Physics 403. Segev BenZvi. Parameter Estimation, Correlations, and Error Bars. Department of Physics and Astronomy University of Rochester

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1 Physics 403 Parameter Estimation, Correlations, and Error Bars Segev BenZvi Department of Physics and Astronomy University of Rochester

2 Table of Contents 1 Review of Last Class Best Estimates and Reliability Properties of a Good Estimator 2 Parameter Estimation in Multiple Dimensions Return of the Quadratic Approximation The Hessian Matrix and its Geometrical Interpretation Maximum of the Quadratic Form Covariance 3 Multidimensional Estimators Gaussian Mean and Width Student-t Distribution χ 2 Distribution Segev BenZvi (UR) PHY / 35

3 Best Estimates and Reliability We can identify the best estimator ˆx of a PDF by maximizing p(x D, I ): dp d 2 p = 0, dx ˆx dx 2 < 0 ˆx We assessed the reliability of the estimator by Taylor expanding L = ln p about the best value: ( ˆσ 2 = d 2 L dx 2 ˆx ) 1 This only works when the quadratic approximation is reasonable For an asymmetric PDF, it s better to use a confidence interval when reporting the reliability of an estimate For a multimodal PDF, there could be no single best estimate, and calculating reliability becomes complicated. Don t summarize the PDF, just report the whole thing Segev BenZvi (UR) PHY / 35

4 Example Estimators from Last Class Best estimator of binomial probability p (n successes in N trials): ˆp = n N, ˆσ2 = n(n n) N 3 Arithmetic mean: best estimator of Gaussian with known variance σ 2 : ˆµ = 1 N N x i, i=1 ˆσ 2 = σ2 N Weighted mean: best estimator of Gaussian with different error bars: ˆµ = / N N x i w i w i, ˆσ 2 = i=1 i=1 1 N i=1 w, w i = 1/σi 2 i Reduces to arithmetic result when σ i = σ. Segev BenZvi (UR) PHY / 35

5 Bayesian vs. Frequentist Interpretations Bayesian: given a measurement, we have some confidence that our best estimate of a parameter lies within some range of the data Frequentist: given the true value of the parameters, we have some confidence that our measurement lies within some range of the true value Difference: p(θ D, I ) versus p(d θ, I ). Segev BenZvi (UR) PHY / 35

6 (Frequentist) Properties of a Good Estimator A good estimator should be: 1. Consistent. The estimate tends toward the true value with more data: lim ˆθ = θ N 2. Unbiased. The expectation value is equal to the true value: b = ˆθ θ = dx p(x θ) ˆθ(x) θ = 0 3. Efficient. The variance of the estimator is as small as possible (minimum variance bound, to be discussed): var (ˆθ) = dx p(x θ) (ˆθ(x) ˆθ) 2 MSE = (ˆθ θ) 2 = var (ˆθ) + b 2 It is not always possible to satisfy all three requirements. Segev BenZvi (UR) PHY / 35

7 Table of Contents 1 Review of Last Class Best Estimates and Reliability Properties of a Good Estimator 2 Parameter Estimation in Multiple Dimensions Return of the Quadratic Approximation The Hessian Matrix and its Geometrical Interpretation Maximum of the Quadratic Form Covariance 3 Multidimensional Estimators Gaussian Mean and Width Student-t Distribution χ 2 Distribution Segev BenZvi (UR) PHY / 35

8 Parameter Estimation in Higher Dimensions Moving to more dimensions: x x, p(x D, I ) p(x D, I ) As in the 1D case, the posterior PDF still encodes all the information we need to get the best estimator. The maximum of the PDF gives the best estimate of the quantities x = {x j }. We solve the set of simultaneous equations p x i = 0 {ˆxj } Question: how to we make sure that we re at the maximum and not a minimum or a saddle point? Segev BenZvi (UR) PHY / 35

9 The Quadratic Approximation Revisited It s easier to deal with L = ln p({x j } D, I ), so let s do that. Let s also simplify to 2D, without loss of generality, so that x = (x, y). The maximum of the posterior satisfies L L = 0 and = 0 x ˆx,ŷ y ˆx,ŷ Look at the behavior of L about the maximum using its Taylor expansion: L = L(ˆx, ŷ) L 2 x 2 (x ˆx) L ˆx,ŷ 2 y 2 (y ŷ) 2 ˆx,ŷ + 2 L (x ˆx)(y ŷ) +... x y ˆx,ŷ where the linear terms are zero because we re at the maximum. Segev BenZvi (UR) PHY / 35

