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Naive Bayesian Learning in Social Networks

The DeGroot model of naive social learning assumes that agents only communicate scalar opinions. In practice, agents communicate not only their opinions, but their confidence in such opinions. We propose a model that captures this aspect of communication by incorporating signal informativeness into the naive social learning scenario. Our proposed model captures aspects of both Bayesian and naive learning. Agents in our model combine their neighbors' beliefs using Bayes' rule, but the agents naively assume that their neighbors' beliefs are independent. Depending on the initial beliefs, agents in our model may not reach a consensus, but we show that the agents will reach a consensus under mild continuity and boundedness assumptions on initial beliefs. This eventual consensus can be explicitly computed in terms of each agent's centrality and signal informativeness, allowing joint effects to be precisely understood. We apply our theory to adoption of new technology. In contrast to Banerjee et al. [2018], we show that information about a new technology can be seeded initially in a tightly clustered group without information loss, but only if agents can expressively communicate their beliefs.


INTRODUCTION
This paper introduces an intuitive naive social learning model that extends the DeGroot model when agents are differently informed. In standard naive learning, each agent's belief is a scalar, and each agent updates her beliefs by taking the weighted averages of her neighbors' beliefs with fixed weights. This implicitly assumes that every agent's signal quality is the same, which is often not true in practice. We propose a solution to this problem by representing each agent's belief as a probability distribution over possible states of the world. Signal quality can then be naturally communicated by using Bayes' rule to combine one's belief with the beliefs of one's neighbors. For example, if an agent has a scalar opinion µ and a scalar confidence τ indicating how sure she is of that scalar opinion, her belief can be represented by the Gaussian N(µ, 1/τ ). Our model can also accommodate other forms of information. For example, an agent may believe that either one of the two values is very likely, while other values are less likely. This belief can be represented as a double-peaked distribution, but it cannot be represented in standard DeGroot models.
It is known separately in Bayesian learning and naive learning literatures that higher quality signals and more centrally located agents, respectively, have more influence in social learning. The innovation in our model is to introduce signal quality into the naive learning setting, allowing us to precisely investigate how signal quality and agent centrality interact. The two notions are intertwined in the weighted likelihood function, where the formal expression is at equation (1) in Section 1.1. Informally, the value of this function at θ is the product of the votes of each agent, where an agent's vote is how likely the state value θ is after the agent received the signal compared to before (the signal quality part), and the number of votes is the agent's eigenvector centrality (the centrality part).
Our main theorem is that, under some conditions, long-run beliefs are concentrated near the maximizers of the weighted likelihood function. If we know the probability densities of initial beliefs, we can explicitly calculate the consensus belief. This allows us to analyze the interaction; for example, we can calculate how much more informative a signal has to be to offset the centrality disadvantage. This formula can have practical applications in the analysis of information spread in decentralized networks, such as propagation of rumors or adoption of new technologies. Practitioners interested in empirical results can also use our formula without having to fully grasp the theory in the paper. Some examples of applications are discussed in Section 7.
Rahimian and Jadbabaie [19] propose essentially the same model in this paper in the case where the state space is finite, and they have derived the same concentration result as ours in that case. Our novel contribution is in the analysis of feasibility and convergence of beliefs when the state space is infinite discrete or continuous. We also draw two policy implications from the model with Gaussian beliefs: how to seed opinion leaders for technology adoption, and how to solve the clustered seeding problem introduced by Banerjee et al. [5]. We discuss the relationship between our work and Rahimian and Jadbabaie [19] in Subsection 1.2.
Finally, we contrast the results of our model with that of Banerjee et al. [5]. Their model tries to solve the issue of differently informed agents in DeGroot learning by assuming that each agent can either hold a scalar belief or be uninformed. While their model can capture the spread of awareness of new things, it does not have the notion of signal informativeness that our model provides.
Our model also gives a different prediction than that of Banerjee et al. [5] on how to seed a new technology for widespread adoption. Their model has a clustered seeding problem: there is substantial information loss if the initially informed agents ("seeds") are clustered in a small group of communities. This is a problem because a social planner who wants to maximize the impact of a newly introduced technology should seed it to centrally located opinion leaders, but these leaders are often closely connected to each other. Our model does not have this problem; the influence of each agent depends only on the agent's signal informativeness and centrality. The key modeling difference is that agents in our model can communicate confidence of their beliefs that their neighbors can take into account. Therefore, to solve the clustered seeding problem, the social planner should encourage open and expressive communication between agents.

Contributions
Our agents are in a social network, represented as a strongly connected directed graph. Assume that there is a true state of the world θ * . Each agent gets a signal drawn from a distribution that depends on θ * . She then forms a belief, the posterior probability distribution of θ * given the signal. In each round, each agent has access to beliefs of her neighbors and naively assumes that these all come from independent signals. She then forms the Bayesian posterior of θ * given the signals of her own and her neighbors as her new belief. This gives rise to an update rule that combines her belief with her neighbors' beliefs that she uses in every round.
We prove that our learning process can be viewed as follows. At every time step, each agent passes copies of signals she has to her neighbors, so copies of signals "flow" through the network. Every copy is taken as independent and no copy is discarded, so the number of copies of agent j's signal that agent i has at round t is the number of paths from j to i of length t. Agent i's belief at time t is then the posterior belief conditional on these repeated signals (Prop. 4.7).
We then tackle the important question of what happens to agents' beliefs in the long run. The answer depends on the following weighted likelihood function L (Def. 5.1). Let f (t ) i (θ ) be agent i's belief of θ * in round t and f * (θ ) be the common prior of θ * . Then where i is the eigenvector centrality of agent i. We show that if L(θ 1 ) < L(θ 2 ), then as t → ∞, (Prop. 5.5). In other words, agents' beliefs that the true parameter is θ 2 are much stronger than the corresponding beliefs for θ 1 as time goes on. Thus, in the long run, agents' beliefs should concentrate at values θ that maximize or nearly maximize L.
The form of the weighted likelihood function highlights two factors that determine the quality of learning. An agent who is more centrally located (high i ) and has a more informative signal (high or low f (0) i (θ )/f * (θ )) has more influence over the consensus belief. Note that our consensus belief is fundamentally different from that of DeGroot. In our model, the limit belief is the weighted likelihood maximizer, while in DeGroot, it is a weighted average of beliefs.
The fact that beliefs tend to converge to a point distribution agrees with the following intuition: agents take a growing number of repeated signals as independent, so they become increasingly confident in their beliefs. The fact that the limit belief is tractable and interpretable comes as a surprise, and we believe this to be an attractive feature of our model.
In spite of this intuition and much to our astonishment, the concentration previously discussed turns out not to hold in general if Θ, the state space, is infinite. We discover that even with a single θ max uniquely maximizing L, agents' beliefs may not converge to the point distribution at θ max (Ex. 6.6). We resolve this difficulty by introducing an extra condition: if θ max still maximizes a "perturbed" version of L, then the concentration result holds. We can think of this as θ max being a sufficiently robust maximizer of L. This result requires careful analysis of convergences of infinite sequences, and the generalization of this idea for an arbitrary state space Θ can be found in Proposition 5.7.
When the state space Θ = R is a continuum, for example in the case of Gaussian beliefs, this perturbed L condition is still too strong: perturbing L will almost always shift the maximizer due to the continuity of the space. The correct condition needs to allow the maximizer to shift, but not too much. Furthermore, it is insufficient to consider the density at a single point, since this tells us nothing about densities at nearby points and so cannot establish concentration. Instead, we need to show that for any neighborhood of θ max , agents' beliefs that the true state lies in that neighborhood tend to 1. Our Master Theorem 5.10 pulls together all of these ideas.
With the Master Theorem 5.10, we derive a convergence theorem when the state space is a continuum, provided that the initial beliefs and priors are sufficiently continuous (Def. 6.7). Our main result in this most difficult case is Theorem 6.9. The following is a simplified version of this theorem, simplified from Corollary 6.10.
Assume that the state space Θ = R k . Suppose that the normalized beliefs f (0) i (θ )/f * (θ ) are bounded continuous functions; the weighted likelihood function L attains the maximum L max uniquely at the point θ max ; and L decays in the sense that there is a bounded set S such that sup θ S L(θ ) < L max . Then, as t → ∞, beliefs f (t ) i converge to the point distribution at θ max .
This result states that whenever the initial beliefs are continuous functions that are bounded with respect to the prior and they decay sufficiently fast, agents' beliefs converge to the maximizer of the weighted likelihood function if this maximizer is unique. This result encompasses a wide range of probability distributions in practice. For example, the theorem applies when beliefs are multivariate Gaussian. Therefore, Theorem 1.1 is widely applicable.
The rest of the paper is organized as follows. We discuss related literature in the remaining of this section. Section 2 describes our model. Section 3 discusses mathematical preliminaries. Section 4 gives fundamental definitions, interprets the updated beliefs, and analyzes feasibility of initial conditions. Section 5 proves the "Master Theorem" that establishes concentration of beliefs near the weighted likelihood maximizers. Section 6 uses the Master Theorem to prove convergence of beliefs in the finite, infinite discrete, and R k cases. Section 7 gives applications in the cases of binary, Poisson, and Gaussian beliefs.

