Efficient Sampling for Gaussian Process Inference using Control Variables

Transcription

1 Effent Samplng for Gaussan Proess Inferene usng Control Varables Mhals K. Ttsas, Nel D. Lawrene and Magnus Rattray Shool of Computer Sene, Unversty of Manhester Manhester M 9PL, UK Abstrat Samplng funtons n Gaussan proess (GP) models s hallengng beause of the hghly orrelated posteror dstrbuton. We desrbe an effent Markov han Monte Carlo algorthm for samplng from the posteror proess of the GP model. Ths algorthm uses ontrol varables whh are auxlary funton values that provde a low dmensonal representaton of the funton. At eah teraton, the algorthm proposes new values for the ontrol varables and generates the funton from the ondtonal GP pror. The ontrol varable nput loatons are found by mnmzng an objetve funton. We demonstrate the algorthm on regresson and lassfaton problems and we use t to estmate the parameters of a dfferental equaton model of gene regulaton. Introduton Gaussan proesses (GPs) are used for Bayesan non-parametr estmaton of unobserved or latent funtons. In regresson problems wth Gaussan lkelhoods, nferene n GP models s analytally tratable, whle for lassfaton determnst approxmate nferene algorthms are wdely used [6, 4,, ]. However, n reent applatons of GP models n systems bology [] that requre the estmaton of ordnary dfferental equaton models [,, 8], the development of determnst approxmatons s dffult sne the lkelhood an be hghly omplex. Other applatons of Gaussan proesses where nferene s ntratable arse n spato-temporal models and geostatsts and determnst approxmatons have also been developed there [4]. In ths paper, we onsder Markov han Monte Carlo (MCMC) algorthms for nferene n GP models. An advantage of MCMC over determnst approxmate nferene s that t provdes an arbtrarly prese approxmaton to the posteror dstrbuton n the lmt of long runs. Another advantage s that the samplng sheme wll often not depend on detals of the lkelhood funton, and s therefore very generally applable. In order to beneft from the advantages of MCMC t s neessary to develop an effent samplng strategy. Ths has proved to be partularly dffult n many GP applatons, beause the posteror dstrbuton desrbes a hghly orrelated hgh-dmensonal varable. Thus smple MCMC samplng shemes suh as Gbbs samplng an be very neffent. In ths ontrbuton we desrbe an effent MCMC algorthm for samplng from the posteror proess of a GP model whh onstruts the proposal dstrbutons by utlzng the GP pror. Ths algorthm uses ontrol varables whh are auxlary funton values. At eah teraton, the algorthm proposes new values for the ontrol varables and samples the funton by drawng from the ondtonal GP pror. The ontrol varables are hghly nformatve ponts that provde a low dmensonal representaton of the funton. The ontrol nput loatons are found by mnmzng an objetve funton. The objetve funton used s the expeted least squares error of reonstrutng the funton values from the ontrol varables, where the expetaton s over the GP pror. We demonstrate the proposed MCMC algorthm on regresson and lassfaton problems and ompare t wth two Gbbs samplng shemes. We also apply the algorthm to nferene n a systems