10 The Hessian Matrix As in the 1D case, the quadratic terms in the expansion dominate the behavior near the maximum. Insight: rewrite the quadratic terms in matrix notation: Q = 1 ( ) ( ) ( ) A C x ˆx x ˆx y ŷ 2 C B y ŷ = 1 2 (x ˆx) H(ˆx)(x ˆx) where H(ˆx) is a 2 2 symmetric matrix with components A = 2 L x 2, B = 2 L ˆx,ŷ y 2, C = 2 L ˆx,ŷ x y Note: H( ˆx), the matrix of second derivatives, is called the Hessian matrix of L. ˆx,ŷ Segev BenZvi (UR) PHY / 35

11 Geometrical Interpretation Contour of Q in xy plane is an ellipse centered at (ˆx, ŷ) Orientation and eccentricity are determined by the values of A, B, and C Principal axes correspond to the eigenvectors of H. I.e., if we solve ( A C C B ) ( x y Hx = λx ) = λ ( x y ) we get two eigenvalues λ 1 and λ 2 which are inversely related to the square of the semi-major and semi-minor axes of the ellipse Segev BenZvi (UR) PHY / 35

12 Condition for a Maximum L(ˆx) is a maximum if the quadratic form Q(x ˆx) = Q( x) < 0 x. If H is symmetric, there exists an orthogonal matrix O = ( ) e 1 e 2 such that ( ) O λ1 0 HO = D = 0 λ 2 where e 1 and e 2 are the eigenvectors of H. Therefore, H = ODO, and we can express Q as Q x H x = x (ODO ) x = (O x) D(O x) = x D x = λ 1 (x ˆx) 2 + λ 2 (y ŷ) 2 Q < 0 iff λ 1 and λ 2 are both negative. Segev BenZvi (UR) PHY / 35

13 Condition for a Maximum The eigenvalues of H are given by where λ 1(2) = 1 2 Tr H + ( ) (Tr H) 2 /4 det H Tr H = A + B, det H = AB C 2 Intuition: what happens if the cross term C = 0? Then the principal axes of the ellipse defined by Q are aligned with the x and y axes and the eigenvalues reduce to λ 1 = A, λ 2 = B Analogous to the 1D case, we can associate the error bars on ˆx and ŷ as the inverse root of the diagonal terms of the Hessian, or ( ) 1 ( ) 1 ˆσ x 2 = λ 1 1 = 2 L x 2, ˆσ y 2 = λ 2 1 = 2 L y 2 ˆx,ŷ ˆx,ŷ Segev BenZvi (UR) PHY / 35

14 General Case: C 0 What happens when the off-diagonal term of H is nonzero? Let s work in 2D. If we were only interested in the reliability of ˆx, then we would evaluate the behavior of the marginal distribution about the maximum p(x D, I ) = p(x, y D, I ) dy Using our quadratic approximation, p(x, y D, I ) = exp L exp Q: ( ) 1 p(x D, I ) exp 2 x H x dy ( ) 1 = exp 2 (Ax 2 + By 2 + 2Cxy) dy, where (without loss of generality) we set ˆx = ŷ = 0. Segev BenZvi (UR) PHY / 35

15 General Case: C 0 Solving the Gaussian Integral Factor out terms in x, and explicitly change signs because we know that Q < 0: p(x D, I ) = e 1 2 (Ax2 +By 2 +2Cxy) dy = e 1 2 Ax2 e 1 2 (By 2 +2Cxy) dy = e 1 2 (A+ C2 B ) x 2 e 1 Cx B(y+ 2 B ) 2 where we completed the square: By 2 + 2Cxy = B(y + Cx/B) 2 C 2 x 2 /B, allowing us to rearrange the xy cross term. The remaining integral is a Gaussian integral of form ) exp ( u2 2σ 2 du = σ 2π dy Segev BenZvi (UR) PHY / 35

16 General Case: C 0 Expressions for σ x and σ y Therefore, the marginal distribution becomes ( 2π p(x D, I ) = B exp 1 AB C 2 ) x 2 2 B 2π = ( B exp x 2 ), where σ 2 x = 2σ 2 x B AB C 2 = H yy det H Similarly, if we solve instead for p(y D, I ), we ll find that σ 2 y = A AB C 2 = H xx det H Note: we absorbed a negative sign back into A and B to match the properties of the Hessian. Segev BenZvi (UR) PHY / 35

17 Connection to Variance and Covariance Recall the definition of variance for a 1D PDF: var (x) = (x µ) 2 = dx (x µ) 2 p(x D, I ) This can extended using the 2D PDF σx 2 = (x ˆx) 2 = dx dy (x ˆx) 2 p(x, y D, I ) If we use the quadratic approximation for p(x, y D, I ), we find and similarly, σ 2 x = H yy det H σ 2 y = (y ŷ) 2 = H xx det H, the same expressions we just derived (convince yourself). Segev BenZvi (UR) PHY / 35

