A Bayesian perspective on estimating mean, variance, and standard-deviation from data
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1 Brigham Youg Uiversity BYU ScholarsArchive All Faculty Publicatios A Bayesia perspective o estimatig mea, variace, ad stadard-deviatio from data Travis E. Oliphat Follow this ad additioal works at: Part of the Electrical ad Computer Egieerig Commos BYU ScholarsArchive Citatio Oliphat, Travis E., "A Bayesia perspective o estimatig mea, variace, ad stadard-deviatio from data" (006. All Faculty Publicatios This Peer-Reviewed Article is brought to you for free ad ope access by BYU ScholarsArchive. It has bee accepted for iclusio i All Faculty Publicatios by a authorized admiistrator of BYU ScholarsArchive. For more iformatio, please cotact scholarsarchive@byu.edu.
2 A Bayesia perspective o estimatig mea, variace, ad stadard-deviatio from data Travis E. Oliphat December 5, 006 Abstract After reviewig some classical estimators for mea, variace, ad stadard-deviatio ad showig that u-biased estimates are ot usually desirable, a Bayesia perspective is employed to determie what is kow about mea, variace, ad stadard deviatio give oly that a data set i-fact has a commo mea ad variace. Maximum-etropy is used to argue that the likelihood fuctio i this situatio should be the same as if the data were idepedet ad idetically distributed Gaussia. A oiformative prior is derived for the mea ad variace ad Bayes rule is used to compute the posterior Probability Desity Fuctio (PDF of (µ, σ as well as ( µ, σ i terms of the sufficiet statistics x = i xi ad C = i (xi x. From the joit distributio margials are determied. It is show ( that µ x C is distributed as Studet-t with degrees of freedom, σ C geeralized-gamma with c = ad a =, ad σ C is distributed as is distributed as iverted-gamma with a =. It is suggested to report the mea of these distributios as the estimate (or the peak if is too small for the mea to be defied ad a cofidece iterval surroudig the media. Itroductio A stadard cocept ecoutered by ayoe exposed to data is the idea of computig a mea, a variace, ad a stadard deviatio from the data. This paper will explore various approaches to computig estimates of mea, stadard-deviatio, ad variace from samples ad will coclude by recommedig a Bayesia approach to iferece about these values from data. Typically it is assumed that the data are realizatios of a collectio of idepedet, idetically distributed (i.i.d. radom variables. This radom vector is deoted X = X, X,..., X, ad the joit Probability Desity Fuctio (PDF of X is f X (X = f X (x i.. Traditioal mea estimate Commoly, the mea of X is estimated as the sample average: ˆµ = Oe ca the show i a rather satisfyig fashio that i= X i. i= E ˆµ = E X Var ˆµ = Var X. These statemets are typically used to justify this choice of estimator for the mea as ubiased ad cosistet. Sometimes this estimator is further justified by oticig that it is also the Maximum Likelihood (ML estimate for the mea assumig the oise comes from a expoetial family (e.g. Gaussia.