Related Literature
Our paper is closely related to a growing literature on learning and belief formation in social networks [3,4,6,11,22,23]. Much of this literature focuses on Bayesian learning [1,7,15,18] and DeGroot learning [8,12]. Comparable to Golub and Jackson [12], our model shows that centrally located agents, as measured by eigenvector centralities, have more influence on the consensus belief. However, in our model, eigenvector centralities appear as duplicate agents rather than linear weights in the limit belief.
Specifically, our paper belongs to the quasi-Bayesian learning literature. Eyster and Rabin [9,10] and Rahimian et al. [20] study long-run aggregation of information when agents ignore repetition. Li and Tan [14] study learning where each agent updates both her belief on the state of the world and her belief on her neighbors' beliefs. Jadbabaie et al. [13] also combine DeGroot learning with Bayesian updates. Features that distinguish their model from ours are learning objectives and arrivals of information. Agents in our model receive signals only once at the beginning, rather than at every time step, and Bayesian-update based on their neighbors' beliefs rather than taking weighted averages of distributions. Agents in our model can form and update beliefs even if the true state of the world does not exist, and our limit beliefs are biased by influential agents which do not exist in their model. Lastly, likelihood ratios play a central role in our model; they appear in Rahimian et al. [21] and Molavi et al. [17] in a different context. Rahimian and Jadbabaie [19] propose a model where agents use Bayes' rule to come up with a belief update rule that is optimal for the first round and naively continue to use the same rule for subsequent rounds. This naive agent behavior is the same as ours, and resembles the microfoundation of the DeGroot update rule proposed in DeMarzo et al. [8]. Agents in their model have utilities; they take actions to maximize their utilities and only observe their neighbors' actions. In contrast, agents in our models do not have utilities, do not take actions, and have direct access to their neighbors' beliefs. When the state space is finite, the action space is a probability simplex over the state space, and each agent's utility is the negative Euclidean distance between her action and the point mass at the true state, each agent's action corresponds to her belief. Under these assumptions, their model becomes equivalent to ours when the state space is finite, and they independently derive what we call the naive Bayesian update rule (Def 2.1) in their Section 4. They also independently prove the concentration result in this special case when the state space is finite (Thm. 6.3) in their Theorem 4.
Finally, our work is related to Banerjee et al. [5]. In their model, the informedness of agents is a binary state, informed individuals hold scalar beliefs, and the learning rule is to take the average belief of informed neighbors. This contrasts with our model where informedness is a continuous attribute. Their work identifies the clustered seeding problem discussed in the introduction, which does not occur in our model. See further discussion of this problem in Section 7.3.

MODEL
We discuss the assumptions of our model in this section.

Social Network
Our agents are in a social network, modeled as a directed graph G with N ≥ 2 nodes, labeled 1, 2, . . . , N corresponding to N agents. An edge from i to j is denoted (i, j) and means that there is information flow from agent i to agent j. Assume that G is strongly connected, that is, for every i and j there is a path from i to j. Suppose that there is a self-loop (i, i) at every node, meaning that information flows from every agent to herself. If there is an edge (i, j), we say that i is an in-neighbor of j. Let N (i) denote the set of in-neighbors of i.

Beliefs
Let Θ be the space of parameters θ , modeled as a measure space. We model beliefs as probability density functions on Θ: nonnegative measurable functions f : Θ → R ≥0 such that the integral ∫ θ ∈Θ f (θ ) = 1. These reduce to probability mass functions in a discrete setting, i.e. when Θ is discrete with the counting measure. Note that we model beliefs as probability densities, which is a special case of general probability distributions. For example, a point distribution on R cannot be represented as a density, but most probability distributions in practice are densities. A prior on Θ is any nonnegative measurable function f : Θ → R ≥0 . If the integral ∫ θ ∈Θ f (θ ) diverges, we call the prior f improper. Note that our improper priors are not necessarily limits of proper priors, but simply densities whose integrals diverge and so cannot be normalized. An example of an improper prior is the uninformative prior, f (θ ) ≡ 1, where Θ is an infinite discrete set or an unbounded interval of R.
Suppose that each agent has a common (possibly improper) prior f * for the true state of the world θ * on Θ. Assume that f * (θ ) > 0 for all θ ∈ Θ. This assumption is important in order to state the update rule for beliefs (e.g. in Definition 2.1).

Belief Updates
Agents update their beliefs based on the beliefs of their in-neighbors using the following rule, which will be justified shortly.
Definition 2.1. The naive Bayesian updated belief for agent i with respect to beliefs f 1 , . . . , f N is the belief , which is well-defined whenever the integral in the denominator is nonzero and finite.
It is easily checked that this definition defines a belief. We now provide a justification for this definition. In the justification, we will assume that f * is a probability density function, but the formula in Definition 2.1 also makes sense for improper priors.
Consider the following scenario called an underlying scenario. Suppose that agent i knows a family of probability density functions (p θ,i ) θ ∈Θ , indexed by the parameter θ and the agent i, over a common sample space Ω i . We make the following marginalizability assumption, which is important in order to reason consistently about marginal densities. For any integers n 1 , . . . , n N ≥ 0, let X i, 1 , . . . , X i,n i be random variables drawn independently from p θ,i . We thus have a joint probability density for these random variables Assume that this distribution can be marginalized over θ with positive density, that is is nonzero and finite.
Agent i receives a signal X i drawn from p θ * ,i , the distribution corresponding to the true parameter θ * , without knowing θ * . She then forms a belief f i (θ ) for θ * conditional on the signal X i . Suppose that agent i learns about beliefs f j (θ ) of all her in-neighbors j ∈ N (i), formed in the same way, and wishes to perform a Bayesian update on her own belief. The result will be the belief f (θ ) for θ * conditional on signals X j for every j ∈ N (i). Even though the resulting density seems to depend on X j , we will show in the next proposition that this update can be performed knowing only the beliefs f j but not the private signals X j , with the formula as in Definition 2.1.
In an underlying scenario described above, the updated belief for agent i is the naive Bayesian updated belief with respect to beliefs f 1 , . . . , f N as in Definition 2.1.

