bread and sweets on her way to her grandmother's house; with the blonde Rapunzel (the Witch's adopted daughter and Baker's biological sister), whose tower the Baker's wife finds in the woods; and with Cinderella, who runs into the Baker's wife while fleeing the Prince and whose ball outfit includes gold slippers.


title: Into the Woods (film)

text: After a series of failed attempts and misadventures, the Baker and his wife are finally able to gather the necessary items. After the Witch regains her youth and beauty after drinking the potion, each of the characters receives a "happy ending". The Baker and his wife have a son; Cinderella marries the Prince; Rapunzel is freed from the Witch by the Prince's brother, whom she marries; Jack provides for his mother by stealing riches from the Giant in the sky, courtesy of the beanstalk, and kills the pursuing Giant by cutting down the beanstalk; and Red Riding Hood and her grandmother are saved from the Big Bad Wolf.


title: Into the Woods (film)

text: However, each of the characters learn that their endings don't remain happy: the Baker is worried that he is a poor father to his newborn baby; the Baker's wife lets the Prince temporarily seduce her; Cinderella is disenchanted by her cheating Prince; and the Witch learns that she has lost her powers in exchange for her youth and beauty, after being rejected by Rapunzel, who then runs off with her Prince. The growth of a second beanstalk from the last remaining magic bean allows the Giant's widow to climb down and threaten the kingdom if no one delivers Jack in retribution for killing her husband. The characters attempt to find and protect Jack. In the process, Red Riding Hood's mother and grandmother, Jack's mother and the Baker's wife are killed. The Baker, Cinderella, Jack and Red Riding Hood all blame each other for their individual actions that led to the tragedy, ultimately blaming the Witch for growing the beans in the first place. She curses them all for their inability to accept any responsibility, as well as their refusal to do the "right thing" (handing Jack over). Casting all the remaining beans away, the Witch begs her mother to punish her again, and she abandons the group by melting into a large pit of boiling tar.


title: Into the Woods (film)

text: The remaining characters resolve to kill the threatening Giant's widow, though they discuss the complicated morality of retribution and revenge in the process. They lure the Giant's widow into stepping in the tar pit, where she trips and falls with a tree crushing her. With the Giant's widow dead, the characters move forward with their lives. The Baker, thinking of his wife, is determined to be a good father. Cinderella decides to leave her Prince and help the Baker with Jack and Red Riding Hood, as they are now orphans, and will be moving into the bakery. The Baker comforts his son by telling the story of the film as the movie ends with the Witch's moral ("Children will Listen"), which means children can change due to their parents' actions and behaviors.


title: Into the Woods (film)

text: While it was initially reported that Disney had decided to make some major plot changes for the film version, Stephen Sondheim revealed that this was not the case and that any changes in the film version had been approved by him and James Lapine.


title: Into the Woods (film)

text: The film does differ from the stage production. The songs "I Guess This Is Goodbye", "Maybe They're Magic", "First Midnight" and "Second Midnight" interludes, "Ever After" (Act I finale of the original play), "So Happy", "Into the Woods" Reprise, "Agony" Reprise and "No More" (performed by the Baker) were cut from the film, although both "Ever After" and "No More" are used as instrumentals in the film. Meanwhile, many of the songs in the film have slightly different lyrics than their stage counterparts due to the tweaking of story lines.


title: Into the Woods (film)

text: Other changes include a major reduction of the significant role of the "mysterious man", who manipulates much of the action in the first act and is eventually revealed to be the Baker's father. Additionally, the character of the narrator was cut, and the film is instead narrated by the Baker. The minor role of Cinderella's father was cut; he is instead mentioned as deceased. Due to the film's compressed story line, Rapunzel's pregnancy is eliminated, as is the subplot where the two princes have affairs with Snow White and Sleeping Beauty. In the film, the Giant's wife first attacks during the marriages of Cinderella and Rapunzel to their respective princes; in the stage show, the Giant's wife first attacks the Witch's garden. Rapunzel's ultimate fate is also changed: rather than being killed by the Giant, she refuses to cooperate with the Witch and flees with her prince.


title: Into the Woods (film)

text: Early attempts to adapt "Into the Woods" to film occurred in the early 1990s, with a script by Lowell Ganz and Babaloo Mandel. A reading was held with a cast that included Robin Williams as the Baker, Goldie Hawn as the Baker's Wife, Cher as the Witch, Danny DeVito as the Giant, Steve Martin as the Wolf, and Roseanne as Jack's Mother. By 1991, Columbia Pictures and Jim Henson Productions were also developing a film adaptation with Craig Zadan as producer and Rob Minkoff as director. In 1997, Columbia put the film into turnaround, with Minkoff still attached as director, and Billy Crystal, Meg Ryan, and Susan Sarandon reportedly in talks to star. After the report by "Variety", a film adaptation of "Into the Woods" remained inactive for 15 years.