3 . Traditioal variace estimate The ML estimate for variace assumig Gaussia oise is It is sometimes suggested to use istead to esure that ˆσ ML = ˆσ = (X i ˆµ. i= (X i ˆµ i= E ˆσ = ˆσ. We are supposed to believe that this is preferable to a estimator that istead miimizes some other metric such as the mea-squared error (which icludes both bias ad variace. A good discussio of these cocepts will also metio that if the X i are all ormal radom variables, the ˆµ ad ˆσ are idepedet, ˆµ is ormal, ad ( ˆσ /σ is chi-squared with degrees of freedom. Cofidece itervals ca the be determied from these facts i a straightforward way..3 Stadard-deviatio estimates Typically, stadard-deviatio estimates are obtaied usig ˆσ = ˆσ. Typically, little is the said about the ucertaity of this estimate. Ofte, the square-root of the u-biased variace is take with little justificatio other tha coveiece ad despite the fact that ˆσ is geerally ot a u-biased estimate of σ eve whe ˆσ is..4 Outlie of the paper I this paper, the mea-square error of modified classical estimators for the variace ad stadard-deviatio will be compared. The poit of this compariso will be to elucidate which ormalizatio factor gives the smallest error (uder the hypothesis of ormally-distributed data. While istructive, this compariso does ot ed the discussio as it does ot address the questio of whether or ot the ormalizatio costat should be the oly issue i dispute. As a result, the problem will be addressed from a Bayesia perspective. Uder this perspective, I begi with the assumptio that the data has a commo mea ad variace ad use maximum etropy (with a flat prior to assert that the likelihood fuctio is ormal. Usig a flat prior for µ ad a Jeffrey s prior for σ (ad σ, the posterior probability of (µ, σ ad ( µ, σ is derived. From this joit posterior, the posterior probability for µ, σ, ad σ ca be give which leads to simple rules for a estimate ad cofidece iterval calculatios. Comparig various estimators Assumig the X i come from a stadard ormal populatio with mea µ ad variace σ, three estimators for σ ad σ will be compared i terms of the mea-squared error ad bias: the ubiased estimator: ˆσ UB ad ˆσ UB, the maximum-likelihood estimator: ˆσ ML ad ˆσ ML, ad 3 the Miimum Mea-Squared Estimator (MMSE (amog those of a certai class: ˆσ MMSE ad ˆσ MMSE. All three estimators of both quatities are of the form ˆσ = a ˆσ = (X i ˆµ i= a (X i ˆµ. i=
4 ˆθ For both classes of estimators, the bias, E, ad the mea-square error, E (ˆθ θ = E ˆθ θe ˆθ + θ, will be calculated assumig X i comes from a ormal distributio with mea µ ad variace σ. The idetity ( MSE ˆθ E (ˆθ θ = Var ˆθ + E ˆθ θ will be useful i what follows.. Estimators of variace For all three estimators of variace it is kow that uder the hypothesis of ormally distributed data, ˆσ /aσ is χ ad therefore has mea ad variace (. Cosequetly, E ˆσ = aσ ( E ˆσ 4 = a σ 4 ( + ( = a σ 4 ( E (ˆσ σ = E ˆσ 4 σ E ˆσ + σ 4 = σ 4 a ( a ( +. It ca be show that the maximum-likelihood estimator for σ requires a ML =. The ubiased estimator for σ is obviously a UB =. The miimum mea-square error estimator is foud by differetiatig Eq. (?? ad settig the result equal to zero. This procedure results i a MMSE = +. The three estimators ad their performace are summarized i the followig table: ˆσ E ˆσ MSE ˆσ UB (X i ˆµ σ σ 4 i= ML (X i ˆµ ( σ 4 σ i= MMSE (X i ˆµ σ σ + It is ot difficult to show that for > i= + < <, ad therefore i a mea-square sese, the MMSE ad ML estimators are both better tha the ubiased estimator. This example serves to show a geeral property that improved estimators are usually possible i a mea-square sese by usig biased estimators. Figure shows MSE ˆσ /σ 4 ad E ˆσ /σ for the three estimators whe >. 3 Estimators for σ Estimators for σ are ot ofte discussed, but are ofte used ad should, therefore, receive better treatmet. For ormally distributed data, the maximum likelihood estimator for σ is ˆσ ML = ˆσ ML = (X i ˆµ, i 3