2 bology model where a set of genes s regulated by a transrpton fator proten [8]. Ths provdes an example of a problem wth a non-lnear and non-fatorzed lkelhood funton. Samplng algorthms for Gaussan Proess models In a GP model we assume a set of nputs (x,..., x N ) and a set of funton values f = (f,..., f N ) evaluated at those nputs. A Gaussan proess plaes a pror on f whh s a N-dmensonal Gaussan dstrbuton so that p(f) = N(y µ, K). The mean µ s typally zero and the ovarane matrx K s defned by the kernel funton k(x n, x m ) that depends on parameters θ. GPs are wdely used for supervsed learnng [] n whh ase we have a set of observed pars (y, x ), where =,..., N, and we assume a lkelhood model p(y f) that depends on parameters α. For regresson or lassfaton problems, the latent funton values are evaluated at the observed nputs and the lkelhood fatorzes aordng to p(y f) = N = p(y f ). However, for other type of applatons, suh as modellng latent funtons n ordnary dfferental equatons, the above fatorzaton s not applable. Assumng that we have obtaned sutable values for the model parameters (θ, α) nferene over f s done by applyng Bayes rule: p(f y) p(y f)p(f). () For regresson, where the lkelhood s Gaussan, the above posteror s a Gaussan dstrbuton that an be obtaned usng smple algebra. When the lkelhood p(y f) s non-gaussan, omputatons beome ntratable and we need to arry out approxmate nferene. The MCMC algorthm we onsder s the general Metropols-Hastngs (MH) algorthm []. Suppose we wsh to sample from the posteror n eq. (). The MH algorthm forms a Markov han. We ntalze f () and we onsder a proposal dstrbuton Q(f (t+) f (t) ) that allows us to draw a new state gven the urrent state. The new state s aepted wth probablty mn(, A) where A = p(y f (t+) )p(f (t+) ) Q(f (t) f (t+) ) p(y f (t) )p(f (t) ) Q(f (t+) f (t) ). () To apply ths gener algorthm, we need to hoose the proposal dstrbuton Q. For GP models, fndng a good proposal dstrbuton s hallengng sne f s hgh dmensonal and the posteror dstrbuton an be hghly orrelated. To motvate the algorthm presented n seton., we dsuss two extreme optons for spefyng the proposal dstrbuton Q. One smple way to hoose Q s to set t equal to the GP pror p(f). Ths gves us an ndependent MH algorthm []. However, samplng from the GP pror s very neffent as t s unlkely to obtan a sample that wll ft the data. Thus the Markov han wll get stuk n the same state for thousands of teratons. On the other hand, samplng from the pror s appealng beause any generated sample satsfes the smoothness requrement mposed by the ovarane funton. Funtons drawn from the posteror GP proess should satsfy the same smoothness requrement as well. The other extreme hoe for the proposal, that has been onsdered n [], s to apply Gbbs samplng where we teratvely draw samples from eah posteror ondtonal densty p(f f, y) wth f = f \f. However, Gbbs samplng an be extremely slow for densely dsretzed funtons, as n the regresson problem of Fgure, where the posteror GP proess s hghly orrelated. To larfy ths, note that the varane of the posteror ondtonal p(f f, y) s smaller or equal to the varane of the ondtonal GP pror p(f f ). However, p(f f ) may already have a tny varane aused by the ondtonng on all remanng latent funton values. For the one-dmensonal example n Fgure, Gbbs samplng s pratally not applable. We further study ths ssue n seton 4. A smlar algorthm to Gbbs samplng an be expressed by usng the sequene of the ondtonal denstes p(f f ) as a proposal dstrbuton for the MH algorthm. We all ths algorthm the Gbbs-lke algorthm. Ths algorthm an exhbt a hgh aeptane rate, but t s neffent to sample from hghly orrelated funtons. A smple generalzaton of the Gbbs-lke algorthm that s more approprate for samplng from smooth funtons s to dvde the doman of the funton nto regons and sample the entre funton wthn eah regon by ondtonng on the remanng funton regons. Loal regon samplng teratvely draws eah blok of funtons values f k from Thus we replae the proposal dstrbuton p(f f, y) wth the pror ondtonal p(f f ).