title: Into the Woods (film)

text: After the critical and commercial success of "Chicago" in 2002, director Rob Marshall approached Sondheim with a proposal to adapt one of his musicals. Marshall was initially interested in adapting "Follies" and "", Sondheim suggested "Into the Woods" instead. Marshall concurred, but the project was postponed while he focused on directing "Memoirs of a Geisha" and "Nine". In 2011, Marshall's interest in the project was rekindled when he heard a speech by President Barack Obama on the tenth anniversary of the September 11 attacks to the families of the victims, which seemed to evoke the same message as the musical's most important song, "No One Is Alone". Marshall firmly believed that "Into the Woods" was “a fairy tale for the post-9/11 generation". In January 2012, he approached Walt Disney Pictures—for which he had just directed ""—and pitched the idea to the studio, with Lapine writing the script and Sondheim "expected" to write new songs. Academy Award-winner Dion Beebe, who previously collaborated with Marshall on "Chicago", "Memoirs of a Geisha", and "Nine", served as cinematographer. Sondheim confirmed that a new song had been written for the film in an April 2013 interview with "The Independent".


title: Into the Woods (film)

text: With Disney's backing, a three-day reading of the entire updated screenplay took place in New York in October 2012 under Marshall's direction, with Nina Arianda as the Baker's Wife, Victoria Clark as Cinderella's Mother/Granny/Giant, James Corden as the Baker, Donna Murphy as the Witch, Christine Baranski as Cinderella's Stepmother, Tammy Blanchard as Florinda, Ivan Hernandez as the Wolf, Megan Hilty as Lucinda, Cheyenne Jackson as Rapunzel's Prince, Allison Janney as Jack's Mother, Anna Kendrick as Cinderella, Michael McGrath as Steward/Mysterious Man, Laura Osnes as Rapunzel, Taylor Trensch as Jack, Casey Whyland as Little Red Riding Hood, and Patrick Wilson as Cinderella's Prince. This reading convinced Walt Disney Studios president Sean Bailey to green-light the film despite ongoing concerns about the original musical's dark nature (which Disney executives had begun to understand since Marshall's original pitch). Disney (which self-finances all its films) provided only a relatively small production budget of $50 million (relative to other feature-length fantasy films on its development slate), in turn forcing both cast and crew to accept pay cuts to work on the film.


title: Into the Woods (film)

text: Reports surfaced in January 2013 that Meryl Streep had been cast as the Witch. Streep had instituted a personal "no witch" rule after she turned 40 and was offered three witch roles, but ultimately broke the rule to do a Sondheim role again; she had been in the original production of Sondheim's "The Frogs" as a student at Yale University. During the same month, it was reported that Janney had been confirmed to join the film, but five months later, Tracey Ullman was cast as Jack's Mother instead.


title: Into the Woods (film)

text: In April 2013, Johnny Depp was in final negotiations to join the film. "The Hollywood Reporter" reported that to help make the film on such a tight budget, Depp agreed as a favor to Disney and to Marshall to a "boarding" arrangement, in which he would appear in a minor role for a fee of $1 million, instead of his typical fee of $20 million for a starring role. In May, Corden, who took part in the reading of the screenplay, was in talks to play the role of the Baker. On May 10, 2013, Disney confirmed the casting of Streep, Depp, and Corden as the Witch, the Big Bad Wolf, and the Baker, respectively. That same month, Emily Blunt and Christine Baranski were cast, respectively, as the Baker's Wife and Cinderella's Stepmother. Marshall later confirmed that Blunt was selected for her "warm[th]" and likeability to ensure the emotional impact of the sudden death of the Baker's Wife: "[T]hat's very important for that character because it's the heart of the piece and you really have to love her so when she's gone it should feel like a kick in the gut." After she was cast, Blunt discovered she was pregnant; her costume and choreography had to be adjusted accordingly. Her appearance during production seemed to befit the Baker's Wife; as she explained, "I feel like she would have eaten a lot of carbs working in the bakery."