5 Normalized Mea Ubiased Maximum Likelihood Miimum MSE Normalized R MSE Ubiased Maximum Likelihood Miimum MSE Figure : Normalized mea, E ˆσ /σ ad ormalized root-mea-square error (R-MSE, MSE ˆσ /σ 4, of several estimators of σ. 4
6 ad thus a ML =. The mea ad mea-square error for all three estimators ca be computed by oticig that ˆσ /σ a is χ (a chi radom variable with degrees of freedom. Because we ca coclude that where E χ = Γ ( Γ ( ( Γ Var χ = Γ ( ( aγ E ˆσ = σ Γ ( = t σ a E (ˆσ σ = Var ˆσ + (E ˆσ σ ( Γ ( ( = aσ ( aσ aγ Γ ( + σ Γ ( = σ a ( at + t = Γ ( Γ (. From these expressios, the ubiased estimator will result if a UB = /t while the miimum measquare estimator ca be foud by differetiatig with respect to a the expressio for mea-square error ad solvig for a. The result is a MMSE = t / (. The followig table summarizes the estimators ad their performace. UB ML MMSE ˆσ E ˆσ MSE ˆσ (X i ˆµ σ σ t t i= (X i ˆµ t σ σ t i= t (X i ˆµ t σ σ t i= It ca be show (or observed from the plot below that t < t < t. Therefore, comparig the estimators o the basis of mea-squared error results i the MMSE ad the ML estimator outperformig the ubiased estimator. Figure shows plots of E ˆσ /σ ad MSE ˆσ /σ to give some idea of the small-sample performace of these differet estimators o ormal data. 4 Bayesia Perspective Give data {x, x, x 3,..., x }, the task is to fid the mea µ, variace σ = v ad stadard-deviatio σ of these data. As stated the problem does t have a solutio. More iformatio is eeded i order to work 5
7 Normalized Mea Ubiased Maximum Likelihood Miimum MSE Normalized R MSE Ubiased Maximum Likelihood Miimum MSE Figure : Normalized mea, E ˆσ /σ ad ormalized root-mea-square error (R-MSE, MSE ˆσ /σ, of several estimators of σ. 6
8 towards a aswer. First, assume that data has a commo mea ad a commo variace. The priciple of maximum etropy ca the be applied uder these costraits (usig a flat igorace prior to choose the distributio f (X µ, σ = (π / σ exp σ (x i µ. which adds the least amout of iformatio to the problem other tha the assumptio of a commo µ ad σ. Notice that we ca use maximum etropy (with a flat igorace distributio so that etropy is f (x log f (x dx to justify the commo assumptio of ormal i.i.d. data. Usig Bayes rule we fid that f (µ, σ X = f (X µ, σ f (µ, σ f (X = D f (X µ, σ f (µ, σ where D is a ormalizig costat. This distributio tells us all the iformatio that is available about µ ad σ give the data X. We ca use this joit-pdf to estimate µ ad/or σ ad to report cofidece i the estimates. 4. Choosig the prior f (µ, σ Cetral to solvig this problem is choosig the prior kowledge for µ ad σ. Because we ca ormalize the radom variables usig Z µ σ to obtai zero-mea, uit variace radom variables, µ is a locatio parameter, ad σ is a scale parameter. Followig Jayes s reasoig, we choose the prior which expresses complete igorace except for the fact that µ is a locatio parameter ad σ is a scale parameter. I other words, we cosider a ew problem with data x which is shifted ad scaled versio of the old data. The prior i both of these case should be the same fuctio. However the prior has adjusted accordig to well-established rules. This defies a expressio that the prior should satisfy: f (µ, σ = af (µ + b, aσ where a > 0 ad b is a arbitrary real umber. The prior that satisfies this trasformatio equatio is the so-called Jeffrey s prior. f (µ, σ = cost σ. This prior is improper i the sese that it is ot ormalizable by itself. However, whe used to fid the posterior a total ormalizatio costat ca be foud. Specifically, f (µ, σ X = D exp σ+ σ (x i µ i ( = D exp σ+ σ x i µ x i + µ i i = D exp (µ x + C σ+ σ / i where x = i x i C = x x = i x i ( x i = (x i x i i 7