P
. By definition, the updated belief is where the second equality is Bayes' rule and the last equality holds because X j 's are conditionally independent given any θ . Note that the integral in the denominator is nonzero and finite due to the marginalizability assumption. Because p(θ ) = f * (θ ) and substituting these into our first equation gives the desired formula.

Belief Dynamics
Here we describe the evolution of beliefs. Agents share a common prior f * of θ * , which may be improper, and agent i starts off with initial belief f (0) i of θ * . The functions f * , f (0) 1 , . . . , f (0) N are the initial condition of the model. The subtlety is that not all initial conditions lead to well-defined updates. Nevertheless, in Section 4.3, we will provide simple checks that guarantee the existence of updated beliefs. For t = 0, 1, 2, . . . , let f (t ) i be agent i's belief of θ * at round t. Agents update beliefs as follows. For t ≥ 0, f (t +1) i is the naive Bayesian updated belief (Def. 2.1) for agent i with respect to beliefs f We call an initial condition feasible if the updated beliefs are well-defined in every round, that is, the integral in Definition 2.1 is nonzero and finite. Moreover, an initial condition has an underlying scenario if it arises from an underlying scenario in the above justification of Definition 2.1. Proposition 4.7 shows that an initial condition with an underlying scenario is always feasible.
For an initial condition with an underlying scenario, note that although the update rule is rationally justified, it is only optimal for the first round. In later rounds, an optimal update needs to take into account that the agents' beliefs are already updated at least once and so no longer independent. Still, our naive agents use the same update rule for every round. This modeling choice formalizes what we said earlier that people use Bayes' rule to learn, but they ignore repeated information.
It may seem that using Bayes' rule on the entire distribution requires agents to communicate an infinite amount of information. Nevertheless, if conjugate priors are used (say, Gaussian beliefs and Gaussian signals, or Beta beliefs and Binomial signals), then agents only need to keep track of a finite number of distribution parameters. Moreover, our model is robust under discretization in the following sense. Suppose that agents can only learn about other agents' beliefs at a fixed finite set of points. By Proposition 5.4, the ratio of beliefs f (t ) is completely determined by the initial ratios f (0) j (θ 1 )/f (0) j (θ 2 ) for 1 ≤ j ≤ N , so the ratios of beliefs at these fixed points are preserved and hence so are the approximate shapes of beliefs. These ratios also dictate convergence of beliefs towards the maximizer of the weighted likelihood function (Sec. 5). Therefore, assuming that we choose points close to the maximizer and beliefs are sufficiently continuous, the discretized beliefs will converge to the approximate maximizer. We name our model naive Bayesian for two reasons. First, our model combines features from naive learning and Bayesian learning. Second, our agents assume that distributions of neighbors are independent conditional on the state, which is also the assumption made in Naive Bayes classifiers in machine learning.

MATHEMATICAL PRELIMINARIES
To analyze our model, we require the following mathematics. We review Perron-Frobenius theory, which helps us determine the behavior of large powers of matrices and establish convergence of beliefs, and the theory of L p spaces, which helps us bound functions and establish feasibility of initial conditions.

Perron-Frobenius Theory
Let G be the graph of our model, and let A be the adjacency matrix of G: A i j = 1 if there is an edge (i, j) and 0 otherwise. The Perron-Frobenius theory provides us with properties of eigenvalues and eigenvectors of A. For an adjacency matrix, a power of the matrix counts the number of paths between each pair of nodes. Hence we would expect that for a strongly connected social network with self-loops, the corresponding adjacency matrix should be primitive. This is formalized in the following lemma.
A is a primitive matrix.

P
. By strong connectedness, for every i and j, there is a path P i j from i to j. Let k be the maximum length of all P i j 's. We can extend a path to arbitrary lengths by using self-loops, so there is a path from any i to any j of length k. Because the (i, j) entry of A k is the number of paths from i to j of length k (see e.g. Lemma 4.3), A k has positive entries. Therefore A is primitive.
The following theorem from Meyer [16, Chapter 8], a version of Perron-Frobenius for primitive matrices, characterizes the largest eigenvalue of a primitive matrix and the growth rate of the matrix's powers. Finally, the limit lim k→∞ M k /r k exists and is given by This limit P is called the Perron projection of M.
The following lemma will be useful when we reason that the entries of A k tend to infinity. 3.4. The spectral radius of A is greater than 1, that is, ρ(A) > 1.

P
. Let r = ρ(A). Lemma 3.2 implies that A is primitive, so by Theorem 3.3, r is an eigenvalue of A, and there is an eigenvector associated with r with positive entries. Pick an arbitrary edge (i, j) with i j. Then We now define the notion of centrality alluded to in the introduction. Eigenvector centrality is a standard notion that measures the influences of agents. To justify this, note that by definition, Call i the score of agent i. Then this equation says that one's score is large if one's information can reach many agents; moreover, reaching agents with a larger score contributes more to one's score. This exact behavior characterizes influence: one's influence is large if one talks to many people, and talking to more influential people contributes more to one's influence than talking to less influential people. So it is appropriate to say that eigenvector centrality is a measure of influence.

L p Spaces
The following inequality, Hölder's inequality, is fundamental in the study of L p spaces. Note that the version presented here has a different parametrization of exponents from the usual formulation.
In particular, if f i ∈ L 1 for all i, then the weighted geometric mean f We also consider the following "weighted" L p spaces, where the weight is the common prior, which will be useful in the study of initial conditions in Section 4.3.
Definition 3.7. For p ∈ [1, ∞], define the L p space weighted by the prior, L p * , as follows. If p ∈ [1, ∞), the space consists of measurable functions f : This is just like the L p norm but weighted by the prior. The space L ∞ * is the same as L ∞ and consists of measurable functions f : This is exactly the same as the L ∞ norm.
We now present a version of Hölder's inequality on these weighted L p spaces, as well as an interpolation result.

UNDERSTANDING THE MODEL
In Section 4.1, we introduce some preliminary definitions that will be useful in understanding the model in Sections 4.2 and 4.3. Then we interpret the updated beliefs in Section 4.2. Finally, we find the conditions that guarantee that an initial condition is feasible in Section 4.3.

Preliminary Definitions
We introduce the notion of normalized beliefs.
Definition 4.1. The normalized belief of agent i at round t is the function We can interpret this as how large a a belief is compared to the prior. This quantity will appear in the weighted likelihood function in Definition 5.1 and play a central role in our analysis. Note that the normalized belief is well-defined because f * (θ ) > 0 by assumption, and it has the property that it is in L 1 * with norm 1: (t ) i 1, * = 1. However, it need not be a belief, as the integral With normalized beliefs, the update rule in Definition 2.1 assumes the simpler form . (3) We now introduce the notation for the number of paths between each pair of nodes. i j be the number of paths from i to j of length t. By convention, there is one path of length 0 from a node to itself and no path of length 0 from a node to a different node.
The following simple combinatorial lemma characterizes the number of paths. It is completely standard but we include it here for completeness.
The numbers of paths satisfy the recurrence P (t +1) . The recurrence follows by considering that a path from i to j of length t + 1 can be broken down into paths from i to k of length t and from k to j of length 1 for some k. Let P (t ) be the matrix whose (i, j) entry is P (t ) i j . The recurrence can be written in the form P (t +1) = P (t ) A, with the initial condition P (0) = I . It follows by induction that Finally, we formally define the notion of k-feasibility and feasibility.