3 the ondtonal GP pror p(f t+ k k ), where f k = f \ f k. However, ths sheme s stll neffent to sample from hghly orrelated funtons sne the varane of the proposal dstrbuton an be very small lose to the boundares between neghbourng funton regons. The desrpton of ths algorthm s gven n the supplementary materal. In the next seton we dsuss an algorthm usng ontrol varables that an effently sample from hghly orrelated funtons. f (t). Samplng usng ontrol varables Let f be a set of M auxlary funton values that are evaluated at nputs X and drawn from the GP pror. We all f the ontrol varables and ther meanng s analogous to the auxlary ndung varables used n sparse GP models []. To ompute the posteror p(f y) based on ontrol varables we use the expresson p(f y) = p(f f, y)p(f y)df. () f Assumng that f s hghly nformatve about f, so that p(f f, y) p(f f ), we an approxmately sample from p(f y) n a two-stage manner: frstly sample the ontrol varables from p(f y) and then generate f from the ondtonal pror p(f f ). Ths sheme an allow us to ntrodue a MH ), that wll mm samplng from p(f y), and always sample f from the ondtonal pror p(f f ). The whole proposal dstrbuton takes the form algorthm, where we need to spefy only a proposal dstrbuton q(f (t+) Q(f (t+), f (t+) f (t), f (t) ) = p(f (t+) f (t+) f (t) )q(f (t+) f (t) ). (4) Eah proposed sample s aepted wth probablty mn(, A) where A s gven by A = p(y f (t+) )p(f (t+) ). p(y f (t) )p(f (t) ) (t) q(f f (t+) ) q(f (t+) f (t) ). () The usefulness of the above samplng sheme stems from the fat that the ontrol varables an form a low-dmensonal representaton of the funton. Assumng that these varables are muh fewer than the ponts n f, the samplng s manly arred out n the low dmensonal spae. In seton. we desrbe how to selet the number M of ontrol varables and the nputs X so as f beomes hghly nformatve about f. In the remander of ths seton we dsuss how we set the proposal dstrbuton q(f (t+) f (t) ). A sutable hoe for q s to use a Gaussan dstrbuton wth dagonal or full ovarane matrx. The ovarane matrx an be adapted durng the burn-n phase of MCMC n order to nrease the aeptane rate. Although ths sheme s general, t has pratal lmtatons. Frstly, tunng a full ovarane matrx s tme onsumng and n our ase ths adapton proess must be arred out smultaneously wth searhng for an approprate set of ontrol varables. Also, sne the terms nvolvng p(f ) do not anel out n the aeptane probablty n eq. (), usng a dagonal ovarane for the q dstrbuton has the rsk of proposng ontrol varables that may not satsfy the GP pror smoothness requrement. To avod these problems, we defne q by utlzng the GP pror. Aordng to eq. () a sutable hoe for q must mm the samplng from the posteror p(f y). Gven that the ontrol ponts are far apart from eah other, Gbbs samplng n the ontrol varables spae an be effent. However, teratvely samplng f from the ondtonal posteror p(f f, y) p(y f )p(f f ), where f = f \ f s ntratable for non-gaussan lkelhoods. An attratve alternatve s to use a Gbbs-lke algorthm where eah f s drawn from the ondtonal GP pror p(f (t+) f (t) ) and s aepted usng the MH step. More spefally, the proposal dstrbuton draws a new f (t+) for a ertan ontrol varable from p(f (t+) f (t) ) and generates the funton f (t+) from p(f (t+) f (t+), f (t) ). The sample (f (t+), f (t+) ) s aepted usng the MH step. Ths sheme of samplng the ontrol varables one-at-a-tme and resamplng f s terated between dfferent ontrol varables. A omplete teraton of the algorthm onssts of a full san over all ontrol varables. The aeptane probablty A n eq. () beomes the lkelhood rato and the pror smoothness requrement s always satsfed. The teraton between dfferent ontrol varables s llustrated n Fgure. Ths s beause we need to ntegrate out f n order to ompute p(y f ).

