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Evaluating probabilities under high-dimensional latent variable models
- Harriet Johns
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1 Evaluating probabilities under ig-dimensional latent variable models Iain Murray and Ruslan alakutdinov Department of Computer cience University of oronto oronto, ON. M5 3G4. Canada. Abstract We present a simple new Monte Carlo algoritm for evaluating probabilities of observations in complex latent variable models, suc as Deep Belief Networks. Wile te metod is based on Markov cains, estimates based on sort runs are formally unbiased. In expectation, te log probability of a test set will be underestimated, and tis could form te basis of a probabilistic bound. e metod is muc ceaper tan gold-standard annealing-based metods and only sligtly more expensive tan te ceapest Monte Carlo metods. We give examples of te new metod substantially improving simple variational bounds at modest extra cost. Introduction Latent variable models capture underlying structure in data by explaining observations as part of a more complex, partially observed system. A large number of probabilistic latent variable models ave been developed, most of wic express a joint distribution P (v, ) over observed quantities v and teir unobserved counterparts. Altoug it is by no means te only way to evaluate a model, a natural question to ask is wat probability P (v) is assigned to a test observation?. In some models te latent variables associated wit a test input can be easily summed out: P (v) = P (v, ). As an example, standard mixture models ave a single discrete mixture component indicator for eac data point; te joint probability P (v, ) can be explicitly evaluated for eac setting of te latent variable. More complex grapical models explain data troug te combination of many latent variables. is provides ricer representations, but provides greater computational callenges. In particular, marginalizing out many latent variables can require complex integrals or exponentially large sums. One popular latent variable model, te Restricted Boltzmann Macine (RBM), is unusual in tat te posterior over iddens P ( v) is fully-factored, wic allows efficient evaluation of P (v) up to a constant. Almost all oter latent variable models ave posterior dependencies amongst latent variables, even if tey are independent a priori. Our current work is motivated by recent work on evaluating RBMs and teir generalization to Deep Belief Networks (DBNs) []. For bot types of models, a single constant was accurately approximated so tat P (v, ) could be evaluated point-wise. For RBMs, te remaining sum over idden variables was performed analytically. For DBNs, test probabilities were lower-bounded troug a variational tecnique. Peraps surprisingly, te bound was unable to reveal any significant improvement over RBMs in an experiment on MNI digits. It was unclear weter tis was due to looseness of te bound, or to tere being no difference in performance. A more accurate metod for summing over latent variables would enable better and broader evaluation of DBNs. In section 2 we consider existing Monte Carlo metods. ome of tem are certainly
2 more accurate, but proibitively expensive for evaluating large test sets. We ten develop a new ceap Monte Carlo procedure for evaluating latent variable models in section 3. Like te variational metod used previously, our metod is unlikely to spuriously over-state test-set performance. Our presentation is for general latent variable models, owever for a running example, we use DBNs (see section 4 and [2]). e benefits of our new approac are demonstrated in section 5. 2 Probability of observations as a normalizing constant e probability of a data vector, P (v), is te normalizing constant relating te posterior over idden variables to te joint distribution in Bayes rule, P ( v) = P (, v)/p (v). A large literature on computing normalizing constants exists in pysics, statistics and computer science. In principle, tere are many metods tat could be applied to evaluating te probability assigned to data by a latent variable model. We review a subset of tese metods, wit notation and intuitions tat will elp motivate and explain our new algoritm. In wat follows, all auxiliary distributions Q and transition operators are conditioned on te current test case v, tis is not sown in te notation to reduce clutter. Furter, all of tese metods assume tat we can evaluate P (, v). Grapical models wit undirected connections will require te separate estimation of a single constant as in []. 2. Importance sampling Importance sampling can in principle find te normalizing constant of any distribution. e algoritm involves averaging a simple ratio under samples from some convenient tractable distribution over te idden variables, Q(). Provided Q() 0 wenever P (, v) 0, we obtain: P (, v) Q() P ( (s), v ) P (v) = Q() Q ( (s)), (s) Q ( (s)). () Importance sampling relies on te sampling distribution Q() being similar to te target distribution P ( v). pecifically, te variance of te estimator is an α-divergence between te distributions [3]. Finding a tractable Q() wit small divergence is difficult in ig-dimensional problems. 2.2 e Harmonic mean metod Using Q()=P ( v) in () gives an estimator tat requires knowing P (v). As an alternative, te armonic mean metod, also called te reciprocal metod, gives an unbiased estimate of /P (v): P (v) = P () P (v) = P ( v) P (v ) P ( v (s)), (s) P ( (s) v). (2) In practice correlated samples from MCMC are used; ten te estimator is asymptotically unbiased. It was clear from te original paper and its discussion tat te armonic mean estimator can beave very poorly [4]. amples in te tails of te posterior ave large weigts, wic makes it easy to construct distributions were te estimator as infinite variance. A finite set of samples will rarely include any extremely large weigts, so te estimator s empirical variance can be misleadingly low. In many problems, te estimate of /P (v) will be an underestimate wit ig probability. at is, te metod will overestimate P (v) and often give no indication tat it as done so. ometimes te estimator will ave manageable variance. Also, more expensive versions of te estimator exist wit lower variance. However, it is still prone to overestimate test probabilities: If / ˆP HME (v) is te Harmonic Mean Estimator in (2), Jensen s inequality gives P (v) = / E [ / ˆP HME (v) ] E [ ˆPHME (v) ]. imilarly log P (v) will be overestimated in expectation. Hence te average of a large number of test log probabilities is igly likely to be an overestimate. Despite tese problems te estimator as received significant attention in statistics, and as been used for evaluating latent variable models in recent macine learning literature [5, 6]. is is understandable: all of te existing, more accurate metods are arder to implement and take considerably longer to run. In tis paper we propose a metod tat is nearly as easy to use as te armonic mean metod, but wit better properties.