title: Into the Woods (film)

text: Also in May, Jake Gyllenhaal and Chris Pine entered negotiations to play the Princes. Gyllenhaal dropped out due to scheduling conflicts with another film, "Nightcrawler", and was replaced by Billy Magnussen. One month later, Anna Kendrick began talks to play Cinderella. In June 2013, Walt Disney Studios publicly announced that the film had been greenlighted, and scheduled a release date for Christmas Day 2014. In July, MacKenzie Mauzy, Tammy Blanchard, Lucy Punch and Daniel Huttlestone joined the cast. Mauzy later said she had auditioned for Cinderella but did not get the part. Marshall saw her audition tape and brought her back in for Rapunzel, after recognizing "the 'vulnerability' and 'emotion' Mauzy could bring to Rapunzel after she read just one line", as he recalled. In early August, Sophia Grace Brownlee's representatives announced that she had been cast as Little Red Riding Hood. The announcement of Brownlee's casting, which was widely reported but never confirmed by Disney, was criticized as "a stunt" and was met with concern due to her age and the sexual undertones between Little Red and the Wolf. The film's synopsis and the official casting of Streep, Depp, Kendrick, Pine, and Blunt were revealed at the D23 Expo on August 10, 2013. On September 16, 2013, Lilla Crawford was confirmed as playing the character of Little Red Riding Hood, despite previous reports suggesting Brownlee. Crawford auditioned for Marshall via Skype, who offered her the role within two hours. Crawford was on a flight to London the next day as a result. Later on, Dominic Brownlee spoke about the withdrawal of his daughter Sophia Grace from the movie: "After careful consideration, we, the parents of Sophia Grace, felt that as rehearsals progressed that she was too young for this part. It was a joint decision between us and the director and producer of "Into the Woods" to withdraw Sophia Grace from the film." The casting of Richard Glover, Frances de la Tour, Simon Russell Beale, Joanna Riding, and Annette Crosbie in other roles was separately announced later on September 16.


title: Into the Woods (film)

text: In July 2013, before filming began, Marshall put the cast through six weeks of rehearsal on a soundstage and blocked their scenes. In August, the cast visited Angel Recording Studios to record their parts in Sondheim's presence. Over 90% of the vocal tracks in the film are from the recording studio sessions; the rest were recorded on location or on set. The advantage of blocking and rehearsing all scenes first was that the cast members could then precisely calibrate their voices in the recording studio to the planned appearance of each scene when it was filmed, thereby minimizing the disconnection between vocals and choreography typical of music videos.


title: Into the Woods (film)

text: Principal photography took place at London's Shepperton Studios in September 2013, with additional filming at Dover Castle, Hambleden, Waverley Abbey, and Richmond Park. A forest of ancient pine trees in Windsor Great Park was used for many of the scenes in the woods. Marshall struggled with how to stage the melodramatic duet "Agony" in the forest until discovering online that Windsor had an artificial waterfall at Virginia Water Lake, which turned out to be the perfect setting. The production was shot digitally, using Arri Alexa cameras configured in a two-camera setup, and footage was edited together in Avid Media Composer. The exterior of Byfleet Manor in Surrey served as Cinderella's home.


title: Into the Woods (film)

text: The filmmakers spent a whole day shooting scenes involving Rapunzel's hair being climbed. Mauzy said the filmmakers wanted to take advantage of her blonde hair, and that the top of Rapunzel's hair was her real hair; makeup artists only braided it into the extension. This hair extension was engineered by hair designer Peter King. After testing loose, flowing hair that King found "uncontrollable", he decided to have 27 wefts of real hair woven into a 30-foot braid, a design inspired by an Arthur Rackham illustration of Rapunzel. To bring in enough real Russian hair strands for the extension, King and his team worked with several distributors from Germany and England. The hair-braiding process required three people, each holding a separate strand and weaving in and out. King also dyed the wefts to match Mauzy's hair color, and blended together six different shades from ash and strawberry to create realistic gradations and highlights. Between scenes, Mauzy had to "wrap [the hair] around her arm like huge rolls of wool", King recalled. A stuntman was used to shoot the hair-climbing scenes, with thin rope and metal rings the only tools concealed in the braid to hold a climbing person's weight.


title: Into the Woods (film)

text: The film's final shot, which essentially merges into and links back to its first shot, actually transitions digitally between three shots: a Technocrane on location lifting as high as possible into the sky, an aerial drone flying down a valley in Wales, and a shot of an overcast sky in Manhattan, New York City. Filming concluded on November 27, 2013.


title: Into the Woods (film)

text: On July 14, 2014, Steve Baldwin posted on a social networking site that reshoots were made during all of July. The next month, however, Marshall denied the film went through reshoots. Instead, they spent three days shooting new material that had been cut and readded to the script after Disney screened the movie. For his role as the Wolf, Depp worked closely with the film's costume designer, Colleen Atwood, to create a Tex Avery-inspired costume, complete with zoot suit and fedora.


title: Into the Woods (film)

text: As noted above, the majority of the songs were pre-recorded by the cast. Music producer Mike Higham, who had worked with Sondheim on "Sweeney Todd": "The Demon Barber of Fleet Street", recorded the film's score with the London Symphony Orchestra and London Philharmonic. Music supervisor and conductor Paul Gemignani instructed the actors how to sing, including singing live to a camera on set, to prerecorded music in studio, and with a live studio orchestra. Jonathan Tunick reorchestrated Sondheim's music. The key of the song, "Hello, Little Girl," was altered to suit Depp's lower vocal range. Of the song's musical arrangement, Higham said, "we emphasized the woodwinds to make it feel a little lighter, especially the flutes. And we just made it a little jazzier — played more on the walking bass line. Inherently, when it has a jazz feel, it just feels lighter." The soundtrack was released by Walt Disney Records on December 15, 2014.