9 D = = 0 σ + C π Γ (. ( exp α + C σ dαdσ / This joit posterior PDF tells the whole story about µ ad σ if oly samples costraied to have the same µ ad σ are give. Usig this joit PDF we ca compute ay desired probability. Notice that > or else D 0 which is expressig the fact that with = there is o iformatio about σ whatever. Later, will be eeded the joit posterior PDF of µ ad v = σ which is f (µ, v X = G f (X µ, v f (µ, v where f (µ, v = cost ν so that we are just as uiformed about v as about σ. The, G = G = 4. Margial distributios 0 + v ( exp (µ x + C v/ C π Γ ( = D. The joit distributios provide all of the iformatio available about the parameters of iterest usig the data ad the assumptios. Notice that these distributios oly deped o the data (specifically, the statistics x ad C, ad ca be used easily to compute cofidece itervals. We ca itegrate out oe of the variables ad get just the margial desity fuctio of µ or σ separately. f (µ X = = D exp 0 σ+ Γ ( Γ ( πc (µ x + C σ dσ / + (µ x C / ( so that µ x C is Studet-t distributed with degrees of freedom. We aturally eed > for this distributio to provide iformatio. Whe = we have a improper distributio for µ proportioal to x µ. For other cases we ca deduce: E µ X = x dµdv Var µ X = C 3 arg max f (µ X µ = x. > 3 The margial distributio of σ is f (σ X = D σ + exp α + C σ / dα π = D σ exp C σ σ > 0 C = exp C σ Γ ( σ > 0. σ 8
10 σ Thus, C is geeralized gamma distributed with shape parameters c = ad a =. If =, the distributio reduces to a improper distributio proportioal to /σ (i.e. we have o additioal iformatio about σ other tha what we started with. For other values of we ca fid: Γ ( E σ X = Γ ( C >, Var σ X = ( 3 Γ Γ ( C > 3. arg max σ f (σ X = C. This distributio does ot have a well-defied mea uless > ad it does ot have a well-defied variace uless > 3. Fially, the margial distributio of v = σ is f (v X = = ( C Γ ( G v + exp α + C ν/ exp C v (+/ v dα v > 0 ν Whe =, this also reduces to a improper distributio proportioal to /v. For other values of, a iverted gamma distributio with a =. Useful parameters of this distributio are E σ X = 3 C > 3 Var σ X = arg max f ( σ X = σ C ( 3 ( 5 + C. > 5 Notice that this distributio does ot have a well-defied mea uless > 3 ad does ot have a well-defied variace uless > 5. To illustrate, the posterior probabilities for various umbers of samples, Figures 3, 4, 5 show ormalized plots of the Studet-t, geeralized Gamma, ad iverted gamma distributios for =3, 0, ad 50, correspodig to the mea, stadard-deviatio, ad variace of the data sample. 4.3 Gaussia approximatios The margial posterior distributios for µ, σ, ad σ all approach Normal distributios as. I particular, the posterior distributio for µ approaches a ormal distributio with mea x ad variace C. The posterior distributio for σ approaches a ormal distributio with mea C ad variace C. Fially, the posterior distributio for σ approaches a ormal distributio with mea C ad variace C. 4.4 Joit MAP estimators Joit Maximum A-Posterior (MAP estimators are sometimes useful Because µ X ad σ X (ad similarly µ X ad σ X are ot idepedet, the joit MAP estimator ca produce differet results tha the margial MAP estimators. These estimators miimize the joitly-uiform loss fuctio. To fid the joit estimator we solve ˆµ, ˆσ = arg max f (µ, σ X µ,σ = arg mi log f (µ, σ X µ,σ = arg mi µ,σ ( + log σ + (µ x + C σ / C is 9
11 Studet-t PDF with - DOF (Mea =3 =0 = (t (, (/ f (/ t=( x/c Figure 3: Graph of the posterior PDF for the mea for several values of. The fuctio f µ (t, ν = Γ( ν+ πνγ( ν + x ν ν+ is the PDF of the Studet-t distributio with ν Degrees Of Freedom (DOF. 0
12 Geeralized gamma PDF with c=-, a=(-/ (Std. =3 =0 = (s / (/,, ( / / f (/ s= (/C Figure 4: Graph of the posterior PDF for the stadard-deviatio for several values of. The fuctio f σ (s, c, a = c xca Γ(a exp ( x c x > 0 is the PDF of the geeralized gamma distributio with shape parameters c ad a.
13 Iverse gamma PDF with a=(-/ (Variace =3 =0 = (v /, ( / / (3/ f v= (/C Figure 5: Graph of the posterior PDF for the stadard-deviatio for several values of. The fuctio f σ (s, a = Γ(a x a exp ( x x > 0 is the PDF of the iverse gamma distributio with shape parameter a.