3 2.3 Importance sampling based on Markov cains Paradoxically, introducing auxiliary variables and making a distribution muc iger-dimensional tan it was before, can elp find an approximating Q distribution tat closely matces te target distribution. As an example we give a partial review of Annealed Importance ampling (AI) [7], a special case of a larger family of equential Monte Carlo (MC) metods (see, e.g., [8]). ome of tis teory will be needed in te new metod we present in section 3. Annealing algoritms start wit a sample from some tractable distribution P. teps are taken wit a series of operators 2, 3,...,, wose stationary distributions, P s, are cooled towards te distribution of interest. e probability over te resulting sequence H = { (), (2),... () } is: ( Q AI (H) = P () ) ( s (s) (s )). (3) o compute importance weigts, we need to define a target distribution on te same state-space: P AI (H) = P ( () v ) ( s (s ) (s)). (4) Because () as marginal P ( v) = P (, v)/p (v), P AI (H) as our target, P (v), as its normalizing constant. e operators are te reverse operators, of tose used to define Q AI. For any transition operator tat leaves a distribution P ( v) stationary, tere is a unique corresponding reverse operator, wic is defined for any point in te support of P : ( ) = ( ) P ( v) ( ) P ( v) = ( ) P ( v) P (. (5) v) e sum in te denominator is known because leaves te posterior stationary. Operators tat are teir own reverse operator are said to satisfy detailed balance and are also known as reversible. Many transition operators used in practice, suc as Metropolis Hastings, are reversible. Non-reversible operators are usually composed from a sequence of reversible operations, suc as te component updates in a Gibbs sampler. e reverse of tese (so-called) non-reversible operators is constructed from te same reversible base operations, but applied in reverse order. e definitions above allow us to write: Q AI (H) = P AI (H) Q AI(H) P AI (H) = P AI(H) P ( ) () P ( () v ) = P AI (H) P (v) [ ( ) P () ] P ( (), v ) Ps ( (s) ) Ps ( (s ) ) s ( (s) (s )) s ( (s ) (s)) P AI(H) P (v). w(h) We can usually evaluate te Ps, wic are unnormalized versions of te stationary distributions of te Markov cain operators. erefore te AI importance weigt w(h) = / [ ] is tractable as long as we can evaluate P (, v). e AI importance weigt provides an unbiased estimate: [ ] E QAI(H) w(h) = P (v) P AI (H) = P (v). (7) H As wit standard importance sampling, te variance of te estimator depends on a divergence between P AI and Q AI. is can be made small, at large computational expense, by using undreds or tousands of steps, allowing te neigboring intermediate distributions P s () to be close. 2.4 Cib-style estimators Bayes rule implies tat for any special idden state, P (v) = P (, v)/p ( v). (8) is trivial identity suggests a family of estimators introduced by Cib [9]. First, we coose a particular idden state, usually one wit ig posterior probability, and ten estimate P ( v). We would like to obtain an estimator tat is based on a sequence of states H ={ (), (2),..., () } generated by a Markov cain tat explores te posterior distribution P ( v). e most naive estimate of P ( v) is te fraction of states in H tat are equal to te special state s I((s) = )/. (6)
4 Obviously tis estimator is impractical as it equals zero wit ig probability wen applied to igdimensional problems. A Rao Blackwellized version of tis estimator, ˆp(H), replaces te indicator function wit te probability of transitioning from (s) to te special state under a Markov cain transition operator tat leaves te posterior stationary. is can be derived directly from te operator s stationary condition: P ( v) = ( )P ( v) ˆp(H) ( (s) ), { (s) } P(H), (9) were P(H) is te joint distribution arising from steps of a Markov cain. If te cain as stationary distribution P ( v) and could be initialized at equilibrium so tat P(H) = P ( () ) v ( (s) (s )), (0) ten ˆp(H) would be an unbiased estimate of P ( v). For ergodic cains te stationary distribution is acieved asymptotically and te estimator is consistent regardless of ow it is initialized. If is a Gibbs sampling transition operator, te only way of moving from to is to draw eac element of in turn. If updates are made in index order from to M, te move as probability: M ( ) = P ( ) j :(j ), (j+):m. () j= Equations (9, ) ave been used in scemes for monitoring te convergence of Gibbs samplers [0]. It is wort empasizing tat we ave only outlined te simplest possible sceme inspired by Cib s general approac. For some Markov cains, tere are tecnical problems wit te above construction, wic require an extension explained in te appendix. Moreover te approac above is not wat Cib recommended. In fact, [] explicitly favors a more elaborate procedure involving sampling from a sequence of distributions. is opens up te possibility of many sopisticated developments, e.g. [2, 3]. However, our focus in tis work is on obtaining more useful results from simple ceap metods. ere are also well-known problems wit te Cib approac [4], to wic we will return. 3 A new estimator for evaluating latent-variable models We start wit te simplest Cib-inspired estimator based on equations (8,9,). Like many Markov cain Monte Carlo algoritms, (9) provides only (asymptotic) unbiasedness. For our purposes tis is not sufficient. Jensen s inequality tells us P (v) = P (, v) P ( v) = P [ (, v) P ( ] E[ˆp(H)] E, v). (2) ˆp(H) at is, we will overestimate te probability of a visible vector in expectation. Jensen s inequality also says tat we will overestimate log P (v) in expectation. Ideally we would like an accurate estimate of log P (v). However, if we must suffer some bias, ten a lower bound tat does not overstate performance will usually be preferred. An underestimate of P (v) would result from overestimating P ( v). e probability of te special state will often be overestimated in practice if we initialize our Markov cain at. ere are, owever, simple counter-examples were tis does not appen. Instead we describe a construction based on a sequence of Markov steps starting at tat does ave te desired effect. We draw a state sequence from te following carefully designed distribution, using te algoritm in figure : Q(H) = ( (s) ) s =s+ ( (s ) (s ) ) s s = ( (s ) (s +) ). (3) If te initial state were drawn from P ( v) instead of ( (s) ), ten te algoritm would give a sample from an equilibrium sequence wit distribution P(H) defined in (0). is can be cecked by repeated substitution of (5). is allows us to express Q in terms of P, as we did for AI: Q(H) = ( (s) ) P ( (s) v ) P(H) = P ( v) [ ( (s))] P(H). (4)
5 Inputs: v, observed test vector, a (preferably ig posterior probability) idden state, number of Markov cain steps, Markov cain operator tat leaves P ( v) stationary (4). Draw s Uniform({,... }) 2. Draw (s) ( (s) ) 3. for s = (s + ) : 4. Draw (s ) ( (s ) ) (s ) 5. for s = (s ) : : 6. Draw (s ) ( (s ) ) (s +) / 7. P (v) P (v, ) ( (s ) ) s = (3) (2) () Figure : Algoritm for te proposed metod. e grapical model sows Q(H s = 3) for = 4. At eac generated state ( (s ) ) is evaluated (step 7), rougly doubling te cost of sampling. e reverse operator, e, was defined in section 2.3. e quantity in square brackets is te estimator for P ( v) given in (9). e expectation of te reciprocal of tis quantity under draws from Q(H) is exactly te quantity needed to compute P (v): / E Q(H) [ ( (s))] = P ( P(H) = v) P ( v). (5) Altoug we are using te simple estimator from (9), by drawing H from a carefully constructed Markov cain procedure, te estimator is now unbiased in P (v). is is not an asymptotic result. As long as no division by zero as occurred in te above equations, te estimator is unbiased in P (v) for finite runs of te Markov cain. Jensen s implies tat log P (v) is underestimated in expectation. Neal noted tat Cibs metod will return incorrect answers in cases were te Markov cain does not mix well amongst modes [4]. Our new proposed metod will suffer from te same problem. Even if no transition probabilities are exactly zero, unbiasedness does not exclude being on a particular side of te correct answer wit very ig probability. Poor mixing may cause P ( v) to be overestimated wit ig probability, wic would result in an underestimate of P (v), i.e., an overly conservative estimate of test performance. e variance of te estimator is generally unknown, as it depends on te (generally unavailable) auto-covariance structure of te Markov cain. We can note