Interpretation of Updated Beliefs
We interpret what agents are doing in our model. Our goal is to show that each agent aggregates information where another agent's initial belief is weighted by the number of paths from that agent to the aggregating agent. This means that an agent who has many paths to other agents, i.e. an agent with high centrality, will be influential in our model. This result will lead to conditions for feasible initial conditions in Section 4.3 and show concentration of beliefs in Section 5.
To carry out this program, we first give an explicit condition for k-feasibility and an explicit formula for the updated beliefs in the next proposition.
. The proof is essentially by induction on k. Any initial condition is certainly 0-feasible.
And for any i, N j=1 This proves the base case k = 0. Assume that the proposition holds for k. For a (k + 1)-feasible initial condition, the updated beliefs given by equation (3) are well-defined. By the induction hypothesis, this equals where the second equality holds because the exponent of (0) So the formula holds. Finally, the denominator of the right-hand expression shows that N j=1 ( . By the induction hypothesis, the initial condition is k-feasible, and we must show that it is (k + 1)-feasible. The right-hand expression of equation (4) is well-defined, so by running the same argument as above backwards, it can be written as the expression for (k+1) i (θ ) in the update rule. Therefore (k+1) i is well-defined and the initial condition is (k + 1)-feasible.
We can now prove that an initial condition with an underlying scenario (see Section 2.3) is feasible, as a result of the next proposition.
N has an underlying scenario. For any integers n 1 , . . . , n N ≥ 0, the function the ordered tuple of n i independent copies of X i for 1 ≤ i ≤ N . By the marginalizability assumption, the joint density of this tuple is and it is nonzero and finite.
As a corollary of the previous two propositions, we will show that an initial condition with an underlying scenario is feasible, as well as give an explicit formula for the updated beliefs. This is important as it gives us insight into the update process. An initial condition f * , f (0) 1 , . . . , f (0) N with an underlying scenario is feasible. If agent i receives signal X i in the underlying scenario and I (t ) i is the ordered tuple consisting of P (t ) ji independent copies of X j for each j, then In other words, the belief of agent i in round t for each parameter θ is the belief for that parameter conditional on the information I By Proposition 4.5, this quantity equals f (t ) i (θ ). Proposition 4.7 gives us an intuition of the process. At every time step, each agent passes copies of signals she has to her neighbors, so copies of signals "flow" through the network. Every copy is taken as independent and no copy is discarded, so the number of copies of agent j's signal that agent i has at round t is the number of paths from j to i of length t. Moreover, we can interpret the normalized belief i )/p(θ ) by Proposition 4.7, this quantity is larger or smaller than 1 if learning I (t ) i increases or decreases the probability density at θ , respectively.
From Proposition 4.7, we can qualitatively predict how our model behaves in the long term. Because repeated signals in I (t ) i are taken as independent, agents should become overconfident over time. Furthermore, because the numbers of repeated signals are the numbers of paths, agents with more paths to other agents (those with higher eigenvector centralities) should be more influential. Our results on concentration and convergence of beliefs in Sections 5 and 6 confirm these predictions.

Initial Conditions
We now build on the work of the previous section to investigate containment relationships between classes of initial conditions and present simple sufficient conditions for feasibility.
Let P and IP denote classes of proper and improper initial conditions. Let F and F k denote classes of feasible and k-feasible initial conditions. Let U denote the class of initial conditions with underlying scenarios.
The notion of properness may be of interest because it tells us whether the prior can be realized as a probability density. On the one hand, a prior which is a probability density seems to correspond to real-world situations, but on the other hand, the "uninformative prior" f * ≡ 1 is a natural choice in many situations but is often improper. Note that an initial condition with an underlying scenario is proper by definition, i.e. U ⊆ P.
It is obvious that F k+1 ⊆ F k , F ⊆ F k and U ⊆ P. Proposition 4.7 shows the nontrivial containment U ⊆ F . Thus, the classes are contained in one another as shown in the diagram in Figure 1. We now show that there is Θ and G such that all regions in the diagram are nonempty. In particular, these containments are strict, and so the classes do not collapse into one another. For example, it is not sufficient to show that an initial condition is feasible by checking 1-feasibility. Region The following considerations will help verify the table. An initial condition is in P if ∫ θ f * < ∞ and is in IP otherwise. For any t ≥ 1, there are 2 t −1 paths from any node i to any node j. By Proposition 4.5, an initial condition is in F k if and only if ∫ is nonzero and finite for all 1 ≤ t ≤ k, and it is in F if and only if this is true for all t ≥ 1. These checks are straightforward using the fact that ∞ θ =1 θ −p is finite for p > 1 and infinite for p ≤ 1. To check that the example for (F ∩ P) \ U does not have an underlying scenario, use Proposition 4.6 and the fact that Now we develop a sufficient condition for feasibility and show that it is "individually optimal, " in the sense that will be explained in Proposition 4.13. We first take care of the degeneracy issue. Alternatively, an initial condition is degenerate if the product of initial beliefs vanishes almost everywhere. We might expect that degeneracy is a problem for performing updates as the denominator in Definition 2.1 might be zero. The following proposition makes this rigorous. P 4.11. A feasible initial condition is nondegenerate.

P
. Pick any node i and pick k such that there is a path from any j to i of length k. Let n j = P (k) ji > 0. Then Proposition 4.5 implies that ( (0) 1 ) n 1 . . . ( (0) N ) n N ∈ L 1 * with nonzero norm. If the initial condition is degenerate, then this function vanishes almost everywhere and must have zero norm, contradiction. Therefore the initial condition is nondegenerate.
Conversely, by assuming nondegeneracy, we can prove feasibility by requiring some form of boundedness. Specifically, it suffices for the normalized beliefs to be in L p * for p large; this is stated precisely below.
One might ask whether the requirement in the previous proposition can be further weakened. Unfortunately, the following proposition shows that the hypothesis (0) i ∈ L p * in Proposition 4.12 is already the optimal one to put on each (0) i individually. P 4.13. Fix f * . Let S ⊆ L 1 * be a set with the following property: any nondegenerate initial P . Pick any f ∈ S and 1 ≤ p < ∞. We must show that f ∈ L p * . If f vanishes almost everywhere, then clearly f ∈ L p * . So suppose otherwise. An initial condition with (0) i = f is well-defined (i.e. f (0) i has integral 1) because f ∈ L 1 * , and it is nondegenerate, so it is feasible by hypothesis. We show that this feasibility forces f to be in L p * . By Proposition 4.5, the function * for any i and t. By Lemma 4.3, these P (t ) ji 's are entries of A t , which go to infinity as t → ∞ by Theorem 3.3 and Lemma 3.4. So we obtain a sequence p t → ∞ as t → ∞ such that f ∈ L p t * . By Lemma 3.9, we can interpolate to get f ∈ L p * .
Finally, Proposition 4.12 can be slightly weakened to give a sufficient condition for feasibility that is easy to check in practice in the next corollary. This weaker version is used to establish feasibility in e.g. Corollary 6.10 and Theorem 1.1.
i ∈ L ∞ , that is, (0) i is bounded almost everywhere.