4 Fgure : Vsualzaton of teratng between ontrol varables. The red sold lne s the urrent f (t), the blue lne s the proposed f (t+), the red rles are the urrent ontrol varables f (t) whle the damond (n magenta) s the proposed ontrol varable f (t+). The blue sold vertal lne represents the dstrbuton p(f (t+) (wth two-standard error bars) and the shaded area shows the effetve proposal p(f t+ f (t) ). f (t) ) Although the ontrol varables are sampled one-at-at-tme, f an stll be drawn wth a onsderable varane. To larfy ths, note that when the ontrol varable f hanges the effetve proposal dstrbuton for f s p(f t+ f (t) ) = p(f t+ f f (t+) (t+), f (t) )p(f (t+) f (t) )df (t+), whh s the ondtonal GP pror gven all the ontrol ponts apart from the urrent pont f. Ths ondtonal pror an have onsderable varane lose to f and n all regons that are not lose to the remanng ontrol varables. As llustrated n Fgure, the teraton over dfferent ontrol varables allow f to be drawn wth a onsderable varane everywhere n the nput spae.. Seleton of the ontrol varables To apply the prevous algorthm we need to selet the number, M, of the ontrol ponts and the assoated nputs X. X must be hosen so that knowledge of f an determne f wth small error. The predton of f gven f s equal to K f, K, f whh s the mean of the ondtonal pror p(f f ). A sutable way to searh over X s to mnmze the reonstruton error f K f, K, f averaged over any possble value of (f, f ): G(X ) = f K f, K, f p(f f )p(f )dfdf = Tr(K f,f K f, K, Kf,). T f,f The quantty nsde the trae s the ovarane of p(f f ) and thus G(X ) s the total varane of ths dstrbuton. We an mnmze G(X ) w.r.t. X usng ontnuous optmzaton smlarly to the approah n []. Note that when G(X ) beomes zero, p(f f ) beomes a delta funton. To fnd the number M of ontrol ponts we mnmze G(X ) by nrementally addng ontrol varables untl the total varane of p(f f ) beomes smaller than a ertan perentage of the total varane of the pror p(f). % was the threshold used n all our experments. Then we start the smulaton and we observe the aeptane rate of the Markov han. Aordng to standard heursts [] whh suggest that desrable aeptane rates of MH algorthms are around /4, we requre a full teraton of the algorthm (a omplete san over the ontrol varables) to have an aeptane rate larger than /4. When for the urrent set of ontrol nputs X the han has a low aeptane rate, t means that the varane of p(f f ) s stll too hgh and we need to add more ontrol ponts n order to further redue G(X ). The proess of observng the aeptane rate and addng ontrol varables s ontnued untl we reah the desrable aeptane rate. When the tranng nputs X are plaed unformly n the spae, and the kernel funton s statonary, the mnmzaton of G plaes X n a regular grd. In general, the mnmzaton of G plaes the ontrol nputs lose to the lusters of the nput data n suh a way that the kernel funton s taken nto aount. Ths suggests that G an also be used for learnng ndung varables n sparse GP models n a unsupervsed fashon, where the observed outputs y are not nvolved.

5 Applatons We onsder two applatons where exat nferene s ntratable due to a non-lnear lkelhood funton: lassfaton and parameter estmaton n a dfferental equaton model of gene regulaton. Classfaton: Determnst nferene methods for GP lassfaton are desrbed n [6, 4, 7]. Among these approahes, the Expetaton-Propagaton (EP) algorthm [9] s found to be the most effent [6]. Our MCMC mplementaton onfrms these fndngs sne samplng usng ontrol varables gave smlar lassfaton auray to EP. Transrptonal regulaton: We onsder a small bologal sub-system where a set of target genes are regulated by one transrpton fator (TF) proten. Ordnary dfferental equatons (ODEs) an provde an useful framework for modellng the dynams n these bologal networks [,,, 8]. The onentraton of the TF and the gene spef knet parameters are typally unknown and need to be estmated by makng use of a set of observed gene expresson levels. We use a GP pror to model the unobserved TF atvty, as proposed n [8], and apply full Bayesan nferene based on the MCMC algorthm presented prevously. Bareno et al. [] ntrodue a lnear ODE model for gene atvaton from TF. Ths approah was extended n [, 8] to aount for non-lnear models. The general form of the ODE model for transrpton regulaton wth a sngle TF has the form dy j (t) = B j + S j g(f(t)) D j y j (t), (6) dt where the hangng level of a gene j s expresson, y j (t), s gven by a ombnaton of basal transrpton rate, B j, senstvty, S j, to ts governng TF s atvty, f(t), and the deay rate of the mrna, D j. The dfferental equaton an be solved for y j (t) gvng y j (t) = B j D j + A j e Djt + S j e Djt t g(f(u))e Dju du, (7) where A j term arses from the ntal ondton. Due to the non-lnearty of the g funton that transforms the TF, the ntegral n the above expresson s not analytally obtaned. However, numeral ntegraton an be used to aurately approxmate the ntegral wth a dense grd (u ) P = of ponts n the tme axs and evaluatng the funton at the grd ponts f p = f(u p ). In ths ase the ntegral n the above equaton an be wrtten P t p= w pg(f p )e Djup where the weghts w p arse from the numeral ntegraton method used and, for example, an be gven by the omposte Smpson rule. The TF onentraton f(t) n the above system of ODEs s a latent funton that needs to be