title: Into the Woods (film)

text: It was initially reported that the film would feature two new songs: a duet for the Baker and his Wife, "Rainbows"—originally written for a 1992 film adaptation that was never made—and a new song for the Witch, "She'll Be Back". But neither song appears in the film: "Rainbows" was cut before shooting began and "She'll Be Back", though filmed, was cut on the grounds that it slowed the story down. "She'll Be Back" was included as a bonus feature on the film's home media release.


title: Into the Woods (film)

text: The first official company presentation took place at the 2013 Disney D23 expo. The official teaser trailer debuted on July 31, 2014. A featurette was released showing behind-the-scenes clips and the vocals of Streep, Kendrick, Blunt and others. A second trailer was released on November 6, 2014.


title: Into the Woods (film)

text: The film held its world premiere at the Ziegfeld Theatre in New York City on December 8, 2014. It was released theatrically in the United States on December 25, 2014.


title: Into the Woods (film)

text: Walt Disney Studios Home Entertainment released the film on Blu-ray, DVD, and digital download in North America on March 24, 2015. The film debuted in second place on the home media charts behind "". The home media version includes Sondheim and Lapine's original song "She'll Be Back". A sing-along version of the film was released on Disney+ on October 14, 2022.


title: Into the Woods (film)

text: "Into the Woods" grossed $128 million in North America and $85.1 million in other territories, for a worldwide total of $213.1 million, against a budget of $50 million.


title: Into the Woods (film)

text: "Into the Woods" began playing across North American theaters on December 24, 2014, and earned $1.1 million from late-night Christmas Eve showings and $15.08 million on opening day (including previews) from 2,440 theaters. Its opening-day gross was the fourth-biggest Christmas Day debut and the sixth-biggest Christmas Day gross ever. The film was one of four films put into wide release on December 25, 2014, the other three being Universal Pictures' "Unbroken" (3,131 theaters), Paramount Pictures' "The Gambler" (2,478 theaters), and TWC's "Big Eyes" (1,307 theaters). It earned $31.1 million in its traditional three-day opening ($46.1 million including its Christmas Day gross), debuting at #2 at the box office behind " and" achieving the highest opening weekend for a film based on a Broadway musical (previously held by "Mamma Mia!"). The film's $3.5 million debut in Japan marked the largest opening for a 21st-century live-action musical film.


title: Into the Woods (film)

text: The review aggregation website Rotten Tomatoes reports an approval rating of 71% based on 225 reviews, with an average rating of 6.60/10. The site's consensus reads: "On the whole, this Disney adaptation of the Sondheim classic sits comfortably at the corner of Hollywood and Broadway—even if it darkens to its detriment in the final act." Another review aggregator, Metacritic, calculate a weighted average score of 69 out of 100, based on 41 critics, indicating "generally favorable reviews". In CinemaScore polls conducted during the opening weekend, audiences gave the film an average grade of "B" on an A+ to F scale.


title: Into the Woods (film)

text: Streep was immensely applauded for her performance as the Witch, with many critics and audiences believing she was the film's heart. Lou Lumenick of the "New York Post" called her and Blunt's portrayals "two of the greatest female performances of the year". Susan Wloszczyna for RogerEbert.com praised Streep's performances of the songs "Stay with Me" and "Last Midnight", and summed her performance up as leaving "practically everyone else in the dust and by design."


title: Into the Woods (film)

text: "The Hollywood Reporter" wrote that the film "benefits from respect for the source material, enticing production values and a populous gallery of sharp character portraits from a delightful cast". Stephen Holden of "The New York Times" lauded the film, writing; ""Into the Woods", the splendid Disney screen adaptation of the Stephen Sondheim-James Lapine musical, infuses new vitality into the tired marketing concept of entertainment for 'children of all ages'". Leonard Maltin called the movie "one of the year's best films". Similarly, Pete Hammond of Deadline praised the film as "the most dazzling movie musical since Marshall's own Chicago." and praised the performance of the cast, particularly Streep. Lou Lumerick of the New York Post called the film "this century's best musical" and lauded the performances of Streep and Blunt as the best female performances of the year. Scott Mendelson of "Forbes" gave a positive review declaring the film "Rob Marshall's best movie ever" and praised it for its genuine entertainment and strong cast performances. Richard Corliss of "Time" gave a positive review, stating that the film was a "smart, appealing, upside-down children's story for adults of all ages". Gregory Ellwood of HitFix tapped Streep as an Academy Award contender in the Best Supporting Actress category, and also praised the performance of Chris Pine.