CONCENTRATION OF BELIEFS
With the tools we have built up, we can now start seriously analyzing the model. Assume from this point onwards that the initial condition is feasible. Our goal in this section is to establish the following remarkable result about concentration of beliefs. In the long term, beliefs of agents in our model tend to concentrate at points of Θ which maximize the weighted likelihood function L (Def. 5.1). Specifically, for a set A ⊆ Θ such that the values of L on this set are dominated in some sense by L(θ ′ ) for another θ ′ ∈ Θ, We can think of this as saying that the agents' beliefs are being concentrated more and more near the maximizer of L, and so the beliefs on any set that stays clear of the maximizer tend to zero. The precise statement is in the Master Theorem 5.10. Once we have this result, it will become relatively easy to show convergence of the agents' beliefs towards maximizers of L in various settings; this will be done in Section 6.

Weighted Likelihood
We first define the weighted likelihood function mentioned in the introduction.
Definition 5.1. Assume that the initial condition is feasible. The weighted likelihood function is defined for each θ ∈ Θ by

Recall that (0)
i is the normalized beliefs defined in Definition 4.1 and that the eigenvector centrality i can be thought of as the influence of agent i (Sec. 3.1). So this function aggregates the normalized beliefs of agents while being biased by the agents' influences. To see how this function is being "biased, " the unbiased likelihood of each parameter θ given the initial beliefs of agents formed according to an underlying scenario (Sec. 2.3) is given by where c is a normalizing constant. Under optimal information aggregation, the posterior belief is proportional to L unbiased . To derive this, simply do the computation analogous to the one in Proposition 2.2. Notice that L and L unbiased are different in two ways. First, each (0) i factor in L is to the power of i , suggesting that L is biased by the centrality of each agent as already pointed out. Second, there is an extra factor of f * , the prior, in L unbiased . This is because agents in our model become so confident in their opinions that the prior is disregarded entirely, so this factor does not appear in L.
We may wonder what properties the weighted likelihood function L has. The following curious proposition gives a partial answer: L is almost bounded, that is, almost in L ∞ . This result is not used anywhere.
j ) P (t ) j i ∈ L 1 * for any i and t. The idea is that we can average these functions together to get the desired result. Recall that i i = 1. By Lemma 3.8, we can take the weighted geometric mean for all i weighted by i to get N j=1 By Lemma 4.3, P (t ) ji is the (j, i) entry of A t , so the quantity in the exponent is the jth entry of A t . Because A = r where r = ρ(A), this quantity is r t j . We conclude that L ∈ L r t * for all t ≥ 0. Because r > 1 by Lemma 3.4, Lemma 3.9 implies that L ∈ L p * for all 1 ≤ p < ∞. Its norms are nonzero because the initial condition is nondegenerate (Prop. 4.11).
We now show that the weighted likelihood function guides the convergence of the ratio of beliefs at two different points (Prop. 5.5). We can think of this as a "pointwise" concentration result. It is much weaker than the actual concentration result we seek (except when Θ is finite; see Theorem 6.3). Nonetheless, it is the basis of the proof of the Master Theorem 5.10. The next lemma deals with the degeneracy of the weighted likelihood function, i.e. the characterization of when L(θ ) = 0. It is used to avoid some issues of dividing by zero. The next proposition gives an explicit formula for the ratio of beliefs at two different points, highlighting the terms that become small in the limit. It is a crucial component that is used in Section 5.2 to prove the Master Theorem 5.10. i, j → 0 as t → ∞ such that for any θ 1 , θ 2 ∈ Θ with L(θ 2 ) > 0, From this proposition, we can begin to see the weighted likelihood function take shape: in the formula above, if the prior factors in front are ignored and the small terms ε (t ) i, j disregarded, what is left is the ratio L(θ 1 )/L(θ 2 ) raised to some power.

P
. The proof is essentially by direct computation. Note that f (t ) i (θ 2 ) > 0 by Lemma 5.3, so the formula makes sense. By Proposition 4.5 for θ 1 and θ 2 ,

By Lemma 4.3, P (t )
ji is the (j, i) entry of A t . By Theorem 3.3, this quantity equals i, j → 0 as t → ∞. The result follows by letting ε i, j w ⊤ /w i and substituting these all in.
We can now show the "pointwise" concentration of beliefs at maximizers of L. This hints at the actual concentration result in the next subsection. P 5.5. If θ 1 , θ 2 ∈ Θ are such that L(θ 1 ) < L(θ 2 ), then as t → ∞, The proof is by directly applying Proposition 5.4. As t → ∞, the expression in the bracket converges to L(θ 1 )/L(θ 2 ) < 1, so this expression is smaller than some α < 1 for large t. By Lemma 3.4, the exponent of this expression goes to infinity, so f (t ) i (θ 1 )/f (t ) i (θ 2 ) → 0 as t → ∞.