estmated. Addtonally, the knet parameters of eah gene α j = (B j, D j, S j, A j ) are unknown and also need to be estmated. To nfer these quanttes we use mrna measurements (obtaned from mroarray experments) of N target genes at T dfferent tme steps. Let y jt denote the observed gene expresson level of gene j at tme t and let y = {y jt } ollet together all these observatons. Assumng a Gaussan nose for the observed gene expressons the lkelhood of our data has the form N T p(y f, {α j } N j=) = p(y jt f p Pt, α j ), (8) j= t= where eah probablty densty n the above produt s a Gaussan wth mean gven by eq. (7) and f p Pt denotes the TF values up to tme t. Note that ths lkelhood s non-gaussan due to the non-lnearty of g. Further, ths lkelhood does not have a fatorzed form, as n the regresson and lassfaton ases, sne an observed gene expresson depends on the proten onentraton atvty n all prevous tmes ponts. Also note that the dsretzaton of the TF n P tme ponts orresponds to a very dense grd, whle the gene expresson measurements are sparse,.e. P T. To apply full Bayesan nferene n the above model, we need to defne pror dstrbutons over all unknown quanttes. The proten onentraton f s a postve quantty, thus a sutable pror s to onsder a GP pror for log f. The knet parameters of eah gene are all postve salars. Those parameters are gven vague gamma prors. Samplng the GP funton s done exatly as desrbed n seton ; we have only to plug n the lkelhood from eq. (8) n the MH step. Samplng from the knet parameters s arred usng Gaussan proposal dstrbutons wth dagonal ovarane matres that sample the postve knet parameters n the log spae.

6 KL(real empral) gbbs regon ontrol MCMC teratons x 4 KL(real empral) 6 4 Gbbs regon ontrol dmenson (a) (b) () number of ontrol varables orrcoef ontrol dmenson Fgure : (a) shows the evoluton of the KL dvergene (aganst the number of MCMC teratons) between the true posteror and the emprally estmated posterors for a -dmensonal regresson dataset. (b) shows the mean values wth one-standard error bars of the KL dvergene (aganst the nput dmenson) between the true posteror and the emprally estmated posterors. () plots the number of ontrol varables together wth the average orrelaton oeffent of the GP pror. 4 Experments In the frst experment we ompare Gbbs samplng (Gbbs), samplng usng loal regons (regon) (see the supplementary fle) and samplng usng ontrol varables (ontrol) n standard regresson problems of vared nput dmensons. The performane of the algorthms an be aurately assessed by omputng the KL dvergenes between the exat Gaussan posteror p(f y) and the Gaussans obtaned by MCMC. We fx the number of tranng ponts to N = and we vary the nput dmenson d from to. The tranng nputs X were hosen randomly nsde the unt hyperube [, ] d. Thus, we an study the behavor of the algorthms w.r.t. the amount of orrelaton n the posteror GP proess whh depends on how densely the funton s sampled. The larger the dmenson, the sparser the funton s sampled. The outputs Y were hosen by randomly produng a GP funton usng the squared-exponental kernel σf xm xn exp( l ), where (σ f, l ) = (, ) and then addng nose wth varane σ =.9. The burn-n perod was 4 teratons. For a ertan dmenson d the algorthms were ntalzed to the same state obtaned by randomly drawng from the GP pror. The parameters (σf, l, σ ) were fxed to the values that generated the data. The expermental setup was repeated tmes so as to obtan onfdene ntervals. We used thnned samples (by keepng one sample every teratons) to alulate the means and ovaranes of the -dmensonal posteror Gaussans. Fgure (a) shows the KL dvergene aganst the number of MCMC teratons for the -dmensonal nput dataset. It seems that for tranng ponts and dmensons, the funton values are stll hghly orrelated and thus Gbbs takes muh longer for the KL dvergene to drop to zero. Fgure (b) shows the KL dvergene aganst the nput dmenson after fxng the number of teratons to be 4. Clearly Gbbs s very neffent n low dmensons beause of the hghly orrelated posteror. As dmenson nreases and the funtons beome sparsely sampled, Gbbs mproves and eventually the KL dvergenes approahes zero. The regon algorthm works better than Gbbs but n low dmensons t also suffers from the problem of hgh orrelaton. For the ontrol algorthm we observe that the KL dvergene s very lose to zero for all dmensons. Fgure () shows the nrease n the number of ontrol varables used as the nput dmenson nreases. The same plot shows the derease of the average orrelaton oeffent of the GP pror as the nput dmenson nreases. Ths s very ntutve, sne one should expet the number of ontrol varables to nrease as the funton values beome more ndependent. Next we onsder two GP lassfaton problems for whh exat nferene s ntratable. We used the Wsonsn Breast Caner (WBC) and the Pma Indans Dabetes (PID) bnary lassfaton datasets. The frst onssts of 68 examples (9 nput dmensons) and the seond of 768 examples (8 dmensons). % of the examples were used for testng n eah ase. The MCMC samplers were run for 4 teratons (thnned to one sample every fve teratons) after a burn-n of 4 teratons. The hyperparameters were fxed to those obtaned by EP. Fgures (a) and (b) shows For Gbbs we used 4 teratons sne the regon and ontrol algorthms requre addtonal teratons durng the adapton phase.