title: Into the Woods (film)

text: Richard Lawson of "Vanity Fair", Karen D'Souza of the "San Jose Mercury News", and Dana Stevens of "Slate" all published critical reviews of the film. All three found much to like in the acting (especially Blunt and Pine's performances), but also concluded that after the various cuts and changes, the film had failed to adequately preserve the power of the dark existentialist message at the heart of the original musical's second act. Stevens characterized the result as a "generic dystopian bummer," while Lawson criticized the film as a "dutiful but perfunctory adaptation" that lacked "genuine heart". Paul Katz of "The Huffington Post" felt the change in tone between the last two acts was too abrupt, and also criticized the film's faithfulness to the stage musical. Conversely, Alyssa Rosenberg of "The Washington Post" disliked the performances by Streep and Depp, while simultaneously calling the film's first two acts a "surprising delight".


title: James Franco filmography

text: James Franco is an American actor, filmmaker, and college instructor. He began acting on television, guest-starring in "Pacific Blue" (1997). He landed his breakthrough role in the comedy-drama television series "Freaks and Geeks" (1999–2000). After his film debut in "Never Been Kissed" (1999), Franco won a Golden Globe Award for Best Actor – Miniseries or Television Film and was nominated for Screen Actors Guild Award and Primetime Emmy Award in the same categories for playing the eponymous actor in the 2001 television biopic "James Dean". He went on to play Harry Osborn in the superhero film "Spider-Man" (2002), and reprised the role in its sequels "Spider-Man 2" (2004) and "Spider-Man 3" (2007). For the last of the three, he garnered a nomination for the Saturn Award for Best Supporting Actor. His only screen appearance of 2003 was in the ballet film "The Company". Franco directed and starred in the comedy "The Ape" (2005).


title: James Franco filmography

text: After playing one of the titular roles in the romantic drama "Tristan & Isolde" (2006), Franco starred in the Tony Bill-directed war drama "Flyboys" (2006). Two years later, he played against type in the action-comedy film "Pineapple Express", and earned critical acclaim for portraying Scott Smith in the biographical film "Milk" alongside Sean Penn. For the former, he was nominated for a Golden Globe Award for Best Actor – Comedy. Franco portrayed the trapped canyoneer Aron Ralston in "127 Hours" (2010), a survival drama, which earned him nominations for an Academy Award, BAFTA Award, Screen Actors Guild Award and Golden Globe Award, all for Best Actor. Franco appeared in four films in 2011, including the poorly-received fantasy film "Your Highness", and the science fiction film "Rise of the Planet of the Apes "(2011), a critical and commercial success.


title: James Franco filmography

text: Franco had six roles in 2012 none of which had much success except the crime-comedy film "Spring Breakers", in which he played a gangster to highly positive reviews"." The following year, Franco played the titular role in the fantasy film "Oz the Great and Powerful", and the disaster film "This Is the End "saw him play a fictional version of himself. For the first one, he was nominated for the Teen Choice Award for Choice Movie Actor - Fantasy. Also in 2013, he directed and starred in the drama "As I Lay Dying". He starred in the action thriller "Good People" (2014), an adaptation of Marcus Sakey's 2008 novel of the same name. In the 2014 controversial satirical comedy "The Interview", he was seen as a journalist instructed to assassinate a North Korean leader. He had nine film releases in 2015, most of which failed financially except the animated film "The Little Prince", a modest commercial success. In 2017, Franco directed and starred in "The Disaster Artist" as Tommy Wiseau, for which he won a Golden Globe Award for Best Actor – Motion Picture Musical or Comedy.


title: Kumkum Bhagya

text: Kumkum Bhagya () (from 2021 onwards) is an Indian Hindi-language drama television series produced by Ekta Kapoor under Balaji Telefilms. It premiered on 15 April 2014 on Zee TV and it is digitally available on ZEE5. It formerly starred Sriti Jha and Shabir Ahluwalia. Currently, it stars Mugdha Chaphekar and Krishna Kaul as the second generation leads.


title: Kumkum Bhagya

text: The series started as the story of Arora sisters Pragya and Bulbul, and their mother Sarla Arora's hope to see them married. Pragya gets married to a rockstar, Abhishek "Abhi" Mehra while Bulbul gets married to Abhi's friend, Purab. Bulbul sadly dies. Abhi and Pragya's married life continues with several challenges. Abhi blames Pragya for his grandmother's death and they get separated.


title: Kumkum Bhagya

text: Pragya now has a daughter with Abhi named Kiara. She is living with King Singh, a famous rapper in London. Later, she comes back to Delhi and Abhi and Pragya reconcile. Pragya gives birth to twin daughters, Prachi and Rhea. Kiara is killed by enemies. Pragya and Abhi are heartbroken and blame each other for Kiara's death and separate, with Pragya taking one twin while Abhi keeps the other.