Master Theorem
This is a technical section; its aim is to prove the Master Theorem 5.10 which establishes concentration of beliefs in very general settings. In order to do this, subtle issues of convergence arise which can be best dealt with using ideas from measure theory (e.g. Lebesgue's dominated convergence theorem). Even in the relatively benign setting Θ = N, these issues are not so easy to circumnavigate. We first present the general idea of the Master Theorem. Suppose A ⊆ Θ and θ ′ ∈ Θ are such that L(θ ) < L(θ ′ ) for all θ ∈ A. Thus, θ ′ "dominates" all of A in the sense of the weighted likelihood function. We hope that ∫ To do so, we need to control the convergence in Proposition 5.5; the convergence needs to be "uniform" in some way. It turns out that the correct hypothesis is that the inequality L(θ ) < L(θ ′ ) holds when the exponents in L are slightly perturbed: As explained in Section 1.1, we can think of this as θ ′ dominating A robustly, i.e. the domination does not fail even if the centralities of agents are being slightly perturbed. Later we will offer an example where this robustness condition fails and concentration also fails (Ex. 6.6), which shows that this hypothesis cannot be omitted.
After obtaining the bound (5), if Θ is discrete, i.e. Θ = N, then the obvious bound f (t ) i (θ ′ ) ≤ 1 finishes the proof. But this is not so easy if Θ is continuous, i.e. Θ = R k . In fact, if concentration is to be true, it is likely that f (t ) i (θ ′ ) → ∞ as t → ∞ if θ ′ maximizes L, and so we cannot get from (5) to the desired result directly. The second idea is to pick θ ′ that is not a maximizer of L and show that f (t ) i (θ ′ ) → 0 (which is likely to happen if concentration near the maximizer is to occur), which will then allow us to get the desired result. To do this, we find a set B such that L(θ ) > L(θ ′ ) for all θ ∈ B, i.e. now the set B dominates θ ′ instead. Then if the convergence is similarly controlled, we should get ∫ Putting all of this together completes the proof. To summarize, the mechanics of the Master Theorem are a two-step domination process. Suppose that we have sets A and B and a point θ ′ , such that θ ′ dominates A and B dominates θ ′ , and both dominations are robust. Then the latter domination forces the density at θ ′ to go to zero, which, by the former domination, in turn forces the density on A to go to zero. To see how these sets might be picked in actual situations, suppose Θ = R and L is continuous with a unique peak at 0. Let A = [1,2]. Then it might be possible to pick θ ′ = 1/2 and B = [−1/4, 1/4], i.e. we want to pick B to be very close to 0, and θ ′ to be somewhere between A and B. This selection process will be further automated in Section 6 to get results like Theorem 1.1 that can be directly used in practice.
In what follows, we implement the program outlined above. First we formalize the notion of this "robust domination" which allows us to control convergence. The following lemma is a purely analytical statement that contains the essential ingredients to carrying out the "θ ′ dominates A" part; then Proposition 5.7 will actually carry it out. Throughout, we assume familiarity with Lebesgue's dominated convergence theorem and Fatou's lemma. L 5.6. Let S be a measure space. For t ∈ Z ≥0 and 1 ≤ i ≤ N , let a i , c : S → R ≥0 be measurable functions, r t > 0 and ε (t ) i ≥ −1. As t → ∞, r t → ∞ and, for every i, ε (t ) Suppose that for every t, the integral Then, as t → ∞, In particular, if A(x) < 1 almost everywhere, then I t → 0 as t → ∞.
Note that the form of I t is reminiscent of the expression in Proposition 5.4 and indeed this is what the lemma will be used for; but this abstraction in terms of these various variables is necessary in order for the proof of the lemma to not become too large and convoluted.
. The first part of the proof is to show that if we perturb the exponents of a i 's a little, the result will still be close to A(x); specifically, it will be A(x) raised to some power in a fixed range. To do this, the condition A(x)a i (x) δ i ≤ 1 will be of help, along with a series of algebraic manipulations.
All statements in this paragraph are meant to hold for almost every x ∈ S. For any 0 ≤ < δ i /2, This allows us to perturb one exponent. To perturb many exponents at once, note that for any Now we extend to the case where the perturbations can be negative. Let 0 < δ ′ < 1 be such that This final inequality is what we want: both upper and lower bounds for when all exponents are perturbed at once, regardless of signs.
The second part of the proof is to use this bound to establish convergence. Let T be such that for every i and t ≥ T , ε (t ) i < δ ′ . Then, for t ≥ T , by the previous paragraph, ∫ Note that, as written, the right-hand expression may be infinite. The final step is Lebesgue's dominated convergence. Let e. Thus f T serves as the "Lebesgue dominant" in this argument. By Lebesgue's dominated convergence theorem, c1 A(x )=1 ∈ L 1 and ∫ as t → ∞, so I t converges to this same limit.
To better understand how the proof of Lemma 5.6 works, one might want to carry it out in the case where S = N with counting measure; in that case, the lemma is a statement about convergence of infinite series. The proof in this special case does not require Lebesgue's dominated convergence, but still requires some thought on how to estimate the tails of the series. The authors of this paper proved the lemma for this special case first before generalizing it to the present form; the final step is the only part of the argument that needs nontrivial change.
Once we have Lemma 5.6, Proposition 5.7 follows immediately, although it may be a little tedious to check that all conditions are satisfied. P 5.7. Let A ⊆ Θ be a measurable set. Let θ ′ ∈ Θ be such that L(θ ′ ) > 0 and L(θ ) ≤ L(θ ′ ) for almost every θ ∈ A. Moreover, for each i, there is If the set {θ ∈ A : L(θ ) = L(θ ′ )} has measure zero, then M = 0.
P . All we have to do is to directly apply Lemma 5.6. Apply Proposition 5.4 to get the expression into the form Then apply Lemma 5.6 with S = A, The hypotheses of the lemma are readily verified using the hypotheses of the proposition and Lemma 3.4.
The previous proposition completes the "θ ′ dominates A" part of the program. To carry out the "B dominates θ ′ " part, we prove the following Lemma 5.8 and Proposition 5.9, which can be thought of as analogous versions of Lemma 5.6 and Proposition 5.7 but with reversed inequalities and slight changes in details. In particular, the part that changes is the final step of the lemma where here we use Fatou's lemma to conclude instead. i , A and I t be as in Lemma 5.6 with the following changes. The inequalities involving A are reversed: A(x) ≥ 1 for almost every x ∈ S, and for every i, there is δ i > 0 such that A(x)a i (x) δ i ≥ 1 for almost every x ∈ S. Moreover, the integral I t may be infinite. If A(x) = 1 almost everywhere, then as t → ∞, where the limit can be infinite. Otherwise, I t → ∞ as t → ∞.

P
. The first part of the proof stays the same. By an analogous argument to the one in Lemma 5.6 with reversed inequalities, there are 0 < δ ′ < 1 and T such that, for t ≥ T , ∫ x ∈S To carry out the second part of the proof, use the following argument. The result is immediate if A(x) = 1 almost everywhere. Otherwise, the right-hand integrand converges pointwise to a function that is infinite on a set of positive measure. By Fatou's lemma, the right-hand integral converges to infinity. Then I t → ∞ as t → ∞.
One can prove the following proposition by applying Lemma 5.8 in the same way that one applies Lemma 5.6 to prove Proposition 5.7. So we omit the proof of the following proposition. From here it is easy to combine Propositions 5.7 and 5.9 into the Master Theorem. Recall that the various technical hypotheses of the theorem should be intuitively read as "θ ′ dominates A, and B dominates θ ′ robustly in the sense of the weighted likelihood function, " as explained earlier in this section.  T  ). Let A, B ⊆ Θ be measurable sets, where B has positive measure. Let θ ′ ∈ Θ be such that L(θ ′ ) > 0, L(θ ) ≤ L(θ ′ ) for almost every θ ∈ A, and L(θ ) ≥ L(θ ′ ) for almost every θ ∈ B. Moreover, for each i, there are where M 1 and M 2 are limits in the two respective propositions. Because B has positive measure, M 2 > 0. Our final hypothesis implies that either M 1 = 0 or M 2 = ∞, so M 1 /M 2 = 0. Then our result follows from ∫ θ ∈B f (t ) i (θ ) ≤ 1.

CONVERGENCE OF BELIEFS
We apply the Master Theorem 5.10 to show convergence of beliefs of agents toward a point distribution at the maximizer or near-maximizer of L. We establish separate results for the finite case (Sec. 6.2), the infinite discrete case (Sec. 6.3) and the R k case (Sec. 6.4). Our results in the R k case are the basis of Theorem 1.1. Results in this section will be used in applications in Section 7. We first explain the general idea of this section. Recall that to use the Master Theorem, for a given set A, we need to find a point θ ′ that dominates A and a set B that dominates θ ′ . What this section aims to do is the following. First, for a given set A, the choice of θ ′ and B will be automated and abstracted away from the user. Second, even the choice of A will be abstracted away, leaving only the conclusion that "beliefs converge to a point distribution, " at least in the case of a unique maximizer. The general idea is not hard, but the details can be quite formidable.

Preliminaries
First we need some preliminaries. This subsection presents the technical details in carrying out the "automatically choosing θ ′ and B" part. Let L ess sup = ess sup θ ∈Θ L(θ ) ≤ ∞, that is, L ess sup is the smallest M for which {θ ∈ Θ : L(θ ) > M } has measure zero. Note that we can replace "ess sup" by "sup" if Θ is infinite discrete or "max" if Θ is finite. The following proposition gives sufficient conditions for beliefs to concentrate in places where L is near L ess sup . To better understand the above proposition, it will be useful to understand what roles the various hypotheses play. The "bounded" and "bounded below by a positive number" hypotheses ensure that the gaps between A and θ ′ and θ ′ and B are robust as needed in the Master Theorem 5.10. Now we note that S dominates Θ \ S; however, we still need to pick θ ′ and B out of S. There are two ways to do this depending on the situation. First, think of Θ = R. In this case, if L is constant on S, it will be hard to pick B that dominates θ ′ robustly; but we can do this otherwise. This is condition (1) in the proposition. Second, think of Θ = N. In this case, even if S is a single point (and so L is constant on it), it will be easy to pick θ ′ and B; but this case does not satisfy (1). This is where condition (2) comes in. Putting all these details together gives us the proposition.