14 ad thus, 0 = (ˆµ x ˆσ / 0 = + ˆσ (ˆµ x + C ˆσ 3. / Solvig these simultaeously gives ˆµ JMAP = x, ˆσ JMAP = + C. The joit estimator for µ ad v ca also be foud i the same way. + ˆµ, ˆv = arg mi log v + (µ x + C. µ,v v/ Differetiatig results i Solvig these simultaeously gives 0 = ˆµ x ˆv/ 0 = + ν (ˆµ x + C v. / ˆµ JMAP = x ˆσ JMAP = + C. While the estimator for the mea is u-iterestig, a wide variety of ormalizatio costats to C show up i this aalysis. Usig these estimates, requires a particular devotio to maximizig f (µ, σ V istead of other estimatio approaches. The most useful approach to uderstadig estimates of µ, σ, ad σ is to determie cofidece itervals which is the subject of the ext sectio. 4.5 Cofidece itervals Oe of the advatages of the Bayesia perspective is that it automatically provides a method to obtai practical cofidece itervals for the estimates. With the probability desity fuctio give, cofidece itervals ca be costructed by fidig a area straddlig the mea (or the peak, or the media with equal areas o either side. Give the ature of the cofidece iterval as a area there is some aesthetic value i choosig the media as the middle value to surroud Geeral case Suppose there is a parameter with probability desity fuctio f (θ ad cumulative distributio fuctio F (θ. How is a cofidece iterval, a, b, costructed about the mea, peak, ad/or media. The iterval should be such that the probability of θ lyig withi the rage is α 00 percet, where α is a give parameter. I other words, the area uder f (θ over the cofidece iterval should be α. Suppose ˆθ is the positio about which it is desired a equal-area cofidece iterval. The, the two ed poits of the iterval ca be calculated from { P a θ ˆθ } = α } P {ˆθ θ b = α. 3
15 These state that (ˆθ F F (a = α (ˆθ F (b F = α so that (ˆθ a = F F α (ˆθ b = F F + α. (ˆθ For the cofidece iterval about the media, F = so that for that importat case. α a = F + α b = F 4.5. Mea For the case of the mea, we have see that the distributio of freedom. As a result, F µ (q = x + C F t (q; µ X x C/( is Studet-t with degrees of where Ft (q; ν is the iverse cumulative distributio fuctio (cdf of the Studet-t distributio with ν degrees of freedom. I additio, the mea, the media, ad the peak are all the same value. Note also that because the Studet-t distributio is symmetric: ( ( Ft q; ν = Ft + q; ν so that ( C + α a = x F t ( C + α b = x + F t ; ; Stadard deviatio For the case of the stadard deviatio, we have see that the distributio of (σ X gamma with c = ad a =. Therefore F σ (q = ( C F q; C is geeralized where F (q; a is the iverse cumulative distributio fuctio (cdf of the geeralized gamma distributio with parameters c = ad a: F (q; a = { Γ a, Γ (a q } / where Γ ( a, Γ (a, y = y. As a result: ( C a = F (F ˆσ ( C b = F (F ˆσ C ; C ; α ; + α ; 4
16 where F (x; a = Γ ( a, x /Γ (a is the cumulative distributio fuctio (cdf of the geeralized gamma with parameter a ad c =. Whe usig the media as the ceter poit, these expressios simplify to: ( C α a = F ;, ( C + α b = F ;. If the peak of the distributio is used as the ceter poit, the ( ( C a = F F ; ( ( C b = F F ; Variace We have see that the distributio of ( σ X C F σ (q = C F I α ; + α ; is iverted gamma with a =. Therefore, ( q; where F I (q; a is the iverse cdf of the iverted gamma distributio with parameter a: where Γ ( a, Γ (a, y = y. Therefore, F I (q; a = Γ a, Γ (a q a = C ( F I (F I b = C F I ˆσ C ; ( (F I ˆσ C ; α ; + α ; where F I (x; a = Γ ( a, x /Γ (a is the cdf of the iverted gamma with parameter a. Agai, if the media is used as the ceter poit, the these expressio simplify to a = C ( α F I ;, b = C ( + α F I ;. If the peak of the margial distributio is used as the ceter poit, the equatios are a = C ( ( F I F I + ; α ; b = C ( ( F I F I + ; + α ; Notice that the cofidece iterval for the variace is the square of the cofidece iterval for the stadard deviatio whe the media of the distributio is used i both cases. Also, care must be take for large α ad small that oe of the argumets to the iverse cdf are egative. Such a situatio, idicates that symmetry aroud the peak is impossible. Therefore, either the media should be used as the middle poit, or the area should be take from 0 to a upper boud, b. 5