title: Kumkum Bhagya

text: Pragya and Abhi's twins, Prachi and Rhea have now grown up. A love triangle is formed between Prachi, Rhea and Abhi's friend's son, Ranbir Kohli who is the college heartthrob. Prachi is the replica of Pragya who is sweet, caring and honest whereas Rhea turns out to be selfish, cunning and over pampered by her aunt Alia. Tanu returns and later marries Abhi who was supposed to be marrying Pragya. Due to a misunderstanding, Abhi cuts ties with Pragya. Prachi and Ranbir get married secretly with only Pragya’s blessings. The Kohlis disown the newly married couple. Pragya gets into an accident and is presumed to be dead.


title: Kumkum Bhagya

text: Pragya survives and comes back from Australia with the intention of taking revenge from Abhi as she was tricked into thinking that Abhi was the one who tried to kill her by Tanu and Alia. Prachi and Ranbir's happily married life continues and they soon get accepted back in the Kohli mansion. Rhea gets married to Siddharth, Ranbir's elder cousin brother. Abhi and Pragya clear their misunderstandings and reunite. They meet with an accident and are in a coma. Ranbir accuses Prachi of extramarital affair with Siddharth due to a misunderstanding created by Rhea and they get separated. Abhi and Pragya die in their coma.


title: Kumkum Bhagya

text: Sid and Rhea are now divorced. Ranbir turns out to be a successful businessman whereas Sushma who is Pragya's mentor tries to console a heartbroken Prachi. Prachi decides to move on and focus on her career with Sushma's help but gets to know that she is pregnant with Ranbir's child. However, Rhea and Ranbir get married illegally as Ranbir thinks Prachi has moved on. Prachi witnesses their wedding and decides to fight for her self-esteem. For this she returns to the Kohli House to regain her right as Ranbir's legal wife and to throw Rhea out.


title: Kumkum Bhagya

text: The series is a love story produced by Ekta Kapoor and Shobha Kapoor under the banner Balaji Telefilms that airs on Zee TV. Ekta Kapoor considers the letter "K" to be lucky and has named many of her productions starting with "K". Talking about introducing a fresh concept on prime time, Zee TV Programming head Namit Sharma said, "It takes one epic show to replace another, so you can imagine. It is a passionate drama of love, a far cry from regular saas-bahu soaps and has an ensemble cast of some very talented actors."


title: Kumkum Bhagya

text: Ekta Kapoor added, "While two of my shows ("Pavitra Rishta" and "Jodha Akbar") are already on-air and doing immensely well on Zee TV, it's a pleasure to extend my partnership with Zee to a third show. "Pavitra Rishta" has had a run of over five years now and the journey has been most gratifying for me as a producer. While the audiences who are hooked on to "Pavitra Rishta" will continue to enjoy the show at 6:30 PM, "Kumkum Bhagya" will open at 9 PM, drawing maximum eyeballs. It is so contemporary in its treatment that viewers will be able to instantly spot characters out of their own lives in the show. It's a story of passionate love that anyone with a soul would be able to easily relate to".


title: Kumkum Bhagya

text: Speaking of his comeback as an actor, Shabbir Ahluwalia, the protagonist of the show says, "With me producing shows, it was getting very difficult to manage time for acting assignments. But now, I have decided to concentrate on acting. You cannot be away from what you love doing for long. The role appealed to me a great deal. This is the first time that I will be playing a rockstar on television and in order to look my part, I had to ensure that I did not just get my costume and look right, but also get that attitude right that is required to portray a rockstar onscreen."


title: Kumkum Bhagya

text: The production and airing of the show was halted indefinitely in late March 2020 due to the COVID-19 outbreak in India. Because of the outbreak, the filming of television series and films was halted on 19 March 2020 and expected to resume on 1 April 2020 but could not and the series was last broadcast on 24 March 2020 when the remaining episodes were aired. After three months, the production and filming of the series resumed on 29 June 2020 and airing to resume from 13 July 2020.