P
. Note that S has positive measure by the definition of L ess sup . Apply the Master Theorem 5.10 with A = Θ \ S. In the second case, we can pick any θ ′ ∈ T and let B = T . Now consider the first case. First suppose that there is θ 1 ∈ S that minimizes L in S. In this case we pick θ ′ = θ 1 ; now we need to pick B. Because L is not constant a.e. on S, {θ ∈ Θ : L(θ ) > L(θ 1 )} has positive measure. By continuity of measure from below, there is M ′ > L(θ 1 ) such that U = {θ ∈ Θ : L(θ ) > M ′ } has positive measure. Let B = U . Finally, suppose that there is no minimizer of L on S. Pick a sequence θ i ∈ S such that L(θ i ) → inf θ ∈S L(θ ). Define S i = {θ ∈ Θ : L(θ ) ≥ L(θ i )}. Then S = ∪ i S i . By subadditivity of measure, one of S i has positive measure. Then for this i, pick j such that L(θ j ) < L(θ i ). Let θ ′ = θ j and B = S i .
The result now follows by applying the Master Theorem to A, B and θ ′ just chosen. To show the existence of δ i > 0, use the fact that L(θ )/L(θ ′ ) ≤ α < 1 for all θ ∈ A and (0) i is bounded a.e. on A.
The existence of δ ′ i > 0 is similar; in case (1) we use the fact that L(θ )/L(θ ′ ) ≥ β > 1 for all θ ∈ B and (0) i is bounded below a positive number a.e. on S. We now present the following example of a social network, which will be used in Examples 6.4 and 6.6. What this example is trying achieve is a network consisting of two people, but where the strength of information flow on each edge can vary, i.e. a "weighted" social network. We can generalize our model to allow for weighted edges, but we choose not to: it turns out that, at least in this simple case, the weighted edges can be simulated by our unweighted network by using teams of people to represent one person in the weighted case. The details are below. x (θ ) 1/2 (0) (θ ) 1/2 . By Proposition 4.5, we can compute

Finite Case
We now apply Proposition 6.1 to show convergence in the case where Θ is finite. The theorem in this case is quite easy and we can appeal to either case (2) of Proposition 6.1 or just Proposition 5.5. T 6.3. Assume that Θ is finite and the initial condition is feasible. Let θ max be the set of points that maximize L. Then as t → ∞, In particular, if θ max consists of a single point, then f (t ) i converges to the point distribution at that point.

Infinite Discrete Case
We now move on to the theorem for the infinite discrete case, which follows from the second case of Proposition 6.1. This is the first nontrivial case and contains hypotheses that the the initial conditions have to verify in order for beliefs to converge. T 6.5. Assume that Θ is infinite discrete and the initial condition is feasible. (1) Let L attain the maximum L max at the set of points θ max ∅. Suppose that sup θ θ max L(θ ) < L max and (0) i is bounded on Θ \ θ max . Then as t → ∞, In particular, if θ max consists of a single point, then f (t ) i converges to the point distribution at that point.
(2) If L does not attain a maximum, let The theorem is essentially saying the following. Note first that there are two separate cases. If maximizers of L exist, then, assuming a "positive gap" condition and a "boundedness" condition, beliefs converge to those maximizers. But if maximizers of L do not exist (for example, if L(θ ) increases to but is never equal to a positive number), then all that we can say is that the beliefs concentrate at points where L are near the supremum, with a boundedness condition similar to the first case.
We focus on the first case, i.e. when maximizers of L exist, and assume a unique maximizer. The interesting question is whether these "positive gap" and "boundedness" conditions, which can be traced back to the conditions in Proposition 6.1 and then the L(θ ) i (θ ′ ) δ i "robust domination" hypothesis of the Master Theorem 5.10, are necessary. If they are, then, as alluded to in Section 5.2, this shows that the robust domination hypotheses of the Master Theorem cannot be omitted; in particular, they are not just an artifact of how we proved that theorem. The following example shows that neither of the two hypotheses of the first case can be omitted. These examples are rather delicate. Specifically, we know that the exponent in Proposition 5.4 grows like r t ; but in order to construct these examples, the error term ε (t ) i, j has to be estimated as well. This gives rise to expressions like 4 θ + 2 θ +1 . Example 6.6. Counterexamples to the conclusion of the first case of Theorem 6.5 that satisfy only the first and second hypotheses, respectively.
(1) This example satisfies the positive gap, but not the boundedness hypothesis. Consider the (3, 1)-graph of Example 6.2. Let Θ = Z ≥0 , 0 < α < 1, The idea is to set things up in such a way that at round t, the parameter θ = t holds a nontrivial fraction of beliefs. Then L(θ ) = αL(0) for all θ ≥ 1, so 0 uniquely maximizes L and the positive gap condition is satisfied. We now compute f (t ) for all t ≥ 1, so that beliefs do not converge to a point distribution at 0 and the conclusion of the theorem does not hold.
(2) This example satisfies the boundedness, but not the positive gap hypothesis. Same as the previous example but with the initial condition It is easily seen that the boundedness condition is satisfied. Then L(θ ) = α 4 −θ L(0) for all θ ≥ 1, so 0 uniquely maximizes L. Similarly, beliefs do not converge to a point distribution at 0 because, for t ≥ 1, Again, we set things up in such a way that the term with θ = t keeps the denominator large in round t.

R k Case
Finally, we deal with convergence in the most difficult case, Θ = R k . Convergence here depends on the topology of R k , so we need to work with functions that are sufficiently continuous. Concepts from real analysis will be freely used. Let A denote the closure of A and B r (a) the open ball of radius r around a.
First we introduce the notion of piecewise continuity. Exactly why this is needed can be explained as follows. If the initial beliefs are continuous, then this continuity allows us to establish convergence quite easily. However, we may also be interested in densities that are not continuous, e.g. a density that is 1 on [0, 1] and 0 elsewhere. But this density is actually continuous enough that we can work with it; specifically, it is continuous on each of (−∞, 0), (0, 1) and (1, ∞). Thus we turn to the notion of piecewise continuity, which should include most densities in practice. There is no universally agreed definition of piecewise continuity, so we define our own in Definition 6.7. On R k , this notion essentially means that there are disjoint open sets on each of which the function is continuous. Moreover, these open sets must "fill up Θ, " in the sense that the portion of Θ not contained in any of these sets has measure zero. Two extra conditions are needed in Definition 6.7 in order to avoid pathologies. The first condition excludes cramming open sets into small spaces, e.g. if the function is continuous on each of (1/(n + 1), 1/n), this can cause problems near 0. The second condition excludes functions with pathological behaviors at the boundaries, e.g. think of sin(1/x) near 0.
has measure zero; only finitely many U i 's intersect any given bounded set V ⊆ R k ; and for each i, the restriction f : U i → R ≥0 can be extended to a continuous function f i : Note that the continuous extension mentioned in the last part of the above definition has the interval in the extended real line [0, ∞] as its range. This means that we allow functions such as 1/x to be piecewise continuous because it has a well-defined limit at 0, even though that limit is infinite. The next lemma shows that multiple functions play well together with regard to this definition: we can choose the open sets in the definition to be the same for all functions. i is piecewise continuous. The following theorem can be regarded as the most significant result in this paper. Recall that we allow initial beliefs to be piecewise continuous, which should include most situations in practice. Moreover, the parameter region Θ can be any subset of R k ; e.g. Θ = (0, ∞) is allowed. However, in the case that Θ is a proper subset of R k , the point that the beliefs converge to may be outside of Θ, but it must be a limit point of Θ. This is not too surprising: if the initial beliefs are 1/x 2 on (0, ∞), then it is likely that the beliefs will converge to 0, which is actually outside of the parameter set. Nevertheless, our theorem is able to account for cases like this. and that the statement is false if this expression is negative. Thus it is as if agent i casts votes of log(x i /(1 −x i )) in support of the statement and gets i votes. The expression log(x i /(1 −x i )), called the log-odds, transforms probability in (0, 1) to the real line. Moreover, we can see that the strength of an agent's vote indeed comes from her confidence and her centrality.
A similar analysis holds when agents are trying to decide between a finite number of alternatives. In this case, if agent i believes in alternative j with probability x i, j , then it is as if she casts votes of log x i, j in support of alternative j. The alternative with the most votes becomes the consensus.