one positive property: for te ideal Markov cain operator tat mixes in one step, te estimator as zero variance and gives te correct answer immediately. Altoug tis extreme will not actually occur, it does indicate tat on easy problems, good answers can be returned more quickly tan by AI. 4 Deep Belief Networks In tis section we provide a brief overview of Deep Belief Networks (DBNs), recently introduced by [2]. DBNs are probabilistic generative models, tat can contain many layers of idden variables. Eac layer captures strong ig-order correlations between te activities of idden features in te layer below. e top two layers of te DBN model form a Restricted Boltzmann Macine (RBM) wic is an undirected grapical model, but te lower layers form a directed generative model. e original paper introduced a greedy, layer-by-layer unsupervised learning algoritm tat consists of learning a stack of RBMs one layer at a time. Consider a DBN model wit two layers of idden features. e model s joint distribution is: H P (v,, 2 ) = P (v ) P ( 2, ), (6) were P (v ) represents a sigmoid belief network, and P (, 2 ) is te joint distribution defined by te second layer RBM. By explicitly summing out 2, we can easily evaluate an unnormalized probability P (v, )=ZP (v, ). Using an approximating factorial posterior distribution Q( v),
6 Estimated est Log probability AI Estimator MNI digits Our Proposed Estimator Estimate of Variational Lower Bound Number of Markov cain steps Estimated est Log probability AI Estimator Our Proposed Estimator Estimate of Variational Lower Bound Image Patces Number of Markov cain steps Figure 2: AI, our proposed estimator and a variational metod were used to sum over te idden states for eac of 50 randomly sampled test cases to estimate teir average log probability. e tree metods sared te same AI estimate of a single global normalization constant Z. obtained as a byproduct of te greedy learning procedure, and an AI estimate of te model s partition function Z, [] proposed obtaining an estimate of a variational lower bound: log P (v) Q( v) log P (v, ) log Z + H(Q( v)). (7) e entropy term H( ) can be computed analytically, since Q is factorial, and te expectation term was estimated by a simple Monte Carlo approximation: Q( v) log P (v, ) log P (v, (s) ), were (s) Q( v). (8).. Instead of te variational approac, we could also adopt AI to estimate P (v, ). is would be computationally very expensive, since we would need to run AI for eac test case. In te next section we sow tat variational lower bounds can be quite loose. Running AI on te entire test set, containing many tousands of test cases, is computationally too demanding. Our proposed estimator requires te same single AI estimate of Z as te variational metod, so tat we can evaluate P (v, ). It ten provides better estimates of log P (v) by approximately summing over for eac test case in a reasonable amount of computer time. 5 Experimental Results We present experimental results on two datasets: te MNI digits and a dataset of image patces, extracted from images of natural scenes taken from te collection of Van Hateren (ttp://lab.pys.rug.nl/imlib/). e MNI dataset contains 60,000 training and 0,000 test images of ten andwritten digits (0 to 9), wit pixels. e image dataset consisted of 30,000 training and 20,000 test patces. e raw image intensities were preprocessed and witened as described in [5]. Gibbs sampling was used as a Markov cain transition operator trougout. All log probabilities quoted use natural logaritms, giving values in nats. 5. MNI digits In our first experiment we used a deep belief network (DBN) taken from []. e network ad two idden layers wit 500 and 2000 idden units, and was greedily trained by learning a stack of two RBMs one layer at a time. Eac RBM was trained using te Contrastive Divergence (CD) learning rule. e estimate of te lower bound on te average test log probability, using (7), was o estimate ow loose te variational bound is, we randomly sampled 50 test cases, 5 of eac class, and ran AI for eac test case to estimate te true test log probability. Computationally, tis is equivalent to estimating 50 additional partition functions. Figure 2, left panel, sows te results. e estimate of te variational bound was per test case, wereas te estimate of te true test log probability using AI was Our proposed estimator, averaged over 0 runs, provided an answer of e special state for eac test example v was obtained by first sampling from te approximating distribution Q( v), and ten performing deterministic ill-climbing in log p(v, ) to get to a local mode.
7 AI used a and-tuned temperature scedule designed to equalize te variance of te intermediate log weigts [7]. We needed 0,000 intermediate distributions to get stable results, wic took about 3.6 days on a Pentium Xeon 3.00GHz macine, wereas for our proposed estimator we only used =40, wic took about 50 minutes. For a more direct comparison we tried giving AI 50 minutes, wic allows 00 temperatures. is run gave an estimate of 89.59, wic is lower tan te lower bound and tells us noting. Giving AI ten times more time, 000 temperatures, gave is is iger tan te lower bound, but still worse tan our estimator at = 40, or even = 5. Finally, using our proposed estimator, te average test log probability on te entire MNI test data was e difference of about 2 nats sows tat te variational bound in [] was rater tigt, altoug a very small improvement of te DBN over te RBM is now revealed. 5.2 Image Patces In our second experiment we trained a two-layer DBN model on te image patces of natural scenes. e first layer RBM ad 2000 idden units and 400 Gaussian visible units. e second layer represented a semi-restricted Boltzmann macine (RBM) wit 500 idden and 2000 visible units. e RBM contained visible-to-visible connections, and was trained using Contrastive Divergence togeter wit mean-field. Details of training can be found in [5]. e overall DBN model can be viewed as a directed ierarcy of Markov random fields wit idden-to-idden connections. o estimate te model s partition function, we used AI wit 5,000 intermediate distributions and 00 annealing runs. e estimated lower bound on te average test log probability (see Eq. 7), using a factorial approximate posterior distribution Q( v), wic we also get as a byproduct of te greedy learning algoritm, was e estimate of te true test log probability, using our proposed estimator, was In contrast to te model trained on MNI, te difference of over 20 nats sows tat, for model comparison purposes, te variational lower bound is quite loose. For comparison, we also trained square ICA and a mixture of factor analyzers (MFA) using code from [6, 7]. quare ICA acieves a test log probability of 55.4, and MFA wit 50 mixture components and a 30-dimensional latent space acieves , clearly outperforming DBNs. 6 Discussion Our new Monte Carlo procedure is formally unbiased in estimating P (v). In practice it is likely to underestimate te (log-)probability of a test set. Altoug te algoritm involves Markov cains, importance sampling underlies te estimator. erefore te metods discussed in [8] could be used to bound te probability of accidentally over-estimating a test set probability. In principle our procedure is a general tecnique for estimating normalizing constants. It would not always be appropriate owever, as it would suffer te problems outlined in [4]. As an example our metod will not succeed in estimating te global normalizing constant of an RBM. For our metod to work well, a state drawn from ( (s) ) sould look like it could be part of an equilibrium sequence H P(H). e details of te algoritm arose by developing existing Monte Carlo estimators, but te starting state (s) could be drawn from any arbitrary distribution: Q var (H) = [ q( (s) ) P(H) = P (v) P ( (s) v) ] q( (s) ) P(H). (9) P ( (s), v) As before te reciprocal of te quantity in square brackets would give an estimate of P (v). If an approximation q() is available tat captures more mass tan ( ), tis generalized estimator could perform better. We are opeful tat our metod will be a natural next step in a variety of situations were improvements are sougt over a deterministic approximation. Acknowledgments is researc was supported by NERC and CFI. Iain Murray was supported by te government of Canada. We tank Geoffrey Hinton and Radford Neal for useful discussions, imon Osindero for providing preprocessed image patces of natural scenes, and te reviewers for useful comments.