title: Kumkum Bhagya

text: Shabbir Ahluwalia, who was also cast in Ekta's TV series "Kayamath", was selected to portray the lead role of Abhishek Prem Mehra. Ahluwalia made his comeback after Zee twenty years. Actress Sriti Jha landed in lead role in the show. Mrunal Thakur was selected to portray the second leading role of Bulbul while Arjit Taneja was cast opposite her. In May 2014, Shikha Singh entered the show as the main antagonist Alia Mehra, while Madhurima Tuli was cast to portray the negative role of Tanu. Later, Tuli had to shoot for her film "Baby" so, this role was later played by Leena Jumani in September 2014. Actor Faisal Rashid was selected to portray the role of Suresh (Pragya's fiancé) who played an important role in April–May 2014, but in May 2015 he quit the series. Actress Supriya Shukla was cast to play the role of Sarla Arora, while Madhu Raja played the role of Daljeet Arora. Other supporting cast members include Samikssha Batnagar, Ankit Mohan, Charu Mehra and Amit Dhawan. Bhatnagar, who portrayed the supporting role of Mitali was replaced by Swati Anand within the month of launch while Anurag Sharma replaced Amit. In 2015, Neel Motawani (who was also cast in other shows of Ekta) was selected to play the negative role of Neel; while in June 2015 Nikhil Arya entered the series playing the role of main male antagonist Nikhil Sood.


title: Kumkum Bhagya

text: In December 2015, Mrunal Thakur opted out from the show and was to be replaced by Kajol Shrivastava, however the character of Bulbul was phased out of the series after Mrunal's departure. In August 2016, Ankit Mohan quit the show so his character left as well. In September 2016, Arjit Taneja quit the show and was replaced by Vin Rana. In October 2016, Nikhil Arya was replaced by Rujut Dahiya. In July 2017, actress Ruchi Savarn entered the show as Purab's new love interest. Mishal Raheja as King Singh and Kaurwakee Vashista as Kiara Mehra entered the show after the leap. Actor Abeer Adil was cast in 2018 as King's cousin Tarun and Roma Bali as Tarun's mother.


title: Kumkum Bhagya

text: In 2019, a twenty-year leap was introduced which several new characters entered in the show and became pivotal characters. In March 2019, Mugdha Chaphekar, Krishna Kaul and Naina Singh were introduced as leads of the second generation who play Prachi, Ranbir and Rhea respectively alongside Sriti and Shabbir. In August 2020, Naina left the show as she wasn't happy playing a negative role anymore and Pooja Banerjee replaced Singh as Rhea. In February 2022, Pooja Banerjee quit the show because she is in the last month of her pregnancy and her due date is March, and she was replaced by Tina Ann Philip.


title: Kumkum Bhagya

text: The series was released on 15 April 2014 on Zee TV. The series also airs on Zee TV HD and Zee TV Asia.


title: Kumkum Bhagya

text: A spin-off series "Kundali Bhagya" premiered on 12 July 2017. The series narrates the story of Pragya's two sisters, Preeta and Srishti and their attempt to find their mother Sarla and how they meet the rich and famous Luthra family. The show stars Supriya Shukla, Shraddha Arya, Dheeraj Dhoopar, Anjum Fakih and Shakti Arora.


title: Kumkum Bhagya

text: A similar series "Bhagya Lakshmi" premiered on 3 August 2021. It traces the love story of Lakshmi and Rishi. It stars Aishwarya Khare and Rohit Suchanti.


title: Kumkum Bhagya

text: The show climbed to Top 5 most-watched shows across all GEC's. After only a few weeks of its launch and was touted as the biggest fiction launch of 2014. The serial is very popular among Indian women. The show also gave tough fight to the long time Hindi GEC leader "Diya Aur Baati Hum" that aired at same slot on Star Plus.


title: Kumkum Bhagya

text: Reruns of Kumkum Bhagya premiered on Zee Anmol from 2017 and received high ratings in the BARC rural chart when Zee Anmol was FTA. It began running on the network again from episode 1 from 10 June 2020, due to the return to DD Free Dish.


title: Kumkum Bhagya

text: The show has often been criticised for its repetitive storyline and showcasing unusual and logicless plots. The writers of the show have received widespread criticism on social media platforms for showcasing unusual drama, bigamy and violating law and orders in the show.


title: Kumkum Bhagya

text: The "Kumkun Bhagya" Soundtrack was written by Nawab Arzoo and composed by Lalit Sen. Lalit Sen had composed the original songs and background score for the show. "Allah Waariyan" is not the song of this show but it was the theme song of Abhi and Pragya. The first original song of the show is "O Rabba Ki Karaan", audience loved the song. The sad version of this song are also released with the title "Tum Ko Mera Dard Dikhna" and "Har Ke Bhi Jitna Hai Pyaar" sung by Kashi Malik. Meherbaanon Ka Shukriya Song was released on the occasion of completing 1000 episodes and for thanking the audience for the love and support.


title: David Alaba

text: David Olatukunbo Alaba (born 24 June 1992) is an Austrian professional footballer who plays as a centre-back or left-back for La Liga club Real Madrid and captains the Austria national team. He is regarded as one of the best left-backs of his generation.