Poisson Beliefs
Now let us consider the case where there are an infinite number of alternatives, e.g. when the underlying state of the world is the number of events in a time period. The initial beliefs can be modeled by Poisson distributions.
Assume that the initial belief of agent i is Pois(λ i ) and the common prior is a flat prior, then the initial condition is feasible. Let λ * = N i =1 λ i i . If λ * is not an integer, then f (t ) i converges to the point distribution at ⌊λ * ⌋. If λ * is an integer, then f (t ) i lies in {λ * − 1, λ * } with probability going to 1 as t → ∞.

P
. Since (0) i is bounded and the initial condition is nondegenerate, the initial condition is feasible by Corollary 4.14. It is easy to compute that By direct computation, L has a unique maximum at ⌊λ * ⌋ if λ * is not an integer, and two maxima {λ * −1, λ * } if λ * is an integer. Since L(θ ) → 0 as θ → ∞ and i is bounded, we can apply Theorem 6.5 to show convergence.
The Poisson parameter λ is the mean of Pois(λ) and is related to the average event rate. Our result shows that agents will reach a consensus at λ * , the weighted geometric mean of the rates corresponding to agents' beliefs, where again the weight of each agent is her centrality. Note that our model indicates that Poisson rates should be combined multiplicatively rather than additively. This accords with intuition: if two connected people believe that there will be 2 and 1000 floods on average next year, respectively, then Proposition 7.2 says that they will come to believe that there will be ⌊ √ 2000⌋ = 44 floods. In contrast, combining rates additively as in DeGroot leads to the counterintuitive answer of 501 floods. Thus, even though each agent carries essentially one piece of information: the event rate, by modeling the beliefs according to the actual underlying parameter space, our model leads us to a natural consensus. A similar situation is the Gaussian belief case in the next section where beliefs converge to the weighted arithmetic mean and agent i also has weight i .

Gaussian Beliefs
Finally, we consider the interesting case of each agent carrying two pieces of information, her belief and her confidence. Suppose that the underlying state of the world is Θ = R with the flat prior f * ≡ 1. Each agent observes the true state of the world θ * with Gaussian noise: agent i's signal µ i is drawn from the normal distribution N(θ * , 1/τ i ), where τ i is the precision known to agent i. Note that this is an example of an underlying scenario explained in Section 2.3, except that the prior in this case is improper. Then agent i's initial belief for θ * (conditional on the signal µ i ) can be computed to be N(µ i , 1/τ i ). P 7.3. Assume that the initial belief of agent i is normally distributed as N(µ i , 1/τ i ) and the common prior is a flat prior, then this initial condition is feasible and f (t ) i converges to the point distribution at

P
. The weighted likelihood function is and we can see that this is proportional to a Gaussian with the maximum at θ max . Since i are bounded and continuous and L decays quickly at infinity, the convergence follows from Corollary 6.10.
If we interpret the initial belief N(µ i , 1/τ i ) as a scalar belief µ i with confidence τ i , then the consensus belief θ max is the weighted average of agents' signals µ i with weights c i proportional to her centrality i and her precision τ i . Therefore, an agent who is more centrally located and has a more informative signal has more influence over the consensus belief. This phenomenon cannot be captured by the standard DeGroot model which has no notion of quality of signals.
We can evaluate quality of learning by the variance of the consensus. This works as follows. The consensus θ max is a function of the signals µ i , and the signals are random, so the consensus can be viewed as a random variable. Specifically, it is an unbiased estimator of the true state θ * , and the lower its variance, the higher the quality of learning. Since µ i ∼ N(θ * , 1/τ i ) are independent, θ max = i c i µ i ∼ N(θ * , S) remains a Gaussian, with variance We now investigate the effect of increasing the precision τ k on the variance S. We find that, all other variables being fixed, the result depends on the threshold V k = 2 i k 2 i τ i / i k i τ i of the centrality k . By direct computation, if k ≤ V k , then increasing τ k always decreases S, while if k > V k , then increasing τ k decreases S only if τ k ≥ ( k − V k ) i k i τ i / 2 k and increases S otherwise. Thus, increasing the precision τ k usually decreases the variance and improves the quality of learning, except when the centrality k is high and the precision τ k is low. This result has an implication for social planners who want to encourage good learning. The social planner cannot control the network structure k but she might be able to control the signal precision τ k . If an agent is central but relatively uninformed, increasing the precision of that agent can reduce learning quality. An intuitive explanation might be that the agent now has some confidence to use her centrality to impose her less informed opinion on the consensus. Only when that central agent is sufficiently informed will additional precision improve learning quality. Therefore, if a social planner wants to seed centrally located agents (opinion leaders) with new technologies, she must invest enough resources to make those agents well informed; otherwise, the effort can backfire.
Note that we assume independence in the agents' update rule, not in the initial signals themselves. Therefore, our framework can deal with arbitrarily correlated initial signals. In the case of Gaussian beliefs, if we assume that the correlation of initial signals of agents i and j is ρ i j , then the consensus θ max is still unbiased with variance This variance is increasing in every correlation ρ i j , which indicates that higher correlations in initial signals reduce learning equality, as expected.
Another point of interest is that our consensus does not depend on the geometry of the network beyond the eigenvector centralities; this is true for all of our analysis from Section 5 onwards and not only for this particular Gaussian case. Let us contrast this with the work of Banerjee et al. [5], which also incorporates quality of signals into DeGroot learning. In their model, the quality of signals is binary (informed/uninformed), and each agent takes the average of signals of only her informed neighbors. They identify the clustered seeding problem: if the initially informed agents are closely connected, some agents will "block" other agents, reducing the influence of the blocked agents. To illustrate this point, consider a social network in the shape of an undirected cycle of length N and assume that only three adjacent agents 1, 2, 3 are initially informed with the same precision. In our model, the three agents' signals have equal weights: c 1 = c 2 = c 3 = 1/3. But in their model, agent 2 is blocked by agents 1 and 3: c 1 = c 3 → 1/2 and c 2 → 0 as the number of agents N → ∞. Note that all three agents do have the same centralities and the same precisions, yet the geometry of the network still allows them to block each other. Why this occurs may be explained as follows: as the "informed segment" of the network expands, an agent who has just become newly informed will get the opinion of agent 1 or 3 first. Surely she will not be very confident because she has just heard of one opinion, but in the model of Banerjee et al. [5], her opinion counts as much as everyone else, and so the opinions of agents 1 and 3 get injected back into the informed segment and reinforce themselves to an inappropriate degree. In contrast, our model circumvents this problem by allowing agents to express their confidence. Thus, a newly informed agent is not very confident and her opinion counts less; and it turns out that this alone is enough to offset the blocking disadvantage and allows the opinion of agent 2 to come through in the consensus. We conclude that the clustered seeding problem occurs when agents can communicate whether they are informed but cannot communicate precisions of their signals. When agents can communicate precisions of signals, even if beliefs are naively updated, the problem disappears. The conclusion is that it is of vital importance that agents be able to communicate opinions freely and expressively, and not just state their beliefs, in order for blocking to not occur.