18 Connection to Variance and Covariance Also recall the definition of covariance: σxy 2 = (x ˆx)(y ŷ) = dx dy (x ˆx)(y ŷ) p(x, y D, I ) = C AB C 2 = H xy det H if we use the quadratic expansion of p(x, y D, I ). Putting it all together: the covariance matrix, defined a couple of weeks ago, is the negative inverse of the Hessian matrix: ( ) σ 2 x σ xy = σ xy σ 2 y ( ) 1 B C AB C 2 = C A ( ) 1 A C = H 1 (ˆx) C B Segev BenZvi (UR) PHY / 35

19 Covariance Matrix Geometric Interpretation C = 0 implies x and y are completely uncorrelated. The contours of the posterior PDF are symmetric As C increases, the PDF becomes more and more elongated For C = ± AB, the contours are infinitely wide in one direction (though the prior on x or y could vanish somewhere) Also, while C = ± AB implies ˆx and ŷ are totally unreliable, the linear correlation y = ±mx (with m = AB) can still be inferred Segev BenZvi (UR) PHY / 35

20 Caution: Using the Correct Error Bar Be careful about calculating the uncertainty on a parameter in a multidimensional PDF Right: σii 2 = H 1 ii, from marginalization of p(x D, I ) Wrong: get σ 2 ii by holding parameters x j i fixed at their optimal values (underestimate!) See difference in error bars from two procedures at left Reason: when using the Hessian, don t confuse the inverse of the diagonals of H for the diagonals of H 1 Segev BenZvi (UR) PHY / 35

21 Table of Contents 1 Review of Last Class Best Estimates and Reliability Properties of a Good Estimator 2 Parameter Estimation in Multiple Dimensions Return of the Quadratic Approximation The Hessian Matrix and its Geometrical Interpretation Maximum of the Quadratic Form Covariance 3 Multidimensional Estimators Gaussian Mean and Width Student-t Distribution χ 2 Distribution Segev BenZvi (UR) PHY / 35

22 Gaussian PDF: Both µ and σ 2 Unknown Last time we derived best estimators for a Gaussian distribution using p(µ σ, D, I ), i.e., σ was given. Now we have the tools to calculate p(µ D, I ) = 0 p(µ, σ D, I ) dσ. I.e., we can calculate the best estimator for σ 2 not known a priori. First we have to express the joint posterior PDF to a likelihood and prior using Bayes Theorem: p(µ, σ D, I ) p(d µ, σ, I ) p(µ, σ I ) If the data are independent, then by the product rule [ ] p(d µ, σ, I ) = (2πσ 2 ) N/2 exp 1 N 2σ 2 (x i µ) 2 i=1 Segev BenZvi (UR) PHY / 35

23 Gaussian PDF: Priors on µ, σ Now we need to define the prior p(µ, σ I ). Let s assume the priors for µ and σ are independent: p(µ, σ I ) = p(µ I ) p(σ I ) Since µ is a location parameter it makes sense to choose a uniform prior 1 p(µ I ) = µ max µ min Since σ is a scale parameter we ll use a Jeffreys prior: p(σ I ) = 1 σ ln (σ max /σ min ) Let s also assume the prior ranges on µ and σ are large and don t cut off the integration in a weird way Segev BenZvi (UR) PHY / 35

24 Aside: Parameterization of σ Note that we parameterized our width prior in terms of σ, not the variance σ 2. Does the parameterization make a difference? For the Jeffreys prior in σ, p(σ I ) dσ = k dσ σ where k depends on the limits of σ. Now convert to variance ν. Since σ = ν, Therefore, dσ = dν 2 ν p(σ I ) dσ = p(ν I ) dν = k dν 2ν = k dν ν So the Jeffreys prior has the same form if we work in terms of σ or σ 2. Question: would this also be the case for a uniform prior? Segev BenZvi (UR) PHY / 35

25 Posterior PDF of µ Substitute the likelihood and prior into our expression for p(µ D, I ): p(µ D, I ) = 0 p(d µ, σ, I ) p(µ I ) p(σ I ) dσ (2π) N/2 µ ln (σ max /σ min ) σmax σ min σ (N+1) e 1 N 2σ 2 i=1 (x i µ) 2 dσ Let σ = 1/t so that dσ = dt/t 2 : p(µ D, I ) tmax t min t N 1 e t2 N i=1 (x i µ) 2 dt Change variables again so that τ = t (xi µ) 2 : [ N ] N/2 p(µ D, I ) (x i µ) 2 i=1 Segev BenZvi (UR) PHY / 35

26 Best Estimator and Reliability As in past calculations, we maximize L = ln p: [ L = N N ] 2 ln (x i µ) 2 i=1 dl = N N i=1 (x i ˆµ) dµ N ˆµ i=1 (x i ˆµ) = 0 2 This can only be satisfied if the numerator is zero, so ˆµ = x = 1 N N i=1 x i In other words, the best estimate of the PDF is still just the arithmetic mean of the measurements x i Segev BenZvi (UR) PHY / 35