8 References [] Ruslan alakutdinov and Iain Murray. On te quantitative analysis of Deep Belief Networks. In Proceedings of te International Conference on Macine Learning, volume 25, pages , [2] Geoffrey E. Hinton, imon Osindero, and Yee Wye e. A fast learning algoritm for deep belief nets. Neural Computation, 8(7): , [3] om Minka. Divergence measures and message passing. R , Microsoft Researc, [4] Micael A. Newton and Adrian E. Raftery. Approximate Bayesian inference wit te weigted likeliood bootstrap. Journal of te Royal tatistical ociety, eries B (Metodological), 56():3 48, 994. [5] omas L. Griffits, Mark teyvers, David M. Blei, and Josua B. enenbaum. Integrating topics and syntax. In Advances in Neural Information Processing ystems (NIP*7). MI Press, [6] Hanna M. Wallac. opic modeling: beyond bag-of-words. In Proceedings of te 23rd international conference on Macine learning, pages ACM Press New York, NY, UA, [7] Radford M. Neal. Annealed importance sampling. tatistics and Computing, (2):25 39, 200. [8] Pierre Del Moral, Arnaud Doucet, and Ajay Jasra. equential Monte Carlo samplers. Journal of te Royal tatistical ociety B, 68(3): 26, [9] iddarta Cib. Marginal likeliood from te Gibbs output. Journal of te American tatistical Association, 90(432):33 32, December 995. [0] Cristian Ritter and Martin A. anner. Facilitating te Gibbs sampler: te Gibbs stopper and te griddy- Gibbs sampler. Journal of te American tatistical Association, 87(49):86 868, 992. [] iddarta Cib and Ivan Jeliazkov. Marginal likeliood from te Metropolis Hastings output. Journal of te American tatistical Association, 96(453), 200. [2] Antonietta Mira and Geoff Nicolls. Bridge estimation of te probability density at a point. tatistica inica, 4:603 62, [3] Francesco Bartolucci, Luisa caccia, and Antonietta Mira. Efficient Bayes factor estimation from te reversible jump output. Biometrika, 93():4 52, [4] Radford M. Neal. Erroneous results in Marginal likeliood from te Gibbs output, 999. Available from ttp:// radford/cib-letter.tml. [5] imon Osindero and Geoffrey Hinton. Modeling image patces wit a directed ierarcy of Markov random fields. In Advances in Neural Information Processing ystems (NIP*20). MI Press, [6] Aapo Hyvärinen. Fast and robust fixed-point algoritms for independent component analysis. IEEE ransactions on Neural Networks, 0(3): , 999. [7] Zoubin Garamani and Geoffrey E. Hinton. e EM algoritm for mixtures of factor analyzers. ecnical Report CRG-R-96-, University of oronto, 997. [8] Vibav Gogate, Bozena Bidyuk, and Rina Decter. tudies in lower bounding probability of evidence using te Markov inequality. In 23rd Conference on Uncertainty in Artificial Intelligence (UAI), A Real-valued latents and Metropolis Hastings ere are tecnical difficulties wit te original Cib-style approac applied to Metropolis Hastings and continuous latent variables. e continuous version of equation (9), P ( v) = ( )P ( v) d ( (s) ), (s) P(H), (20) doesn t work if is te Metropolis Hastings operator. e Dirac-delta function at = contains a significant part of te integral, wic is ignored by samples from P ( v) wit probability one. Following [], te fix is to instead integrate over te generalized detailed balance relationsip (5). Cib and Jeliazkov implicitly took out te = point from all of teir integrals. We do te same: P ( v) = d ( )P ( v)/ d ( ). (2) e numerator can be estimated as before. As bot integrals omit =, te denominator is less tan one wen contains a delta function. For Metropolis Hastings: ( ) = q(; ) min (, a(; ) ), were a(; ) is an easy-to-compute acceptance ratio. ampling from q(; ) and averaging min(, a(; )) provides an estimate of te denominator. In our importance sampling approac tere is no need to separately approximate an additional quantity. e algoritm in figure still applies if te s are interpreted as probability density functions. If, due to a rejection, is drawn in step 2. ten te sum in step 7. will contain an infinite term giving a trivial underestimate P (v)=0. (teps 3 6 need not be performed in tis case.) On repeated runs, te average estimate is still unbiased, or an underestimate for cains tat can t mix. Alternatively, te variational approac (9) could be applied togeter wit Metropolis Hastings sampling.
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text:
Fit Guide
This style tends to run large. For an accurate fit we recommend purchasing a 1/2 size smaller than you typically purchase. For example, if you normally purchase a size 7 1/2, we recommend purchasing a size 7.
There's no mistaking your sharp fashion sense when you step into the Total Motion® Pointed Toe Pump. Standing tall on a 75mm (3 inch) stiletto heel, this ultra-feminine women's dress shoe features a molded ABS heel for maximum impact resistance and toughness. Additional comfort and support is provided by a layer of retention foam and foam-backed lining. Crafted from full-grain leather, this classic pointed-toe women's pump is easy to clean and maintain. Choose from an array of colors and textures. Pair with a tailored pantsuit, dress, or skirt.