title: David Alaba

text: Alaba started out in Bayern's youth system before being promoted to the reserve team for the 2009–10 season. In January 2011, Alaba joined TSG 1899 Hoffenheim on loan until the end of the 2010–11 season. He returned to Bayern at the start of the 2011–12 season, where he went on to become a regular member of the first-team squad. Alaba made over 400 appearances for Bayern Munich, winning 27 honours including ten Bundesliga titles and two UEFA Champions League titles in 2013 and 2020, both as part of trebles. During his time in Germany, he was named in the UEFA Team of the Year three times. In 2021, Alaba signed for Real Madrid; he won the 2021–22 La Liga, Supercopa and Champions League title in his debut season.


title: David Alaba

text: Alaba is Austria's second youngest player to play for their senior national team, debuting for them in 2009 as a 17-year-old. He has earned over 90 caps and represented his country at UEFA Euro 2016 and UEFA Euro 2020. He was voted Austrian Footballer of the Year on eight occasions (including six consecutive times from 2011 to 2016).


title: David Alaba

text: Born in Vienna, Alaba began his career with SV Aspern, his local club in Aspern, in the Vienna district Donaustadt, before joining the youth setup of FK Austria Wien at age 10. He rose through the ranks quickly, and in April 2008 he was named on the first-team substitutes' bench for an Austrian Bundesliga match. He also played five times for Austria Wien's reserve team, before leaving in summer 2008 to join German Bundesliga side Bayern Munich.


title: David Alaba

text: He started out in Bayern's youth system, playing for the under 17 and under 19 teams, before being promoted to the reserve team for the 2009–10 season. He made his debut in a 3. Liga match against Dynamo Dresden in August 2009 and scored his first professional goal for FC Bayern Munich II on 29 August 2009. He was named in Bayern Munich's squad for the 2009–10 UEFA Champions League, where he was assigned the shirt number 27. In January 2010, it was announced that Alaba would train with the first team for the rest of the 2009–10 season, along with reserve teammates Diego Contento and Mehmet Ekici.


title: David Alaba

text: The trio were named on the first-team substitutes' bench for the first time on 10 February 2010, for a DFB-Pokal game against SpVgg Greuther Fürth, and Alaba came on in the 59th minute, replacing Christian Lell. After one minute on the pitch, and with his second touch of the game, he set up Franck Ribéry to give Bayern a 3–2 lead, in a match they went on to win 6–2. He also became Bayern's youngest ever player in a competitive fixture, at 17 years, 7 months and 8 days old.


title: David Alaba

text: In January 2011, Alaba joined TSG 1899 Hoffenheim on loan until the end of the 2010–11 season. Later that month, he scored his first-ever goal in the Bundesliga in a 2–2 draw with FC St. Pauli.


title: David Alaba

text: Alaba returned to Bayern at the start of the 2011–12 season, where he became a regular member of the first-team squad. On 23 October 2011, Alaba scored his first league goal for Bayern in the 1–2 away loss against Hannover 96. On 20 December, Alaba was named Austrian Footballer of the Year for the first time. During the second half of the 2011–12 Bundesliga season, he established himself as a starter for Bayern. On 25 April 2012, he played in the 2011–12 UEFA Champions League semi-final second leg against Real Madrid, and scored Bayern's first kick of the shootout as they won 3–1 on penalties. However, due to being booked in the semi-final, Alaba was unable to play in the 2012 UEFA Champions League Final through suspension.


title: David Alaba

text: On 5 December, Alaba scored his first Champions League goal in a 4–1 win over BATE Borisov in the 2012–13 season. On 18 December, Alaba was named Austrian Footballer of the Year for the second year running. On 2 April 2013, Alaba scored the seventh-fastest goal (25.02 seconds) in Champions League history to set Bayern on their way to a 2–0 win over Juventus. On 25 May, he played the full 90 minutes at left-back as Bayern beat Borussia Dortmund 2–1 in the 2013 UEFA Champions League Final.


title: David Alaba

text: During the 2013–14 season, he played in the 2013 DFL-Supercup against Borussia Dortmund, the UEFA Super Cup against Chelsea, and two matches in the FIFA Club World Cup; the first against Guangzhou Evergrande and the second against Raja Casablanca. On 2 December 2013, Alaba signed a new contract with Bayern Munich, which would expire in 2018. On 20 December, Alaba was named Austrian Footballer of the Year for the third year running. In January 2014, Alaba was named as the left-back in the UEFA Team of the Year 2013 for the first time.


title: David Alaba

text: On 18 December 2014, Alaba was named Austrian Footballer of the Year for the fourth year running. In January 2015, he was voted as the left-back in the UEFA.com Team of the Year 2014 by users (352,070 votes) for the second consecutive year. On 31 March, Alaba suffered a medial ligament damage in his left knee during Austria's 1–1 friendly draw with Bosnia and Herzegovina. Alaba missed the rest of the season for Bayern because of the injury.