27 Best Estimator and Reliability The second derivative gives the estimate of the width: d 2 L N 2 dµ 2 = N ˆµ i=1 (x i ˆµ) 2 Therefore, setting ˆσ 2 = (d 2 L/dµ 2 ) 1 we find that µ = ˆµ ± S N, where we define S 2 = 1 N N (x i ˆµ) 2 = 1 N i=1 N (x i x) 2 i=1 This is almost the usual definition of sample variance but it s narrower because we divide by 1/N instead of 1/(N 1). Segev BenZvi (UR) PHY / 35

28 Aside: Uniform Distribution in σ Suppose at the beginning of this problem we didn t choose a Jeffreys prior for σ, but a uniform prior such that { constant σ > 0 p(σ I ) = 0 otherwise In this case, the posterior PDF would have been [ N ] (N 1)/2 p(µ D, I ) (x i µ) 2 i=1 and the width estimator would have been the usual sample variance S 2 = 1 N 1 N i=1 (x i ˆµ) 2 = 1 N 1 N (x i x) 2 In other words, the Jeffreys prior gives us a narrower constraint on ˆµ! Segev BenZvi (UR) PHY / 35 i=1

29 Student-t Distribution Let s look back at our PDF but not make the quadratic approximation. First, write where N (x i µ) 2 = N( x µ) 2 + V, i=1 V = N (x i x) 2 i=1 Substituting into the PDF gives p(µ D, I ) [ N( x µ) 2 + V ] N/2 This is the heavy-tailed Student-t distribution, used for estimating µ when σ is unknown and N is small Segev BenZvi (UR) PHY / 35

30 Student-t Distribution Published pseudonymously by William S. Gosset of Guinness Brewery in 1908 [1] t-distributions describe small samples drawn from a normally distributed population Used to estimate the error on a mean when only a few samples N are available, σ unknown Basis of the frequentist t-test to compare two data sets As N large, the tails of the distribution are killed off (Central Limit Theorem) Segev BenZvi (UR) PHY / 35

31 Best Estimate of σ Now that we ve calculate the best estimate of a mean, what s the best estimate of σ given a set of measurements? Start with the posterior PDF p(σ D, I ): p(σ D, I ) = = p(µ, σ D, I ) dµ Plugging in our likelihood and priors gives p(σ D, I ) = p(d µ, σ, I ) p(µ I ) p(σ I ) dµ (2π) N/2 µmax µ ln (σ max /σ min ) σ (N+1) e µ min µmax σ (N+1) e V 2σ 2 e N( x µ) 2 2σ 2 dµ µ min 1 N 2σ 2 i=1 (x i µ) 2 dµ Segev BenZvi (UR) PHY / 35

32 χ 2 Distribution Ignoring all constant terms (including the integral over µ) leaves ( p(σ D, I ) σ N exp V ) 2σ 2 Note that if we had used a uniform prior for σ we would have ( p(σ D, I ) σ (N 1) exp V ) 2σ 2 Let s maximize this expression: L = ln p = (N 1) ln σ V 2σ 2 dl (N 1) = + V dσ ˆσ σ σ 3 = 0 ˆσ 2 = V N 1 = 1 N (x i x) 2 = s 2 N 1 i=1 Segev BenZvi (UR) PHY / 35

33 χ 2 Distribution Taking the second derivative of L gives d 2 L dσ 2 = N 1 ˆσ ˆσ 2 3V ˆσ 4 (N 1)ˆσ2 = ˆσ 4 2(N 1) = ˆσ 2 Therefore, the optimal value of the width is σ = ˆσ ± ˆσ 2(N 1) 3(N 1)ˆσ2 ˆσ 4 Note: with the change of variables X = V /σ 2, we see that ( p(σ D, I ) σ (N 1) exp X ) 2 is the χ 2 ν distribution with ν = 2(N 1). Segev BenZvi (UR) PHY / 35

34 Summary We related the width of a multidimensional distribution the Hessian matrix H to the covariance matrix via [σ 2 ] ij = [ H 1 ] ij Caution: the right way to get the uncertainty on a parameter from a multidimensional distribution is to marginalize p(x, y,... D, I ) The wrong way to get the uncertainty on a parameter from such a distribution is to fix parameters y, z,... at the optimal values and find the uncertainty on x When marginalizing σ in a Gaussian distribution, we obtain the Student-t distribution Whem marginalizing µ in a Gaussian distribution, we obtain the χ 2 2(N 1) distribution Segev BenZvi (UR) PHY / 35

35 References I [1] Student (W.S. Gosset). The Probable Error of a Mean. In: Biometrika 6 (1908), pp Segev BenZvi (UR) PHY / 35