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subset: commoncrawl


text:
Photos from the Chinese Health Newsletter
A quarterly newsletter produced by volunteers from the Chinese community and a sessional editor covering a range of topical issues and up-to-date news on health-related matters.
If you would like to find out more, please visit the project page.
Brainstorming session with volunteers on the 29th Jan 2010


text:
EDISON, NJ – The Edison and Woodbridge Indian community will unite for the 10th Annual India Day Parade on August 10, 2014. The Indian Business Association (IBA) of New Jersey along with many other associations has been organizer of the annual India Day Parade in Edison for the past nine years.
The parade kick off meeting was held on June 19, 2014 at the Executive Suites, Carteret, NJ in with over 150 distinguish business and community leaders. The evening began with singing of the National Anthems of U.S. and India followed by a candle lighting ceremony.
The IBA welcomed the three distinguish communtity leaders in New Jersey, Raj Mukherji, NJ Assemblyman Legislative District 33, Sapana Shah, Edison Township Council member and Vin Gopal, Chairman of Monmouth County Democrats,
Chandrakant Patel, the Chairman of IBA, Manher Shah and Mahesh Shah the Vice Chairman of IBA, and Dhiren Amin, President of IBA welcomed everyone to the meeting and provided an update on the upcoming parade.
"We are very proud to announce that this year we will have one united India Day parade, between Edison and Woodbridge townships. The parade will begin in Edison and conclude in Woodbridge," said Dhiren Amin, President of IBA. "This year's parade will be a grand success celebrating India's culture, tradition and independence. We are expecting more than 20,000 people in support of this parade."
"It's a great pleasure to be here with IBA as we kick off the festivities leading up to a India Day parade in August. There is no better place than NJ to bridge the world's greatest democracy (USA) and the world largest democracy (India)," said Assemblyman Raj Mukherji. "The parade will showcase the strength and power of Indian community abroad."
"I am very excited about this year's India Day parade. The parade is a representation of celebration of India's independence in 1947," said Edison Councilwoman Sapana Shah. "I am looking forward to witness this year's great parade."
For nearly two decades, the Indian Business Association ("IBA") has served the New Jersey Indo-American community. The IBA is proud to continue a history of representing the small businesses on and surrounding Oak Tree Road and serving as a voice for the thousands of Indo-Americans that call New Jersey home.


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Financially troubled Provincetown-Boston Airlines faces a possible shutdown this weekend unless agreement is reached with a buyer or investor, according to President Peter Van Arsdale.
Area newspapers said Van Arsdale said late Wednesday if the Naples-based commuter airline is forced to halt operations, it probably will go out of business.
Van Arsdale said there was a chance a deal would be reached by Friday but said he can't take chances.
"I have to have something in place or there is no way I can continue," he said.
"We can't expect to keep going this way," he said. "We just don't need any more Band-Aid fixes," he said. "We've got to get out of the mode of going up and down like a yo-yo. My feeling is we get it over with, or we have something positive that eliminates the doubt."
PBA failed to meet its payroll two weeks ago, and workers later received 65 percent of their salary. Van Arsdale said they would get the rest Thursday, but another $600,000 payroll is due Saturday and he said he is not sure the employees will be paid.
PBA has been operating under Chapter 11 of the U.S. Bankruptcy Code since March. Chapter 11 allows a company to continue operating under court supervision while it is shielded from creditors and comes up with a plan to pay its debts. That plan must have court approval.


text:
Last Time You Were In Da "Market"?
You hea' dat ALLLLL da time, "market".
It could be for a new/used car, a job interview, dating scene or even da real grocery store market but it's one of those phrases dat we use for practically any kine.
Buying a new/used car - "So you're in da market huh? New or used?"
A job interview - "Da market is so tough out there!"
Dating scene - "You're back on da market?!"
Grocery store - "Babe, you went market today?"
Da last time I was in da "market" for a new truck was back in '03, motorcycle in '06, a girlfriend in....... *cricket noises*......... I'll get back to you on dat one. 😛
Wen was da last time you were in da "market"?
btw, stay tuned for tomorrow's post..... trust me, you'll either love it or piss on it. 😉 😉


text:
Today I am grateful for followers, and specifically for all of you who are followers of my blog -- officially or less so. Thank you for being there, for letting me know that my words have readers, my thoughts have meaning. And thanks to all of you who signed up for my first-ever blog book give-away contest, just ended a week ago.
I follow the blogs of a great number of authors I know and admire and I know it's a commitment, albeit a small one, to follow someone else's blog, especially when, like me, she blogs five times a week. Yet a number of you do it, and often even take the time to comment. Your faithful support is humbling.
So this week a shout goes out to all of you. Thanks, folks. I will try to write words you can enjoy.


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Peace Cereal, Clusters & Flakes, Vanilla Almond, 11 oz (312 g)
Flash Deals are limited time, limited quantity promotions, so act fast! Limited to one item per customer.
Peace Cereal, Clusters & Flakes, Vanilla Almond, 11 oz (312 g)
Expiration Date:?February 2019Expiration Date
The length of time for the expiration date or "best used before" date depends on the type of product, as well as the brand.
Perishable items (such as flax oils or certain probiotics) generally have shorter expiration dates. Although our warehouse is fully air-conditioned, these more fragile items are put in cold storage (freezer or refrigeration unit) for maximum freshness.
Our receiving department does its best to verify and then enter the correct expiration dates for all incoming products. However, discrepancies do occur from time to time. This being said, the exceptionally high turnover at iHerb ensures that our inventory is among the freshest in the industry.
Shipping Weight:?0.95 lbsSwitch to Metric units
- Product Code: GTP-10452
- UPC Code: 075070104521
- Package Quantity: 11 oz
2 x 6.4 x 10.4 in, 0.8 lbsSwitch to Metric units
Product Rank:
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Per Serving
- 240 Calories
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- 220 mg Sodium-9% DV
- 11 g Sugars
- 3 g Fiber
- 5 g Protein
Whole Grain Oat Clusters, Crispy Corn Flakes, Sweet Almonds with Natural Vanilla
Vanilla Almond Naturally Flavored Clusters & Flakes
Good cereal starts with great taste. Many cereals may taste good, but artificial, unhealthy ingredients are often used without the whole grains, fiber and protein which are essential to a healthy breakfast. That's what makes Peace Cereal different.
We add only premium all natural grains, nuts and fruits providing you with essential whole grains, fiber and protein in a naturally delicious cereal. By effectively combining the finest all natural ingredients with uncompromising taste and frequent giving to worthy causes, Peace Cereal truly delivers Peace of Mind in Every Bite!