7 Log lkelhood Log lkelhood MCMC teratons MCMC teratons gbbs ontr ep gbbs ontr ep (a) (b) () Fgure : We show results for GP lassfaton. Log-lkelhood values are shown for MCMC samples obtaned from (a) Gbbs and (b) ontrol appled to the WBC dataset. In () we show the test errors (grey bars) and the average negatve log lkelhoods (blak bars) on the WBC (left) and PID (rght) datasets and ompare wth EP.. p6 sesn Gene frst Repla Deay rates DDB p6 sesn TNFRSFb CIp/p BIK 7 Inferred proten 7. dni Gene 7 yjw Gene Fgure 4: Frst row: The left plot shows the nferred TF onentraton for p; the small plot on top-rght shows the ground-truth proten onentraton obtaned by a Western blot experment []. The mddle plot shows the predted expresson of a gene obtaned by the estmated ODE model; red rosses orrespond to the atual gene expresson measurements. The rght-hand plot shows the estmated deay rates for all target genes used to tran the model. Grey bars dsplay the parameters found by MCMC and blak bars the parameters found n [] usng a lnear ODE model. Seond row: The left plot shows the nferred TF for LexA. Predted expressons of two target genes are shown n the rest two plots. Error bars n all plots orrespond to 9% redblty ntervals. the log-lkelhood for MCMC samples on the WBC dataset, for the Gbbs and ontrol algorthms respetvely. It an be observed that mxng s far superor for the ontrol algorthm and t has also onverged to a muh hgher lkelhood. In Fgure () we ompare the test error and the average negatve log lkelhood n the test data obtaned by the two MCMC algorthms wth the results from EP. The proposed ontrol algorthm shows smlar lassfaton performane to EP, whle the Gbbs algorthm performs sgnfantly worse on both datasets. In the fnal two experments we apply the ontrol algorthm to nfer the proten onentraton of TFs that atvate or repress a set of target genes. The latent funton n these problems s always onedmensonal and densely dsretzed and thus the ontrol algorthm s the only one that an onverge to the GP posteror proess n a reasonable tme. We frst onsder the TF p whh s a tumour repressor atvated durng DNA damage. Seven samples of the expresson levels of fve target genes n three replas are olleted as the raw tme ourse data. The non-lnear atvaton of the proten follows the Mhaels Menten knets nspred response [] that allows saturaton effets to be taken nto aount so as g(f(t)) = f(t) γ j+f(t) n eq.