17 5 Discussio Havig leared that the miimum mea-square estimator for θ from data X is E θ X, oe might be surprised by the fact that the expected value of the posterior margial distributio i this case does ot result i the same estimator as the miimum mea-square estimator over all classes of estimators of the form ac eve though it has the same form. For example, the classic estimator for ˆσ that gives miimum MSE is 3 but the Bayesia miimum mea-square estimator is C. Why are these differet? The differece comes i the subtle distictio betwee the two estimators. The former fids the value of a that miimizes the itegral (ac σ f (ˆσ dˆσ + C while the latter fids the fuctio of X (which happes to be ac (X that miimizes the itegral (g (X σ f ( X, σ dσ dx. This fial itegral icludes a averagig itegral agaist the o-iformative prior as well as a itegral over the data. These are two etirely differet optimizatio problems ad should ot be expected to provide the same result. It is importat to uderstad the full probability distributio of µ, σ, ad σ especially whe the umber of data-samples is small. For example, the variace of the posterior probability distributio for σ is ot eve defied if 5. As a result, it ca be impossible to just report the mea ad variace. A cofidece iterval (or high-desity regio aroud the media of the distributio is always possible. 6 Summary ad Coclusios I this paper, a study of several estimators for the mea, variace, ad stadard-deviatio of data was preseted. I particular, it was show that the ubiased estimator for variace so commoly used is ot typically a good choice (especially for small because usig + as a divisor rather tha shriks the mea-square error of the estimator. I additio, a fully Bayesia perspective o the problem of estimatig a commo mea ad variace from samples was preseted usig maximum etropy ad o-iformative priors. The results provide the posterior coditioal PDF of the mea, stadard-deviatio, ad variace from which estimates ad cofidece itervals ca be calculated. The results also emphasize the poit that calculatig the Bayesia Mea-Square Error (MSE is ot ecessarily the same as other o-bayesia defiitios of MSE because it ivolves aother averagig itegral over the prior iformatio o the quatity to be estimated. The mea of the coditioal PDF miimizes Bayesia MSE. Table summarizes the results for uderstadig µ, σ, ad σ from data assumed to have a commo mea ad variace. The table requires oly the sufficiet statistics It is also very useful to ote that x = C = i= x i (x i x i= ( µ x C is Studet-t distributed with degrees of freedom, σ C is geeralized gamma distributed with shape parameters c = ad a =, ad σ C is iverted gamma with shape parameter a =. These fial facts ca be used to compute estimates ad cofidece itervals from stadardized tables ad distributios. 6
18 Table : Summary of posterior probability distributios for µ, σ, ad σ. PDF: f ( X Mode E Var Γ( µ / Γ( + (µ x C πc C x x 3 > 3 (C/ σ ( / exp Γ( C σ σ σ > 0 C Γ( C > Γ( 3 Γ ( Γ ( σ (C/ ( / exp Γ( C (σ (+/ σ σ > 0 + C 3 C > 3 C ( 3 ( 5 C > 3 > 5 7 Refereces Jayes, E. T., Probability Theory: The Logic of Sciece, Cambridge Uiversity Press, 003. Jeffreys, Sir Harold, Theory of Probability, 3rd editio, Oxford Uiversity Press, 96. 7
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