The Peace Pledge
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- Use only the finest all natural ingredients without compromise
- All ingredients are grown and made without GMOs
- Continually minimize our environmental impact through eco-friendly operations and packaging
- Support our communities through ongoing giving
- Deliver nutritious food that tastes great!
- This pledge is more than a catchy slogan for everyone responsible for Peace Cereal; we live it every day.
Other Ingredients
Corn flour, whole rolled oats, milled cane sugar, almonds, vegetable oil (canola and/or safflower and/or sunflower oil), rice flour, barley malt syrup, cornstarch, brown rice syrup, salt, natural flavor, cinnamon bark, annatto (for color), turmeric (for color), purple carrot juice (for color), ginger root, cardamom seed, black pepper, clove bud.
Contains almonds. May contain other tree nuts, wheat and soy.
|Supplement Facts
|Serving Size: 1 cup (55 g)
|Serving Per Container: About 5
|Amount Per Serving
|%Daily Value*
|Calories From Fat
|Total Fat
|6 g
Saturated Fat
|0.5 g
|Trans Fat
|0 g
|0 mg
|210 mg
|Total Carbohydrate
|42 g
|Dietary Fiber
|3 g
|11 g
|5 g
|Vitamin A
|Vitamin C
*Percent Daily Values are based on a 2,000 calorie diet. Your daily values may be higher or lower depending on your calorie needs:
|Total Fat
|Less than
|65 g
|80 g
|Sat Fat
|Less than
|20 g
|25 g
|Less than
|300 mg
|300 mg
|Less than
|2,400 mg
|2,400 mg
|Total Carbohydrate
|300 g
|375 g
|Dietary Fiber
|25 g
|30 g
|Calories per gram: Fat 9 • Carbohydrates 4 • Protein 4


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Well, where to start.
Hunter Sheets: Red Alert is billed as Truth Rating high and it is not kidding. For anyone who has an interest in the back story of SLA Industries this is a must have. It starts, innocuously enough, with some more info on the Moral Right, SLA Ops internal nemesis de jour. Inspired buy Mary Whitehouse's National Viewers' and Listeners' Association, MR are a relativly recent addition to the World of Progress. Essentially an arm of SLA's internal security aperatus they have supplanted Cloak Division and are rather more interesting as an antagonist. Here we have their latest effort to bring moral fibre to the streets of Downtown Mort, the Virtue Squads. A combination of Football Hooligans, Shin Heads, Neo-Nazis and other brutal thugs lead by the scions of the World of Progres' wealthiest and most powerful families, there is plenty of social commentary here as well as a distinctly Clockwork Orange vibe.
Following on from this are the Hunter Sheets themselves. Five distinctly different targets are presented here, all of whom will provide at least an evening's play and a decent challenge. All bar one also touch on a hidden element of the World of Progress. The detailed system info also provides plenty of material for a GM to work into his own adversaries.
Finally there is the closing story, featuring Intruder, Mr Slayer himself and one of the Huntersheet targets. It's a good read, referencing a number of old, somewhat obscure SLA short stories such as Love is a Stranger (check out the SLA forums at http://nightfall.me/index.php if you are interested) and it's the most Truth heavy section of the whole piece. While it's not the telling of the SLA backstory it does give you enough to see the shape of it and intrigue you as to what is to come.
All in all the Virtue Squads are the highlight of the piece for me. I run SLA as a very street level game and unlike SHIVERs they cannot just be pushed around or outright ignored. The other elements are things I would weave in infrequently to up the wiredness level and keep my players on edge. If you are interested in the secret world of SLA however you are likely to disagree with that assessment.


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Techno Trance Music Viral Video Songs
The How To Create This Sound
Tutorial on Creating A Distorted and Reverb String Sound ~ what the instructor calls programmed, robotic. The lead line is about feeling spacey. If you have never heard this sound before or do not understand this concept of 'trance' I will provide my take on this subject.
Music in all generations can be used somewhat like a hypnotic drug. Like a universal language that puts emotions into another form, like a sound wave, rather than trying to describe in words the inner feeling of a human attempting to communicate.
Technically, the dictionary says: that a musical fugue composition contains a contrapuntal short melody or phrase (the subject) which when taken up successively by another part or instrument develops into a woven stream.
As for a mental description of 'trance' we find, "a state or period of loss of awareness of one's identity, often coupled with flight from one's usual environment, associated with certain forms of hysteria and epilepsy." Thus, you could say the music acts like a mood enhancing drug, changing the mental chemistry, putting the listener into a hypnotic state and perhaps an escape from his day-to-day world.
- DERIVATIVES fuguist |ˈfyoōgist| noun
- ORIGIN late 16th cent.: from French, or from Italian fuga, from Latin fuga 'flight,' related to fugere 'flee.'
As you listen to this short interlude, do you feel yourself sliding into a reverie or dream state? Listen to the repetition and 'even' dynamical quality that sets your brainwaves into the trance state.
Ewan Dobson Time 2 Guitar Riff
Viral Trending Puts Everyone Into This Communal State
While not a science, the art of predicting which YouTube videos will go viral can be entertaining. Watching the numbers soar means you are part of the cultural and societal 'trend.' Today's prediction goes to "Trance," a piece I've already shared with friends on FaceBook. We begin with an introduction to this crisp, clean instrumental melody that grows on you. Based upon a moving techno or electronica sound and reverberation, this rendition calls your attention into the 'trance' state.
The toned-down presentation blows me away. The still atmosphere, the oriental style hat hiding the persona creating a bit of mystery and then the sound of the strings.
The original videos on YouTube have copyright infringement problems. So I'm providing what is available. The second video presentation provides a tutorial on creating this 'trance' reverb guitar sound.
Trance Guitar - Hearing Guitars #2
Sleep Sleep Sleep
On the edge of sleep myself, I keep 'dazing' and dosing off. Who needs narcotics, drugs, or no-doze when we can simply listen to the sounds and watch the screen? The point of the 'trance' music presentation as an introduction and explanation of why many people enjoy this form of instrumentation.
The body picks up on the rhythm, especially those who have a frequency in tune with these robotic suggestions. Very much like the 'tarantella' that 'weaves' the sound in and out, the trance puts someone into this hypnotic state.
Planetary Dreamlike Vocal ~ Trance #3
A Cult
You can search and surf through the "Trance" electronica videos and find favorites that have trended from past and present years. The sound moves one to dance and are favorite tracks played at parties. Psychedelic images and spiraling pictures often shown in these videos enhance the experience. Jimmy Hendrix would love it.