8 (6) where the Mhaels onstant for the jth gene s gven by γ j. Note that sne f(t) s postve the GP pror s plaed on the log f(t). To apply MCMC we dsretze f usng a grd of P = ponts. Durng samplng, 7 ontrol varables were needed to obtan the desrable aeptane rate. Runnng tme was 4 hours for samplng teratons plus 4 burn-n teratons. The frst row of Fgure 4 summarzes the estmated quanttes obtaned from MCMC smulaton. Next we onsder the TF LexA n E.Col that ats as a repressor. In the represson ase there s an analogous Mhaels Menten model [] where the non-lnear funton g takes the form: g(f(t)) = γ j+f(t). Agan the GP pror s plaed on the log of the TF atvty. We appled our method to the same mroarray data onsdered n [] where mrna measurements of 4 target genes are olleted over sx tme ponts. For ths dataset, the expresson of the 4 genes were avalable for T = 6 tmes. The GP funton f was dsretzed usng ponts. The result for the nferred TF profle along wth predtons of two target genes are shown n the seond row of Fgure 4. Our nferred TF profle and reonstruted target gene profles are smlar to those obtaned n []. However, for ertan genes, our model provdes a better ft to the gene profle. Dsusson Gaussan proesses allow for nferene over latent funtons usng a Bayesan estmaton framework. In ths paper, we presented an MCMC algorthm that uses ontrol varables. We showed that ths samplng sheme an effently deal wth hghly orrelated posteror GP proesses. MCMC allows for full Bayesan nferene n the transrpton fator networks applaton. An mportant dreton for future researh wll be salng the models used to muh larger systems of ODEs wth multple nteratng transrpton fators. In suh large systems where MCMC an beome slow a ombnaton of our method wth the fast samplng sheme n [] ould be used to speed up the nferene. Aknowledgments Ths work s funded by EPSRC Grant No EP/F687/ Gaussan Proesses for Systems Identfaton wth Applatons n Systems Bology. Referenes [] U. Alon. An Introduton to Systems Bology: Desgn Prnples of Bologal Cruts. Chapman and Hall/CRC, 6. [] M. Bareno, D. Tomesu, D. Brewer, J. Callard, R. Stark, and M. Hubank. Ranked predton of p targets usng hdden varable dynam modelng. Genome Bology, 7(), 6. [] B. Calderhead, M. Grolam, and N.D. Lawrene. Aeleratng Bayesan Inferene over Nonlnear Dfferental Equatons wth Gaussan Proesses. In Neural Informaton Proessng Systems,, 8. [4] L. Csato and M. Opper. Sparse onlne Gaussan proesses. Neural Computaton, 4:64 668,. [] M. N. Gbbs and D. J. C. MaKay. Varatonal Gaussan proess lassfers. IEEE Transatons on Neural Networks, (6):48 464,. [6] M. Kuss and C. E. Rasmussen. Assessng Approxmate Inferene for Bnary Gaussan Proess Classfaton. Journal of Mahne Learnng Researh, 6:679 74,. [7] N. D. Lawerene, M. Seeger, and R. Herbrh. Fast sparse Gaussan proess methods: the nformatve vetor mahne. In Advanes n Neural Informaton Proessng Systems,. MIT Press,. [8] N. D. Lawrene, G. Sangunett, and M. Rattray. Modellng transrptonal regulaton usng Gaussan proesses. In Advanes n Neural Informaton Proessng Systems, 9. MIT Press, 7. [9] T. Mnka. Expetaton propagaton for approxmate Bayesan nferene. In UAI, pages 6 69,. [] R. M. Neal. Monte Carlo mplementaton of Gaussan proess models for Bayesan regresson and lassfaton. Tehnal report, Dept. of Statsts, Unversty of Toronto, 997. [] C. E. Rasmussen and C. K. I. Wllams. Gaussan Proesses for Mahne Learnng. MIT Press, 6. [] C. P. Robert and G. Casella. Monte Carlo Statstal Methods. Sprnger-Verlag, nd edton, 4. [] S. Rogers, R. Khann, and M. Grolam. Bayesan model-based nferene of transrpton fator atvty. BMC Bonformats, 8(), 6. [4] H. Rue, S. Martno, and N. Chopn. Approxmate Bayesan nferene for latent Gaussan models usng ntegrated nested Laplae approxmatons. NTNU Statsts Preprnt, 7. [] E. Snelson and Z. Ghahraman. Sparse Gaussan proess usng pseudo nputs. In Advanes n Neural Informaton Proessng Systems,. MIT Press, 6. [6] C. K. I. Wllams and D. Barber. Bayesan lassfaton wth Gaussan proesses. IEEE Transatons on Pattern Analyss and Mahne Intellgene, ():4, 998.