The symbolism of the 'road' speaks of going on a 'trip' taking the listener to another space and time. The hypnotic beat provides this escape from reality, even just for ten minutes to enjoy. No drugs necessary.
You will find sky, cloud and airplane images that imply making you "high" while listening.
Others indicate a state of relaxation in fictional landscape scenes or natural settings.
Besides guitar, many other forms and combinations of instrumentation and even cultural flavors may be heard in this electronic music.
Road Trip Trance #4
Did You Have A Favorite?
Listening to these different forms, did you have a favorite Trance?
About the Author
Debby Bruck, CHOM founded Homeopathy World Community social network. Debby believes that homeopathy is the wave of the future that provides hope and healing to those who have tried every other approach. Follow Debby on Twitter.
Disclaimer: This Information is for Educational Purposes Only
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- Human Aura During Sleep and Trance
Clairvoyant observation bears abundant testimony to the fact that when a person falls into a deep slumber, the higher principles in their astral vehicle (Kamic sheath), almost invariably withdraw from the...
- Altered states of consciousness
A well known consciousness investigator into altered states of consciousness, Professor of Psychology at the University of California at Davis, Charles Tart has explored experimental work in parapsychology, ESP, psychokinesis, lucid dreaming, out-of-body and near-death experiences, and similar phenomena.
Many years ago while comparing notes with others in my struggle to play the guitar, I learned a picking style called the "Travis Pick" apparently a product of country guitarist Merle Travis' style. It involved a rapid repetition of the fingers in a particular order on individual strings with the thumb alternating the bass notes on the two upper strings. Combining this picking pattern with a progress of sub-minor notes and chords can produce this type of musical sound. The sub-minors provide such small but such meaningful nuances to the music that we find ourselves listening for them and attempting to predict their tone as the melody grows into more and more complex progressions adeptly demonstrated in the guitar riff video. Music is truly a global thing common to humankind. In my experience, I find far more people who are caught up in the melody of a song...the repetition of the music and just a few who actually hear the lyrics and grasp the meaning and intent of the song. This leaves me wondering how in the world "rap music" ever had a chance of surviving. A very interesting write, Debby, thanks for sharing. Up+! ~WB
I love trance music! I listen to it when I work out. I also enjoyed the videos. I always thought this music was made elctronically only. Completely awesome!
Just followed you on twitter thanks for the heads up
Hello Debby - hope everything is fine with you these days - love this hub subject and your passionate treatment as always - I will post this beauty to my FB music group Let's just talk music or cinema with a direct link back here - and here I am sending you warm wishes and good energy from lake erie time 10:29pm ontario canada
Great hub! I LOVE electronic, trance, drum and bass. That first video "guitar riff" blew me away, If i didnt see it live , i wouldv'e bet it was done with a synth. That vid made my day :)
...okay then - I'll look for you - it is Debby Bruck?
....wow that' a lot of Colin Stewarts and none of them with my pretty face and homely cat - or is it the other way around - lol - okay ...Colin to the rescue ......
...hi Deb - my name is Colin Stewart - same picture as here on FB
..yes DJ Deb I checked out your site and it's marvelous, well like you, and I will definitely go back again - you have an excellent set up there and an eclectic taste in music for sure ....I am so proud of you ..... lake erie time 1:28pm listening to some nice classical music and just finishing up my last cup of coffee for the day
.....so Deb you are also a musicologist? A most definitive hub I would say and excellent research and homework on this one because the epi-man is a master musicologist himself - and appreciates the effort you put into this one - my favorite trance music is traditional Indian classical music - and your eclectic hubs always rock me and roll me ..... completely.
lake erie time 1:29am
Love this one, great advise on some music here. This would be pretty soothing for a person who suffers panic attacks I think. Nice hub, voted up and useful.
Hello Debbie, I enjoyed all the music videos! Hope you're doing well and best of luck with the 30 day challenge! Yikes


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Arya Cohen served as a Campus Recruitment Ambassador (CRA) for City Year at the University of Delaware during her senior year. Arya is from Roslyn, New York and studied psychology and sociology with a social welfare concentration at the University of Delaware. She has submitted an application for City Year New York to serve next year and awaits the decision!
What were your main responsibilities while working as the University of Delaware's CRA?
As CRA at the University of Delaware, it was my role to spread the word about City Year on campus as much as possible. I led 3 interest sessions right before each deadline, contacted Registered Student Organizations (RSO)s with information for their members about City Year, attended networking events and utilized social media to engage with more students. .
Why did you choose to apply to be a CRA?
Over winter session at UD (a 5 week time period where you can intern, study abroad or take classes), I interned at Communities In Schools of Delaware. After that experience, I felt very passionate about education. Applying to be the CRA was a great way for me to have a connection with another non-profit to work towards closing the education achievement gap.
What skills did you take away from your CRA experience? How do you think your CRA year will impact your future studies and career?
Most importantly, I learned how to communicate effectively by practicing my elevator pitch, talking about my experience and background with City Year in three minutes or less. I learned how to network at events andstrengthen my organizational and time management skills, all of which will be extremely important in my future. Being a CRA and representing City Year has made me realize how passionately I feel about non-profit organizations, and specifically how much of a need there is to help students in grade school reach their full potential. Wherever I may end up in the future, I know that I want to be working with like-minded individuals, at an organization that sees a better future for at-risk youth.
What was the highlight of your CRA experience?
Aside from working with my fantastic supervisor, Dawn, and meeting and networking with so many new people, the highlight of my CRA experience was helping coordinate a panel on campus with City Year, Peace Corp and Teach For America. With 75 students in attendance, I think we were really able to reach a new audience of students. It was a great experience for students to hear about various organizations that are all trying to positively contribute to the world.
Adrianna Lucero served with City Year Jacksonville after her Sophomore year at Harvard. After service, she returned to Harvard and became a Campus Recruitment Ambassador for two years. She graduated this spring with a Bachelor of Arts in Social Studies.
What were your responsibilities when serving in Jacksonville as a corps member?
I supported a teacher in a 9th grade reading classroom, and worked one-on-one or in small groups on reading skills with the 7 awesome students on my "focus list". I also assisted with whole-school events and activities and with ELA related-initiatives as the literacy coordinator.
The most unique part of my experience was the fact that it was the start-up year for Jacksonville, so our team and corps did a lot to build connections with the school and community, many of whom had never heard of City Year before. I remember getting to be a part of round tables and discussions with school, business, and political leaders, learning from them and telling them about City Year.
Why did you choose to apply to be a CRA?
I really enjoyed my time with City Year and thought the program would benefit my peers, a lot of whom are passionate about education! Plus, since City Year was founded by Harvard alums, it is really exciting to be recruiting on the campus where it all started.
What advice do you have for alumni who want to recruit for City Year?
Represent City Year beyond just the career fairs and info sessions! I'm not saying you need to wear your red jacket all the time; what I am saying is that I think a successful CRA continues to embody the values and mission of City Year by staying involved with education and service on campus. My most meaningful conversations about City Year were informally talking about the experience with the peers I met in classes about education or in service-related student groups.
Why would you recommend this program to City Year Alumni?
It's a fun way to continue to impact students' lives by making sure there are great people to fill your boots!


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One app that was missing from the carrier software on the Sprint HTC Arrive was Sprint TV (see our hands on with Sprint Radio here).
Luckily, our wait is over as the app is now available in the Marketplace (hit the link here on your phone or use the Sprint hub to get to it). Featuring ESPN, NBC, Disney Channel, ABC, FOX News, you can watch a lot of live or streaming on the go, including some of the latest movies. Some of the features are free, included with your plan, while there are also "premium" features available for some extra cash e.g. Sports Xtra & Sprint TV Xtra which are $5.99 and $9.99 a month, respectively. Oh hey, there's a Looney Tunes option for $4.99 a month too...dibs. [Edit: for some reason we're getting an error when we try to subscribe--this happened on Radio too, could be Sprint's servers still catching up]
Downside? You can't use it over WiFi. Yeah, go figure--they want you to use your "unlimited" data connection.
Over all, the app is well polished and designed and takes advantage of the landscape/keyboard layout of the Arrive quite nicely. Give it a go in the Marketplace and we'll see about a video in the morning.
Thanks, @David_Strait, for the heads up!


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City of Lakes AmeriCorps City of Lakes AmeriCorps
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Mission Statement
The mission of the City of Lakes AmeriCorps program is to help ensure all children succeed both socially and academically regardless of race, culture, ethnicity, native language, gender, poverty level, disability, or any other status.
City of Lakes AmeriCorps is a national service program in partnership with Minneapolis Community Education and the Multilingual Department of Minneapolis Public Schools. We strive to bridge the language gap, increase academic language proficiency and academic achievement for intermediate English Learner students in Minneapolis Public Schools.
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Chairman Marcus Levings, of the Three Affiliated Tribes of the
Fort Berthold Reservation, ND, speaks to President Obama during
the Native American Summit of 11/05/2009 held at Dept. of the
Interior (Photo: Doug Mills, New York Times)
"The Recovery Act broadband projects announced by USDA will open up educational, job and business opportunities to hundreds of thousands of Americans living in rural areas," said Vilsack. "As President Obama said earlier this month, 'America today is on the verge of a broadband-driven Internet era that will unleash innovation, create new jobs and industries, provide consumers with new powerful sources of information, enhance American safety and security, and connect communities in ways that strengthen our democracy.' I am pleased that USDA, with funds provided through the Recovery Act, is leading this effort in rural parts of our country."
Today's announcement provides a $10.95 million loan and a $10.95 million grant to the Reservation Telephone Cooperative to provide service in the remote and rural areas in and around the Fort Berthold Indian Reservation. The project will bring broadband service to homes, businesses, schools and libraries. In addition, service will be provided to tribal government facilities which provide tribal child support, education assistance, employment assistance, health care and public safety services on the reservation. The area has experienced poverty rates as high as 28 percent and a 40 percent unemployment rate.
To date, $1.067 billion has been invested by USDA through the Recovery Act for 68 broadband projects. Together, the projects will make high speed Internet available to an estimated 529,000 households and 96,000 rural business and public facilities across 31 states, one territory and 17 tribal lands and predominantly Alaska Native regions. Many of the areas to be served are extremely remote, sparsely populated and in some cases not connected by roads. Secretary Vilsack and other cabinet officials visited some of the communities that will receive broadband Internet access during a "Rural Tour" last year. Funding is contingent upon the recipient meeting the terms of the loan, grant or loan/grant agreement.
President Obama signed The American Recovery and Reinvestment Act of 2009 into law on Feb. 17, 2009. It is designed to jumpstart the nation's economy, create or save millions of jobs, and help to bridge the digital divide necessary to addressing long-neglected challenges so our country can thrive in the 21st century. The Act includes measures to modernize our nation's infrastructure, enhance energy independence, expand educational opportunities, preserve and improve affordable health care, provide tax relief, and protect those in greatest need.
The Broadband Stimulus PipelineSM – Award # 139
View The Broadband Stimulus Pipeline: Award Locations in a larger map


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Running a Successful Baby-Sitting Business
Almost anyone who loves children can run a baby-sitting business. To run it successfully, though, you need a business plan and a record-keeping system. Follow these brief instructions to get started in the right way.
Begin by making a business plan that sets your goals. Your first goal will be to decide how many children you will care for at one time. Think about how much baby-sitting experience you have and how many kids you have looked after at one time. Don't overwhelm yourself, but don't underestimate your own powers either.
The next goal is setting a rate. The rate you choose should be per hour. To set this rate, you may want to check with your friends who baby-sit, or find out how much your mother and father have paid baby-sitters. Keep in mind that if you set your rates too high, you won't have any customers. On the other hand, if you set them too low, people may take advantage of you or not take you seriously
Finding customers is the third goal. I chose to advertise my business by creating flyers on my computer. The flyer listed all the skills I had come up with while setting my first goal. You also need to know exactly how much money you will spend on supplies and where you will advertise. I limited distribution to a few blocks in my neighborhood. In time, your reputation for doing a good job will be all you need. Chances are, the parents you work for will tell some of their friends, and you'll get free advertising simply for doing good work.
After a few baby-sitting jobs, you'll need to start keeping records. Your record-keeping will include personal profiles of all the different children you baby-sit. The profiles should list names, addresses, phone numbers, favorite toys, and any medical information, such as allergies or medications. By doing this, you will show parents that you really mean business. In addition, this information will help you if you need to contact the parents in an emergency. If a child is being particularly difficult, you can bring out their favorite toy.
Keeping track of your money is easy with a ledger, a book that lists your income and expenses. Use one page to record how much you made from each baby-sitting job (with a brief description of how many hours you worked and the family who hired you). Use another page to record expenses, such as what you spent on advertising supplies.
An organized baby-sitter needs an appointment book. Record the name of the family, the date and the time of each baby-sitting job. Don't forget to leave some time for yourself, too. You'll want to leave time to study or to play with your friends and family.
You should keep up with your business plan and record-keeping when any information changes. If you stick to all the guidelines here, you should be able to run a very successful and prosperous business that will not only help you earn money but assist parents when they need a helping hand.